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On the Problem of the Uniqueness of Fixed Points and Solutions for Quadratic Fractional-Integral Equations on Banach Algebras

Kinga Cichoń

^1,†

Mieczysław Cichoń

^2,*,†

and

Maciej Ciesielski

^1,†

Institute of Mathematics, Faculty of Automatic Control, Robotics and Electrical Engineering, Poznan University of Technology, Piotrowo 3A, 60-965 Poznań, Poland

Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Symmetry 2024, 16(11), 1535; https://doi.org/10.3390/sym16111535

Submission received: 16 October 2024 / Revised: 13 November 2024 / Accepted: 14 November 2024 / Published: 16 November 2024

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

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Abstract

In this paper, we study the problem of the uniqueness of fixed points for operators defined on subspaces of the space of continuous functions

C [a, b]

equipped with norms stronger than the supremum norm. We are particularly interested in Hölder spaces since they are natural ranges of integral operators of fractional order. Our goal is to preserve the expected regularity of the fixed points (solutions of the equations) when investigating their uniqueness, without assuming a contraction condition on the space under study. We claim some symmetry between the case of the obtained results and the case of the classical Banach fixed-point theorem in such spaces, even for operators which are not necessarily contractions in the sense of the norm of these subspaces. This result is of particular interest for the study of quadratic integral equations, and as an application example we prove the uniqueness theorem for such a kind equations with general fractional order integral operators, which are not necessarily contractions, in a suitably constructed generalized Hölder space.

Keywords:

fixed point; Hölder space; Banach algebra; seminorm; uniqueness; generalized fractional integral operator; quadratic integral equation

1. Introduction

For many differential or integral problems, it is important to prove not only the existence of their solutions but also their uniqueness and the method of finding them. The classical method, based on the Banach contraction theorem and the iterative formula

x_{n + 1} = T (x_{n})

, is the basic method. Of course, not only contractions have unique fixed points. In this paper, using quadratic integral equations as an example, we will show how to obtain the uniqueness of their solutions without assuming the contraction condition of the operators under study in the case of Hölder-type spaces and, more generally, spaces with a norm stronger than the supremum norm. We will make use of properties of norm algebras. Surprisingly, such an idea of solving the uniqueness problem for integral or operator problems has not been used and we will realize its usefulness.

We will consider the following general situation: we are looking for the (unique) fixed point for the mappings

{T : (X, ∥ \cdot ∥}_{\infty} {) \to (X, ∥ \cdot ∥}_{\infty})

and some linear subspace Y of the space X equipped with a stronger norm

{∥ \cdot ∥}_{Y}

, i.e., there is a constant

c_{X} > 0

such that

{∥ \cdot ∥}_{Y} \leq c_{X} \cdot {∥ \cdot ∥}_{\infty}

. We are interested in a special case

{∥ \cdot ∥}_{Y} = {∥ \cdot ∥}_{\infty} + S (\cdot)

, where S is a seminorm on Y. As will be clarified later, a prototype for such a class of spaces is the class of Hölder spaces. In particular, it is known that the Hölder spaces arise naturally as interpolation spaces between the domains of the fractional derivative operators, and the space of continuous functions and one of the main goals for the study of differential and integral problems is to obtain maximal regularity results. The main motivations and proof difficulties are apparent in non-linear problems, e.g., for operators that are the pointwise product of classical operators, and such problems will be investigated in this paper (cf. [1,2,3,4,5]).

In addition to the papers already mentioned, it is worth pointing out the papers [6,7,8], which not only justify the study of the regularity of solutions in Hölder spaces but also investigate the uniqueness of solutions of certain evolution equations (using the Banach fixed point theorem). Furthermore, although they are not quadratic equations, the fact that the coefficients are functions from Hölder spaces requires the use of a property of Banach algebra. We will develop the ideas of this article and illustrate them by studying quadratic integral equations of fractional order on Banach algebras (not restricting ourselves, as is usually the case, to the space of continuous functions and the loss regularity of solutions). An interesting introduction to the topic of quadratic weakly singular integral equations in Banach algebras can be found in [9].

Spaces of the type discussed above are, for example, Hölder spaces, and these in turn are natural images of spaces of continuous functions by integral operators of fractional order (cf. [10,11,12,13]). And it is for this reason that such operators will constitute our motivations and illustrations of the results obtained.

Now let us recall that the fixed-point approach to fractional problems via integral-type fractional operators is usually based on the fact that the image of a domain of the fractional operator is a compact subspace of this domain (i.e., by applying compact embedding of Hölder spaces) or even by taking some assumptions guaranteeing that the operator is compact. Both approaches are known and useful (in some cases).

The situation is completely different if we are interested in the problem of the uniqueness of the solution. Both the analytical iterative scheme and the possible application of numerical methods require the existence and uniqueness of a solution. Thus, the Banach fixed-point theorem seems to be the ideal tool. The assumption is that the operator is a contraction (on a complete space).

Note the difference between

(C [a, b], ∥ \cdot ∥_{\infty})

and its subspaces Y equipped with stronger norms

{∥ \cdot ∥}_{Y}

. To simplify the analysis of the problem, we will equip Hölder spaces

H^{α} [a, b]

(

H^{α}

for short) with the norm of this type, i.e., with

{∥ x ∥}_{γ} = {∥ x ∥}_{\infty} + {[x]}_{γ}

, where

{[x]}_{γ} = {sup}_{t \neq s} \frac{| x (t) - x (s) |}{{| t - s |}^{γ}}

. As can easily be seen, the operator can be a contraction by virtue of the norm in

C [a, b]

but not by virtue of the

{∥ \cdot ∥}_{Y}

norm. We will analyze this situation because of the implications and application of other fixed point theorems. Although in this paper we consider the situation where the operator does not satisfy the contraction assumption in the considered space, it is fully symmetric with such a classical case.

And now the question: why are iterated methods not so useful? For integer-order equations, it is very popular approach. The partial answer is presented below. Our main motivation, however, is the study of fractional-order operators, especially quadratic operators. For the Riemann–Liouville fractional operator

T = I^{τ} : H^{α} \to H^{α + τ},

provided that

α + τ < 1

, we obtain for the composition of operators

T^{2} : H^{α + τ} \to H^{α + 2 τ}

whenever

α + 2 τ < 1

. So, we cannot continue the construction of

T^{n}

with

n \to \infty

. Recall that if x satisfies the Hölder condition with exponent

α > 1

, then it is constant. That is, we cannot find the space we are interested in.

So what is the solution used in the papers so far? Obviously, the observation that the target space is contained in the domain of the operator, i.e., case consideration:

T = I^{τ} : H^{α} \to H^{α} .

This allows the study of the compactness of sets in a domain and the application of the Schauder fixed-point theorem in research. Which methods are the simplest? Considering problems in the space of continuous functions

C [a, b]

, Banach algebra and quadratic operators are also well defined. Unfortunately, in such an approach, we lose some regularity of the solutions. Integral operators are improving, so additional properties of the solutions would be expected, including their already mentioned Hölder continuity. So what are the two main problems to be solved? First, to construct a certain Banach space (algebra, for the product of operators) which is an invariant space for the operators under study. The second problem is to change the norm from the supremum norm in

C [a, b]

to some other norm, e.g., the sum of this norm and some seminorm S on a subspace of

C [a, b]

, where S is some seminorm. But, this also makes it much harder to test the contraction condition for operators. We solve the problem in this paper.

2. Preliminaries

The space of continuous functions is on an interval

[a, b]

, where

\infty < a < b < \infty

is denoted by

C [a, b]

. The Banach space of all measurable functions

f : [a, b] \to R

, with norm given by

{∥ f ∥}_{p} = (\int_{I} {| f (t) |}^{p} {d t)}^{1 / p}

, where

1 \leq p < \infty

, is denoted by

(L_{p} [a, b], ∥ \cdot ∥_{p})

. Furthermore, let

A C [a, b]

denote the space of all absolutely continuous functions and

C^{1} [a, b]

(with the norm

{∥ x ∥}_{C^{1}} = {∥ x ∥}_{\infty} + {∥ x^{'} ∥}_{\infty}

) denote the space of continuously differentiable functions on

[a, b]

We say that a function f satisfies the Hölder condition of order

λ \in (0, 1]

on the interval

[a, b]

f \in H^{λ} [a, b]

| f (t + h) - f (t) | \leq {A | h |}^{λ}, t, t + h \in [a, b],

(1)

where A is a constant independent on h. For

λ \in (0, 1)

we denote by

H_{0}^{λ} [a, b] = {f \in H^{λ} [a, b] : f (0) = 0}

The space

H^{λ} [a, b]

is called a Hölder space with a fixed-order

λ

and we call the condition (1) a Hölder condition on

[a, b]

([14]). It is not difficult to show, for

0 < λ_{1} < λ_{2} < 1

, that

H^{1} [a, b] \subset H^{λ_{2}} [a, b] \subset H^{λ_{1}} [a, b] \subset C [a, b] .

Obviously, a function f is Hölder-continuous of order

λ \in (0, 1]

if the following seminorm is finite:

{[f]}_{λ} = sup_{t \neq s} \frac{| f (t) - f (s) |}{{| t - s |}^{λ}} < \infty .

According to this definition, only the case

0 < λ \leq 1

is of interest, because if

λ > 1

, then the space

H^{λ} [a, b]

contains only constant functions. The space

(H^{λ} [a, b], {∥\cdot∥}_{λ})

is a Banach space when equipped with the norm

{∥f∥}_{λ} = {∥ f ∥}_{\infty} + {[f]}_{λ} .

Note that the norms

{∥ \cdot ∥}_{\infty}

and

{∥f∥}_{λ}

are not equivalent in the space of continuous functions. We can anticipate that operators will exhibit different properties when acting between spaces that have different norms! This is important for our approach:

H^{λ} [a, b]

is not a closed subspace of

C [a, b]

with respect to the supremum norm

{∥ \cdot ∥}_{\infty}

Let us recall most typical example, i.e., the Weierstrass function defined by:

f (x) = \sum_{n = 0}^{\infty} a^{n} cos (b^{n} π x),

where

0 < a < 1, b

is an integer,

b \geq 2

, and

a b > 1 + \frac{3 π}{2}

α

-Hölder continuous with

α = - \frac{log (a)}{log (b)} .

It does not satisfy a Hölder condition of order 1, however.

It should be stressed, however, that in addition to Hölder spaces, we are interested in this paper in a broad class of subspaces

X \subset C [a, b]

with a norm stronger than the supremum norm, i.e., of the form

{∥ \cdot ∥}_{X} = {∥ \cdot ∥}_{\infty} + S (\cdot)

, where S is a seminorm on X. Obviously Hölder spaces are representative of such a class of spaces with

S (x) = {[x]}_{λ}

(cf. also

C^{1} [a, b]

). More examples will be given later.

In this paper, we will examine some particularly interesting examples of such spaces and their properties, with a view to investigating the uniqueness of solutions to quadratic integral equations (Section 5).

3. Uniqueness, Iterative Methods, and Fixed Point Theorems

Recall that for invariant domains of operators we can also look for applications of the Banach fixed-point theorem and to solve uniqueness problems.

Although it seems to be well known, we will collect some facts about convergence and continuity of operators on Hölder spaces and on

C [a, b]

. Note that instead of the classical norm

{∥ x ∥}_{H} = | x (a) | + {[x]}_{γ}

on Hölder spaces, we prefer the following (equivalent) one:

{∥ x ∥}_{γ} = {∥ x ∥}_{\infty} + {[x]}_{γ}

, where

{[x]}_{γ} = {sup}_{t \neq s} \frac{| x (t) - x (s) |}{{| t - s |}^{γ}}

(

t, s \in [a, b]

), immediately (cf. [14])

{∥ x ∥}_{\infty} \leq {∥ x ∥}_{γ}

for any

x \in H^{γ} [a, b]

. These norms are not equivalent, as a simple example can show.

Example 1.

Let

a = 0, b = 1

and let

x (t) \equiv 0

x_{n} (t) = 0

for

t \leq 1 - \frac{1}{n}

and

x_{n} (t) = t - (1 - \frac{1}{n})

for

t \in (1 - \frac{1}{n}, 1]

n \in N

Then,

∥ x_{n} {- x ∥}_{\infty} = {sup}_{t \in [0, 1]} | x (t) - x_{n} (t) | = \frac{1}{n} \to 0

n \to \infty

. Nevertheless, by calculating the seminorm

{[x_{n} - x]}_{γ}

(for

γ = 1

), we obtain

{[x_{n} - x]}_{1} = sup_{t \neq s} \frac{| x_{n} (t) - x_{n} (s) |}{| t - s |} \geq \frac{| x_{n} (1) - x_{n} (1 - \frac{1}{n}) |}{\frac{1}{n}} = n \cdot (\frac{1}{n} - 0) = 1

(the choice of

t = 1

and

s = 1 - \frac{1}{n}

in the estimation of the supremum is optimal). Hence, this sequence is convergent in

C [0, 1]

, but not in Hölder norm in

H^{1} [0, 1]

It is clear by definition that insofar as an operator is continuous between Hölder spaces, it is continuous as an operator on a subset of functions satisfying the Hölder condition, treated as a set in the space

C [a, b]

. The inverse does not hold. When studying the uniqueness of fixed points of such operators, it is important to investigate whether it is a contraction between these spaces. Although this fact is well known, for the purpose of this paper it is worth supporting it with an example.

Example 2.

Consider the operator:

T (x) (t) = \frac{1}{3} (x (t) + x (\frac{t}{2})) .

Contraction in

C [0, 1]

. We want to check if this operator is a strict contraction in

C [0, 1]

, where we use the sup-norm. For functions

x, y

, the difference between their images under T is:

| T (x) (t) - T (y) (t) | = \frac{1}{3} (| x (t) - y (t) | + | x (\frac{t}{2}) - y (\frac{t}{2}) |) .

Since both terms on the right-hand side are bounded by

{∥ x - y ∥}_{\infty}

, we obtain:

| T (x) (t) - T (y) (t) | \leq \frac{1}{3} ({∥ x - y ∥}_{\infty} + {∥ x - y ∥}_{\infty}) = \frac{2}{3} {∥ x - y ∥}_{\infty} .

Thus, the sup-norm of the difference is:

{∥ T (x) - T (y) ∥}_{\infty} \leq \frac{2}{3} {∥ x - y ∥}_{\infty} .

Since

\frac{2}{3} < 1

, this shows that the operator T is a strict contraction in

C [0, 1]

, with a contraction constant

\frac{2}{3}

. For any Hölder space, we need to evaluate how the seminorm changes under the action of T. Consider the difference between

T (x) (t_{1})

and

T (x) (t_{2})

| T (x) (t_{1}) - T (x) (t_{2}) | = \frac{1}{3} (| x (t_{1}) - x (t_{2}) | + | x (\frac{t_{1}}{2}) - x (\frac{t_{2}}{2}) |) .

Now, divide this by

| t_{1} - t_{2} |^{α}

\frac{| T (x) (t_{1}) - T (x) (t_{2}) |}{| t_{1} - t_{2} |^{α}} = \frac{1}{3} (\frac{| x (t_{1}) - x (t_{2}) |}{| t_{1} - t_{2} |^{α}} + \frac{| x (\frac{t_{1}}{2}) - x (\frac{t_{2}}{2}) |}{| t_{1} - t_{2} |^{α}}) .

The second term can be simplified by noting that:

|\frac{t_{1}}{2} - \frac{t_{2}}{2}| = \frac{1}{2} | t_{1} - t_{2} | .

So, we have:

\frac{| x (\frac{t_{1}}{2}) - x (\frac{t_{2}}{2}) |}{| t_{1} - t_{2} |^{α}} = \frac{| x (\frac{t_{1}}{2}) - x (\frac{t_{2}}{2}) |}{| \frac{t_{1}}{2} - \frac{t_{2}}{2} |^{α}} \cdot \frac{1}{2^{α}} .

Thus, the Hölder seminorm under the operator becomes:

{[T (x)]}_{α} \leq \frac{1}{3} ({[x]}_{α} + 2^{- α} {[x]}_{α}) = \frac{1}{3} (1 + 2^{- α}) {[x]}_{α} .

Thus, T is a contraction in the Hölder space whenever:

\frac{1}{3} (1 + 2^{- α}) < 1 .

This simplifies to:

2^{- α} < 2,

which is true for any

α \in (0, 1]

. Thus, T is no longer a contraction in Hölder spaces.

As we can see, it is useful to distinguish between two situations: the classical one, when the operator is a contraction between Hölder spaces, and the weaker assumption of the contraction as an operator from

C [a, b]

C [a, b]

. As we will show, in this case we also obtain not only the uniqueness of the fixed point (the solution of the equation) but also a constructive method to obtain it.

Because of the generality of our considerations, we will not limit ourselves here to integral operators of fractional order. They are only an example of problems for which their values lie in Hölder spaces. This is also a prototype of the subspace of the space

C [a, b]

, but with a much stronger norm of the form

∥ \cdot ∥ = {∥ \cdot ∥}_{\infty} + S (\cdot)

, for some seminorm S.

Clearly, as will be clarified, we are not restricted to these classical results. As we will soon see, the norms in the spaces of interest are the sums of the supremum norm and properly defined seminorms. Condensing or contraction operators will be a very typical case.

To be precise, iterative methods can be used when we limit ourselves to the space

C ([a, b])

, i.e.,

T (C ([a, b]) \subset C ([a, b]) .

However,

H^{α} [a, b]

is compactly embedded into

C ([a, b])

, and the norms on these spaces are not the same, so the continuity (or contraction) conditions are different in these cases. The lemma, based on the Schauder estimate and the Arzelà–Ascoli theorem, given below justifies the popularity of methods for investigating the existence of fixed points of operators with values in Hölder spaces by means of compactness methods (but not their uniqueness).

Lemma 1

(cf. [15], for instance). Suppose

0 < γ_{1} < γ_{2}

. Then, the Hölder space

H^{γ_{2}} [a, b]

is compactly embedded to

H^{γ_{1}} [a, b]

and the same conclusion holds true for

H^{γ_{2}} \subset H^{γ_{1}} \subset C [a, b]

This means that we lose some continuity properties of solutions and, since bounded operators with values in

H^{α} [a, b]

are compact in

C ([a, b])

, we turn again to the Schauder fixed-point theorem instead of the Banach fixed-point theorem.

So what is the situation? For the application of the Banach fixed-point theorem it is sufficient that the operator is contractive with respect to the sup-norm in

C [a, b]

and that we have the (only) fixed point of this operator in

C [a, b]

. However, since the operator has values in some subspace X of it (e.g., Hölder spaces), this fixed point must also be an element of that subspace. We will use this situation to outline how this looks from the point of view of fixed-point theory.

Does this mean that such an operator must be a contraction in X because of the norm

∥ \cdot ∥

? Obviously not. In such a case we will be unable to apply the Banach fixed-point theorem in X, so we prove a new result. For motivational reasons, we will prove it immediately for the nonlinear case of the product of certain operators.

Remark 1.

It is worth mentioning at this point the paper [16], in which the authors discuss the problem of applying various fixed-point theorems to fractional order integral equations. The results also cover the problem of the uniqueness of solutions but focus on generalizations of contraction conditions and do not allow for high regularity of solutions (either continuity of solutions only or contraction in the full space norm). However, this points in the direction of further research where the ideas of generalizations of supremum norm contraction can be combined with preserving regularity of solutions. This will be the subject of further research.

4. Contractive Product Operators

Unlike the sum of the operators, studying the contraction property for their product requires some more careful consideration. In this paper, we will pay special attention to operators that are the product of two other, more easily studied operators. Formally, this includes the classical case (we put the fixed operator as one of the operators to be studied), but we will nevertheless deal with this more general case. In this paper, we study the problem of the uniqueness of the solutions of equations and, in practice, fixed points of operators. Thus, we are interested in contractions, since even in the trivial case of the product of two identity operators on

C [a, b]

(which are not contractions), we have the problem

x = I d (x) \cdot I d (x)

, which has no unique solution. So, let us examine the contractions for the product of operators.

Recall (cf. Example 2) that the operator can be a contraction in the supremum norm but not in the Hölder-norm. We will prove that the property of being contractive with respect to the additional seminorm S requires more attention.

Moreover, to check whether the product of two operators

T_{1} (x) (t) \cdot T_{2} (x) (t)

is a contraction with respect to the sup-norm, we need to analyze the behavior of this new operator with respect to the norm

{∥ \cdot ∥}_{\infty}

C [0, 1]

. The pointwise product operator is defined as:

T (x) (t) = T_{1} (x) (t) \cdot T_{2} (x) (t) .

We need to check if this product operator

T (x)

is a strict contraction with respect to the sup-norm, i.e.,

{∥ \cdot ∥}_{\infty}

Now, let us check whether the product operator

T (x) = T_{1} (x) \cdot T_{2} (x)

satisfies this condition. For any two functions

x, y \in C [0, 1]

, we need to compute the difference between their values:

| T (x) (t) - T (y) (t) | = | T_{1} (x) (t) \cdot T_{2} (x) (t) - T_{1} (y) (t) \cdot T_{2} (y) (t) | .

Using the identity

a b - c d = (a - c) b + c (b - d)

, we can rewrite this as:

\begin{matrix} | T_{1} (x) (t) \cdot T_{2} (x) (t) - T_{1} (y) (t) \cdot T_{2} (y) (t) | & = & | (T_{1} (x) (t) - T_{1} (y) (t)) T_{2} (x) (t) \\ + T_{1} (y) (t) (T_{2} (x) (t) - T_{2} (y) (t)) | . \end{matrix}

This is the key expression to analyze. We want to bound the sup-norm of this difference:

{∥ T (x) - T (y) ∥}_{\infty} = sup_{t \in [0, 1]} | T (x) (t) - T (y) (t) | .

Using the triangle inequality, we can bound the right-hand side:

\begin{matrix} | T (x) (t) - T (y) (t) | & \leq & | T_{1} (x) (t) - T_{1} (y) (t) | \cdot | T_{2} (x) (t) | \\ + | T_{1} (y) (t) | \cdot | T_{2} (x) (t) - T_{2} (y) (t) | . \end{matrix}

Now, take the supremum over

t \in [0, 1]

for each term:

\begin{matrix} {∥ T (x) - T (y) ∥}_{\infty} & \leq & sup_{t \in [0, 1]} | T_{1} (x) (t) - T_{1} (y) (t) | \cdot sup_{t \in [0, 1]} | T_{2} (x) (t) | \\ + sup_{t \in [0, 1]} | T_{1} (y) (t) | \cdot sup_{t \in [0, 1]} | T_{2} (x) (t) - T_{2} (y) (t) | . \end{matrix}

This simplifies to:

{∥ T (x) - T (y) ∥}_{\infty} \leq ∥ T_{1} (x) - T_{1} {(y) ∥}_{\infty} \cdot ∥ T_{2} {(x) ∥}_{\infty} + ∥ T_{1} {(y) ∥}_{\infty} \cdot {∥ T_{2} (x) - T_{2} (y) ∥}_{\infty} .

Now, if

T_{1}

and

T_{2}

are contractions individually, then there exist constants

c_{1}, c_{2} \in [0, 1)

such that:

∥ T_{1} (x) - T_{1} {(y) ∥}_{\infty} \leq c_{1} {∥ x - y ∥}_{\infty}, ∥ T_{2} (x) - T_{2} {(y) ∥}_{\infty} \leq c_{2} {∥ x - y ∥}_{\infty} .

Using this in the bound for T, we obtain:

{∥ T (x) - T (y) ∥}_{\infty} \leq c_{1} {∥ x - y ∥}_{\infty} \cdot ∥ T_{2} {(x) ∥}_{\infty} + ∥ T_{1} {(y) ∥}_{\infty} \cdot c_{2} {∥ x - y ∥}_{\infty} .

Thus, the difference becomes:

{∥ T (x) - T (y) ∥}_{\infty} \leq (c_{1} ∥ T_{2} {(x) ∥}_{\infty} + c_{2} {∥ T_{1} (y) ∥}_{\infty}) {∥ x - y ∥}_{\infty} .

If the terms

∥ T_{2} {(x) ∥}_{\infty}

and

∥ T_{1} {(y) ∥}_{\infty}

are bounded, say by a constant M, then we can write:

{∥ T (x) - T (y) ∥}_{\infty} \leq (c_{1} M + c_{2} M) {∥ x - y ∥}_{\infty} = M (c_{1} + c_{2}) {∥ x - y ∥}_{\infty} .

For T to be a strict contraction, we need:

M (c_{1} + c_{2}) < 1 .

This will hold if

M \cdot (c_{1} + c_{2}) < 1

and the functions

T_{1} (x)

and

T_{2} (x)

are bounded by a common constant M that does not blow up the norm.

Recall, that

C [a, b]

is a Banach algebra with respect to the pointwise multiplication, so the operator

T = T_{1} \cdot T_{2}

is well defined. For the sake of completeness, let us summarize the calculations carried out (although the result itself is of course known, cf. [17]):

Lemma 2.

Let

T_{1}, T_{2}

be bounded mappings on

(C [a, b], ∥ \cdot ∥_{\infty})

with

∥ T_{1} ∥_{\infty} \leq M_{1}

∥ T_{2} ∥_{\infty} \leq M_{2}

. Assume that they are Lipschitz with constants

c_{1}, c_{2}

, respectively. If

M_{1} \cdot c_{2} + M_{2} \cdot c_{1} < 1,

then the product

T = T_{1} \cdot T_{2}

is a contraction mapping on

(C [a, b], ∥ \cdot ∥_{\infty})

Here is a slightly more complicated but also more interesting case. When calculating whether a product of two operators

T_{1} (x) (t) \cdot T_{2} (x) (t)

is a contraction in a Hölder space

H^{α} ([a, b])

, the process involves dealing with the Hölder norm instead of the sup-norm. The Hölder norm consists of two parts: the sup-norm

{∥ x ∥}_{\infty}

and the Hölder seminorm

{[x]}_{α}

, which measures the regularity of the function.

To check whether the operator

T (x) (t) = T_{1} (x) (t) \cdot T_{2} (x) (t)

is a contraction in

H^{α} ([a, b])

, we need to check whether:

{∥ T (x) - T (y) ∥}_{α} \leq c {∥ x - y ∥}_{α} for some 0 \leq c < 1 .

This involves examining both the sup-norm and the Höder seminorm separately for the product operator.

The calculation of the sup-norm component

{∥ T (x) - T (y) ∥}_{\infty}

for the product operator is the same as in the previous case. Now, let us focus on the new part: the Hölder seminorm. The Hölder seminorm is given by:

{[T (x)]}_{α} = sup_{t_{1} \neq t_{2}} \frac{| T (x) (t_{1}) - T (x) (t_{2}) |}{| t_{1} - t_{2} |^{α}} .

For the product

T (x) (t) = T_{1} (x) (t) \cdot T_{2} (x) (t)

, we have:

| T (x) (t_{1}) - T (x) (t_{2}) | = | T_{1} (x) (t_{1}) T_{2} (x) (t_{1}) - T_{1} (x) (t_{2}) T_{2} (x) (t_{2}) | .

Using the identity

a b - c d = (a - c) b + c (b - d)

, we rewrite this as:

\begin{matrix} | T_{1} (x) (t_{1}) T_{2} (x) (t_{1}) - T_{1} (x) (t_{2}) T_{2} (x) (t_{2}) | \\ = | (T_{1} (x) (t_{1}) - T_{1} (x) (t_{2})) T_{2} (x) (t_{1}) + T_{1} (x) (t_{2}) (T_{2} (x) (t_{1}) - T_{2} (x) (t_{2})) | . \end{matrix}

Next, divide both sides by

| t_{1} - t_{2} |^{α}

to evaluate the Hölder seminorm:

\begin{matrix} \frac{| T (x) (t_{1}) - T (x) (t_{2}) |}{| t_{1} - t_{2} |^{α}} & \leq & \frac{| T_{1} (x) (t_{1}) - T_{1} (x) (t_{2}) |}{| t_{1} - t_{2} |^{α}} | T_{2} (x) (t_{1}) | \\ + \frac{| T_{2} (x) (t_{1}) - T_{2} (x) (t_{2}) |}{| t_{1} - t_{2} |^{α}} | T_{1} (x) (t_{2}) | . \end{matrix}

Taking the supremum over all

t_{1}, t_{2} \in [0, 1]

, we obtain the bound for the Hölder seminorm:

{[T (x)]}_{α} \leq {[T_{1} (x)]}_{α} ∥ T_{2} {(x) ∥}_{\infty} + {[T_{2} (x)]}_{α} {∥ T_{1} (x) ∥}_{\infty} .

T_{1}

and

T_{2}

are contractions in

H^{α} ([0, 1])

, then there exist constants

c_{1}, c_{2} \in [0, 1)

such that:

{[T_{1} (x)]}_{α} \leq c_{1} {[x]}_{α}, {[T_{2} (x)]}_{α} \leq c_{2} {[x]}_{α} .

Substitute these into the bound for

{[T (x)]}_{α}

{[T (x)]}_{α} \leq c_{1} {[x]}_{α} ∥ T_{2} {(x) ∥}_{\infty} + c_{2} {[x]}_{α} {∥ T_{1} (x) ∥}_{\infty} .

Let us now estimate

{[T (x) - T (y)]}_{α}

. We start with the identity for the product difference

[T (x) - T (y)] (t)

\begin{matrix} T_{1} (x) (t) T_{2} (x) (t) & - & T_{1} (y) (t) T_{2} (y) (t) \\ = (T_{1} (x) (t) - T_{1} (y) (t)) T_{2} (x) (t) + T_{1} (y) (t) (T_{2} (x) (t) - T_{2} (y) (t)), \end{matrix}

and we can obtain the analogous expression for

[T (x) - T (y)] (s)

(

s \neq t

). Subtracting the two expressions, we obtain:

\begin{matrix} (T_{1} (x) (t) T_{2} (x) (t) - T_{1} (y) (t) T_{2} (y) (t)) - (T_{1} (x) (s) T_{2} (x) (s) - T_{1} (y) (s) T_{2} (y) (s)) \\ = ((T_{1} (x) (t) - T_{1} (y) (t)) T_{2} (x) (t) + T_{1} (y) (t) (T_{2} (x) (t) - T_{2} (y) (t))) \\ - ((T_{1} (x) (s) - T_{1} (y) (s)) T_{2} (x) (s) + T_{1} (y) (s) (T_{2} (x) (s) - T_{2} (y) (s))) . \end{matrix}

This formula contains some differences:

\begin{matrix} ((T_{1} (x) (t) - T_{1} (y) (t)) T_{2} (x) (t) - (T_{1} (x) (s) - T_{1} (y) (s)) T_{2} (x) (s)) \\ + (T_{1} (y) (t) (T_{2} (x) (t) - T_{2} (y) (t)) - T_{1} (y) (s) (T_{2} (x) (s) - T_{2} (y) (s))) . \end{matrix}

Estimate the first term

(T_{1} (x) (t) - T_{1} (y) (t)) T_{2} (x) (t) - (T_{1} (x) (s) - T_{1} (y) (s)) T_{2} (x) (s) .

We can decompose this as follows (by adding and subtracting appropriate terms):

\begin{matrix} (T_{1} (x) (t) - T_{1} (y) (t)) T_{2} (x) (t) - (T_{1} (x) (s) - T_{1} (y) (s)) T_{2} (x) (s) \\ = ((T_{1} (x) (t) - T_{1} (y) (t)) - (T_{1} (x) (s) - T_{1} (y) (s))) T_{2} (x) (t) \\ + (T_{1} (x) (s) - T_{1} (y) (s)) (T_{2} (x) (t) - T_{2} (x) (s)) . \end{matrix}

The term

((T_{1} (x) (t) - T_{1} (y) (t)) - (T_{1} (x) (s) - T_{1} (y) (s)))

can be estimated by the Hölder seminorm:

| (T_{1} (x) (t) - T_{1} (y) (t)) - (T_{1} (x) (s) - T_{1} (y) (s)) | \leq {[T_{1} (x) - T_{1} (y)]}_{α} {| t - s |}^{α} .

Moreover,

| T_{2} (x) (t) - T_{2} (x) (s) | \leq {[T_{2} (x)]}_{α} {| t - s |}^{α} .

Thus, the estimate for the first term becomes:

\begin{matrix} | (T_{1} (x) (t) - T_{1} (y) (t)) T_{2} (x) (t) - (T_{1} (x) (s) - T_{1} (y) (s)) T_{2} (x) (s) | \\ \leq {[T_{1} (x) - T_{1} (y)]}_{α} ∥ T_{2} {(x) ∥}_{\infty} {| t - s |}^{α} . \end{matrix}

We need to estimate the second difference in the expression under consideration, namely

T_{1} (y) (t) (T_{2} (x) (t) - T_{2} (y) (t)) - T_{1} (y) (s) (T_{2} (x) (s) - T_{2} (y) (s)) .

Again, we decompose it as:

\begin{matrix} T_{1} (y) (t) (T_{2} (x) (t) - T_{2} (y) (t)) - T_{1} (y) (s) (T_{2} (x) (s) - T_{2} (y) (s)) \\ = (T_{1} (y) (t) - T_{1} (y) (s)) (T_{2} (x) (t) - T_{2} (y) (t)) \\ + T_{1} (y) (s) ((T_{2} (x) (t) - T_{2} (y) (t)) - (T_{2} (x) (s) - T_{2} (y) (s))) . \end{matrix}

The first term

| T_{1} (y) (t) - T_{1} (y) (s) |

is bounded by

{[T_{1} (y)]}_{α} {| t - s |}^{α}

, and the second term

| (T_{2} (x) (t) - T_{2} (y) (t)) - (T_{2} (x) (s) - T_{2} (y) (s)) |

can be controlled by the Hölder seminorm of

T_{2} (x) - T_{2} (y)

, i.e.,

{[T_{2} (x) - T_{2} (y)]}_{α} {| t - s |}^{α}

Thus, we obtain the estimate:

\begin{matrix} | T_{1} (y) (t) (T_{2} (x) (t) - T_{2} (y) (t)) - T_{1} (y) (s) (T_{2} (x) (s) - T_{2} (y) (s)) | \\ \leq ∥ T_{1} {(y) ∥}_{\infty} {[T_{2} (x) - T_{2} (y)]}_{α} {| t - s |}^{α} . \end{matrix}

Combining the two estimates, we obtain:

\begin{matrix} | [T_{1} (x) T_{2} (x) - T_{1} (y) T_{2} (y)] (t) - [T_{1} (x) T_{2} (x) - T_{1} (y) T_{2} (y)] (s) | \\ \leq {[T_{1} (x) - T_{1} (y)]}_{α} ∥ T_{2} {(x) ∥}_{\infty} {| t - s |}^{α} + ∥ T_{1} {(y) ∥}_{\infty} {[T_{2} (x) - T_{2} (y)]}_{α} {| t - s |}^{α} . \end{matrix}

Taking into account the estimates of the supremum norm of the difference

T (x) - T (y)

and the Hölder seminorm, we obtain

{∥ T (x) - T (y) ∥}_{α} \leq (c_{1} ∥ T_{2} {(x) ∥}_{\infty} + c_{2} {∥ T_{1} (y) ∥}_{\infty}) {[x - y]}_{α} .

For the product operator

T (x) = T_{1} (x) \cdot T_{2} (x)

to be a strict contraction in the Hölder space, we need to ensure that:

{∥ T (x) - T (y) ∥}_{α} \leq c {∥ x - y ∥}_{α},

where

0 \leq c < 1

. This will happen if:

c_{1} ∥ T_{2} {(x) ∥}_{\infty} + c_{2} {∥ T_{1} (y) ∥}_{\infty} < 1,

and the terms involving the Hölder seminorm satisfy similar bounds:

c_{1} + c_{2} < 1 .

The contraction condition in Hölder space is more restrictive because it requires both the sup-norm and the Hölder seminorm to be contractions appropriately.

And now the version for the product in Hölder spaces. Note that the operators considered are not necessarily defined on the whole space

C [a, b]

. Stronger assumptions on the subset

C [a, b]

give stronger norm contraction conclusions.

Lemma 3.

Assume that

α \in (0, 1]

. Let

T_{1}, T_{2}

be bounded mappings on

(H^{α} [a, b], ∥ \cdot ∥_{α})

with

∥ T_{1} ∥_{α} \leq M_{1}

∥ T_{2} ∥_{α} \leq M_{2}

. Assume that they are Lipschitz mapping with constants

c_{1}, c_{2}

, respectively. If

M_{1} \cdot c_{1} + M_{2} \cdot c_{2} < 1 and c_{1} + c_{2} < 1,

where constants

c_{1}, c_{2} > 0

are such that

{[T_{1} (x)]}_{α} \leq c_{1} {[x]}_{α}, {[T_{2} (x)]}_{α} \leq c_{2} {[x]}_{α} .

Then, the product

T = T_{1} \cdot T_{2}

is also a contraction mapping on

(H^{α} [a, b], ∥ \cdot ∥_{α})

The study of Hölder spaces was motivated by considerations of integral operators of fractional order. However, we need not restrict ourselves to this case. We can generalize these results to other spaces, covering many classical function spaces but also those which are only constructed as images by successively studied operators.

It is important to note the strength of the assumptions required for the contraction properties of the product in such a space (e.g., conditions for constant contractions, contractibility with respect to seminorms). However, we prove a suitable result since it only requires defining the operator as continuous and bounded on some subspace

C [a, b]

defined by the condition

S (x) < \infty

for

x \in C [a, b]

, and so it can also be useful.

It should be emphasized that we are investigating the situation when the seminorm

S (\cdot)

satisfies an additional condition, which is a generalization of the assumption from the Maligrand–Orlicz lemma, i.e., that for any

x, y \in X

and

l \geq 1

we expect the estimate

S (x \cdot y) \leq l \cdot (∥ x ∥_{\infty} \cdot S (y) + S (x) \cdot ∥ y ∥_{\infty})

(cf. [18]). On the one hand, such a seminorm always exists on subspaces of

C [a, b]

(e.g.,

S (x) \equiv 0

), and there are seminorms that do not satisfy this condition (e.g., [18], Example 1). In general, however, the full characterization of seminorms that satisfy such a condition is not known. We will give interesting examples in Example 3.

Proposition 1.

Let

(X, ∥ \cdot ∥_{X})

X \subset C [a, b]

be closed under pointwise multiplication and equip this space with the norm

{∥ x ∥}_{X} = {∥ x ∥}_{\infty} + S (x),

where S is a given seminorm on X. Assume that for any

x, y \in X

and

l \geq 1

, we have

S (x \cdot y) \leq l \cdot (∥ x ∥_{\infty} \cdot S (y) + S (x) \cdot ∥ y ∥_{\infty}) .

(2)

Let

T_{1}, T_{2}

be bounded mappings on X, with

∥ T_{1} ∥_{X} \leq M_{1}

∥ T_{2} ∥_{X} \leq M_{2}

. Assume that there exist positive constants

c_{1}, c_{2}, d_{1}, d_{2}

such that for any

x, y \in X

∥ T_{1} (x) - T_{1} {(y) ∥}_{\infty} \leq d_{1} \cdot {∥ x - y ∥}_{\infty}, S (T_{1} (x) - T_{1} (y)) \leq c_{1} \cdot S (x - y),

and

∥ T_{2} (x) - T_{2} {(y) ∥}_{\infty} \leq d_{2} \cdot {∥ x - y ∥}_{\infty}, S (T_{2} (x) - T_{2} (y)) \leq c_{2} \cdot S (x - y) .

Assume that

M_{2} \cdot d_{1} + M_{1} \cdot d_{2} < \frac{1}{l}

and

M_{2} \cdot c_{1} + M_{1} \cdot c_{2} < \frac{1}{l}

. Then, the product

T = T_{1} \cdot T_{2}

is a contraction mapping on

(X, ∥ \cdot ∥_{X})

with the contraction constant

M = max (M_{2} \cdot d_{1} + M_{1} \cdot d_{2}, M_{2} \cdot c_{1} \cdot l + M_{1} \cdot c_{2} \cdot l)

Proof.

First of all, we should note that the space X equipped with the norm

{∥ \cdot ∥}_{X}

is a Banach algebra, so the product of operators is well defined (see [18]) and for any

x \in X

T_{1} (x) \cdot T_{2} (x) \in X

Once again, using the identity

a b - c d = (a - c) b + c (b - d)

, we have

\begin{matrix} ∥ T_{1} (x) \cdot T_{2} {(x) ∥}_{X} = {∥ T_{1} (x) \cdot T_{2} (x) ∥}_{\infty} + S (T_{1} (x) \cdot T_{2} (x)) \\ \leq ∥ T_{1} {(x) ∥}_{\infty} \cdot ∥ T_{2} {(x) ∥}_{\infty} + l \cdot ∥ T_{1} {(x) ∥}_{\infty} \cdot S (T_{2} (x)) + l \cdot S (T_{1} (x)) \cdot {∥ T_{2} (x) ∥}_{\infty} \\ \leq l \cdot (∥ T_{1} {(x) ∥}_{\infty} + S (T_{1} (x))) \cdot (∥ T_{2} (x) ∥_{\infty} + S (T_{2} (x))) \\ = l \cdot ∥ T_{1} {(x) ∥}_{X} \cdot {∥ T_{2} (x) ∥}_{X} . \end{matrix}

The product operator is bounded by

l \cdot M_{1} \cdot M_{2}

. Let us investigate the contraction property. Let

x, y \in X

and

t_{1}, t_{2} \in [a, b]

. Then, as S is a seminorm, it satisfies subadditivity and homogeneity property and consequently

\begin{matrix} {∥ T (x) - T (y) ∥}_{X} = {∥ T (x) - T (y) ∥}_{\infty} + S (T (x) - T (y)) \\ \leq & sup_{t \in [a, b]} | T_{1} (x) (t) - T_{1} (y) (t) | \cdot sup_{t \in [a, b]} | T_{2} (x) (t) | \\ + sup_{t \in [a, b]} | T_{1} (y) (t) | \cdot sup_{t \in [a, b]} | T_{2} (x) (t) - T_{2} (y) (t) | \\ + S (T_{1} (x) \cdot T_{2} (x) - T_{1} (y) \cdot T_{2} (y)) \\ \leq & M_{2} \cdot d_{1} {\cdot ∥ x - y ∥}_{\infty} + M_{1} \cdot d_{2} \cdot {∥ x - y ∥}_{\infty} \\ + S ((T_{1} (x) - T_{1} (y)) T_{2} (x) + T_{1} (y) (T_{2} (x) - T_{2} (y))) \\ \leq & M_{2} \cdot d_{1} {\cdot ∥ x - y ∥}_{\infty} + M_{1} \cdot d_{2} \cdot {∥ x - y ∥}_{\infty} \\ + l \cdot [S (T_{1} (x) - T_{1} (y)) \cdot M_{2} + S (T_{2} (x) - T_{2} (y)) \cdot M_{1}] \\ \leq & M_{2} \cdot d_{1} {\cdot ∥ x - y ∥}_{\infty} + M_{1} \cdot d_{2} \cdot {∥ x - y ∥}_{\infty} \\ + M_{2} \cdot c_{1} \cdot l \cdot S (x - y) + M_{1} \cdot c_{2} \cdot l \cdot S (x - y) . \end{matrix}

Thus,

\begin{matrix} {∥ T (x) - T (y) ∥}_{X} & \leq & max (M_{2} \cdot d_{1} + M_{1} \cdot d_{2}, M_{2} \cdot c_{1} + M_{1} \cdot c_{2}) {\cdot (∥ x - y ∥}_{\infty} + S (x - y)) \\ = & max (M_{2} \cdot d_{1} + M_{1} \cdot d_{2}, M_{2} \cdot c_{1} \cdot l + M_{1} \cdot c_{2} \cdot l) \cdot {∥ x - y ∥}_{X} \end{matrix}

and the contraction constant is

M = max (M_{2} \cdot d_{1} + M_{1} \cdot d_{2}, M_{2} \cdot c_{1} \cdot l + M_{1} \cdot c_{2} \cdot l)

. □

It is worth noting that similar studies, which we will still use in this paper, on the consequences of the seminorm property on product continuity, measures of noncompactness and compactness conditions in the spaces under study, have been carried out in [18].

Examples of some classic spaces will be a justification of the usefulness of our result.

Example 3.

Consider the following spaces covered by our approach:

(a) The space

R B V_{p} [0, 1]

of functions with Riesz finite p-variation (for some

p \geq 1

), i.e., with finite seminorms:

V a r_{R}^{p} (x) = sup_{(t_{i})} \sum_{i} \frac{| x (t_{i}) - x (t_{i - 1}) |^{p}}{| t_{i} - t_{i - 1} |^{p}}

for subdivisions

(t_{i})

[0, 1]

. This seminorm satisfies the condition from Proposition 1 for p big enough (cf. [18]).

(b) The space

X = C^{1} [0, 1]

with

S (x) = ∥ x^{'} ∥_{\infty}

. Then,

{S (x \cdot y) = ∥ (x \cdot y)}^{'} ∥_{\infty} = ∥ x^{'} \cdot y + x \cdot y^{'} ∥_{\infty} \leq {∥ y ∥}_{\infty} \cdot S (x) + {∥ x ∥}_{\infty} \cdot S (y) .

B V [0, 1]

when equipped with the norm

∥ x ∥ = {∥ x ∥}_{\infty} + V a r (x, [0, 1])

(with the supremum norm, it is not a complete space). In this case

S (x) = V a r (x, [0, 1])

(cf. [14,18]).

(d) The space

A C [0, 1]

equipped with the norm

{∥ x ∥}_{A C} = {∥ x ∥}_{\infty} + \int_{0}^{1} | x^{'} (t) | d t,

i.e., with a seminorm

S (x) = \int_{0}^{1} | x^{'} (t) | d t

(recall that for

x \in A C [0, 1]

we have

V a r (x) = \int_{0}^{1} | x^{'} (t) | d t

, cf. [14], Theorem 3.19).

Now, a comment that is crucial for the comparison of the results of the paper.

Remark 2.

To study contraction operators on such spaces, we can then use a classical version of the Banach contraction theorem and thus obtain not only the uniqueness of the fixed point, but also the constructive iterative method

x_{n + 1} = T (x_{n})

. Moreover, such a fixed point will be in space X, i.e., we have more regular fixed points, and its additional property is defined by

S (x) < \infty

However, the conditions for its use are quite restrictive (cf. Lemma 3 for the contraction of the product of operators). We will now give a new algorithm for proceeding, in which we retain the advantages of the previous treatment: we have the unique fixed point in X with an iterative method for its computation, but with significantly weaker assumptions (cf. Lemma 2 vs. Lemma 3). This applies to the case of operators defined and continuous over the whole space

C [a, b]

with values in X. We need to prove a new theorem.

The classical case here is the integral operators, and thanks to the Hardy–Littlewood theorem [12], the generalized fractional integral operators are of particular interest, as we will discuss in the next section of this paper and prove some new fixed-point theorems.

Let us consider bounded and continuous operators on

X \subset C [a, b]

, where X is equipped with a norm stronger than the supremum norm but defined on the whole space

C [a, b]

and are contractions on this space (as in Lemma 2). We will show that in this case the contraction condition on X is not necessary to obtain a unique fixed point in X, together with a constructive method for its computation.

With the results obtained earlier, the conclusions of the theorem can be briefly proved.

Theorem 1.

Let

(X, ∥ \cdot ∥_{X})

X \subset C [a, b]

be compact and closed under pointwise multiplication and equip this space with the norm

{∥ x ∥}_{X} = {∥ x ∥}_{\infty} + S (x),

where S is a seminorm on X with

S (x) < \infty

for all

x \in X

. Assume that for any

x, y \in X

and for some

l > 0

we have

S (x \cdot y) \leq l \cdot [∥ x ∥_{\infty} \cdot S (y) + S (x) \cdot ∥ y ∥_{\infty}] .

Let

T_{1}, T_{2}

be bounded and continuous mappings

T_{1}, T_{2} : C [a, b] \to X

and from X to X, with

∥ T_{1} ∥_{\infty} \leq M_{1} < \infty

∥ T_{2} ∥_{\infty} \leq M_{2} < \infty

being contractions with respect to the supremum norm, i.e.,

∥ T_{1} (x) - T_{1} {(y) ∥}_{\infty} \leq d_{1} {\cdot ∥ x - y ∥}_{\infty}, ∥ T_{2} (x) - T_{2} {(y) ∥}_{\infty} \leq d_{2} \cdot {∥ x - y ∥}_{\infty}

with

d_{1}, d_{2} < 1

. Then there exists a unique fixed point

\tilde{x}

T = T_{1} \cdot T_{2}

and

\tilde{x} \in X

and

\tilde{x} = {lim}_{n \to \infty} x_{n}

, where

x_{n + 1} = T (x_{n})

, and

x_{1} \in X

is arbitrary.

Proof.

Existence and uniqueness. First, we need to show that the operator T is well defined with values in X. Since

C [a, b]

is a Banach algebra when equipped with the supremum norm, the pointwise product is continuous on this space. Now note that the condition that the seminorm S satisfies on X guarantees that X equipped with the norm S is a Banach algebra ([18], Theorem 1).

The operators

T_{1}

and

T_{2}

are contractions, so continuous on X. By the continuity of the product in Banach algebras, the values of the operator T lie in the space X and then T is continuous on X. Thus,

T : C [a, b] \to C [a, b]

is well-defined and is a contraction. Moreover,

T : X \to X

is well defined and bounded.

Now note that we can apply the classical Banach fixed point theorem to the map

T : C [a, b] \to C [a, b]

. We obtain a unique fixed point

\tilde{x} = {lim}_{n \to \infty} x_{n}

, where

x_{n + 1} = T (x_{n})

in the supremum norm, so

\tilde{x} \in C [a, b]

. Recall, that

X \subset C [a, b]

Regularity. Now, the improving property of the operators

T_{1}

T_{2}

together with the continuity property of the product on Banach algebras implies that

x_{2}, x_{3}, . . . \in X

. Although X is not closed in

C [a, b]

with respect to the supremum norm, the convergence in

{∥ \cdot ∥}_{X}

is stronger that in

{∥ \cdot ∥}_{\infty}

. But sequence of iterations

(x_{n})

is Cauchy in

{∥ \cdot ∥}_{\infty}

and X is compact in

{∥ \cdot ∥}_{X}

, so must have a convergent subsequence in

{∥ \cdot ∥}_{X}

. Therefore, we have a sequence convergent in the supremum norm with subsequence convergent in

{∥ \cdot ∥}_{X}

. Thus, this subsequence is convergent in the

{∥ \cdot ∥}_{X}

and then the entire sequence is convergent in

{∥ \cdot ∥}_{X}

In this case,

\tilde{x}

is the fixed point of the operator T mapping

C [a, b]

into X, i.e.,

\tilde{x} = T (\tilde{x}) \in X

. Thus,

\tilde{x} \in X

. Note that the operator T does not have to be a contraction in the sense of the norm

{∥ \cdot ∥}_{X}

. □

The above theorem avoids testing the contraction condition for the seminorm S (cf. Lemma 3), which in many cases is difficult to test or does not occur. At the same time, however, we have an iterative method for finding a fixed point without losing its additional regularity, i.e., the property

S (\tilde{x}) < \infty

Although these are well-known facts, this is a good place to recall that neither condensing operators

∥ T (x) - T (y) ∥ < ∥ x - y ∥

(for

x \neq y

) nor nonexpansive operators

∥ T (x) - T (y) ∥ \leq ∥ x - y ∥

should have fixed points. However, the contraction condition is not necessary for the existence and uniqueness of the fixed point. However, we keep the Banach fixed-point theorem as a tool, since an iterative method for obtaining it is also an important goal. There will be a brief discussion of this case in the next section.

For a more detailed discussion of extensions of the Banach fixed-point theorem, the uniqueness of fixed points, and iterative approximations, see [19].

5. Quadratic Problem Applications

It is worth noting that the first quadratic equation to be studied as a result of solving a practical problem is the Chandrasekhar equation [20]

x (t) = 1 + x (t) \cdot \int_{0}^{1} \frac{t}{t + s} φ (s) x (s) d s .

It describes the radiative transfer through the stellar atmosphere. Many more such equations are now being studied (see [1,21]), but it is important to note that the Chandrasekhar equation was initially solved only by iterative methods using a variant of the Banach fixed-point theorem (Argyros [22]). Despite the use of many other methods later on, this method is still relevant because it allows a constructive calculation of the solution. However, this method has been applied to the space of continuous functions, and this and similar problems, especially with fractional order operators, require more regularity in the solution. This is what we propose in this paper.

The problem of the uniqueness of solutions in a subspace of

C [a, b]

is particularly evident when we have to study quadratic integral equations. To establish the goal of the study in this paper, we deal with quadratic equations with two generalized operators of fractional order, defined on an appropriate function space of the type described above. Details will be presented in the final section.

In the case of quadratic problems, first, we have the problem of uniqueness of solutions, which excludes numerical methods. Secondly, in the general case, we do not know whether the product of compact sets is compact, so even the application of theorems based on the compactness of operators is difficult to study. A good solution may therefore be the construction of a suitable invariant space for the operators, and it is also essential that this be a Banach algebra.

It is one more reason to find an invariant space for fractional operators. In such a case we can continue:

T (X) \subset X, T^{2} (X) \subset X, . . . T^{n} (X) \subset X .

This means that for quadratic equations, we expect to find Banach algebras as the domain of the operators T.

Recall that

H^{1} [a, b] \subset . . . \subset H^{α + 2 τ} [a, b] \subset H^{α + τ} [a, b] \subset H^{α} [a, b] \subset C ([a, b]) .

When solving integral equations by the operator method and fixed-point theorems, compactness arguments are very useful.

We can choose here such a class of equations, motivated by, among others, the Gripenberg integral equations ([23] or [1,21])) studied in the context of epidemic models. Let us also mention the quadratic models considered by Nussbaum in [4] or in [24] in the theory of transport. This fractional form of the equation under consideration allows us to show the advantages of the proposed strongly normed subspaces of

C [a, b]

, at the same time being Banach algebras. Our approach includes, as special cases, many previously considered equations (also for a non-quadratic case see [13]).

Since the integral equations considered in Banach algebras have a complicated form, the study of the uniqueness of solutions for such equations requires the use of different tools than for non-quadratic equations. Moreover, we still expect more regular than just continuous solutions.

Since we are motivated by some quadratic equations and are proposing a new fixed-point approach that allows for uniqueness of solutions, we recall an earlier fixed-point theorem by Banaś and Lecko specialized for this case. Note that the fixed-point result does not imply either their uniqueness or iterative methods. As claimed before, a direct use of the Banach fixed-point theorem requires very strong and restrictive assumptions.

To highlight the differences between our approach to applied fixed-point theorems and the previous one, let us first recall the necessary definition. By

M_{X}

, we denote the family of all nonempty and bounded subsets of X and by

N_{X}

, its subfamily consisting of all relatively compact subsets. In the sequel, we use an axiomatic approach to the notion of a measure of noncompactness.

Definition 1

([2,14,17,18]). A mapping

μ : M_{X} \to [0, \infty)

is said to be a measure of noncompactness in X if it satisfies the following conditions:

(i): $μ (A) = 0 \Rightarrow A \in N_{X}$ .
(ii): $A \subset B \Rightarrow μ (A) \leq μ (B) .$
(iii): $μ (\bar{A}) = μ (c o n v A) = μ (A) .$
(iv): $μ (λ A) = | λ | μ (A), f o r λ \in R .$
(v): $μ (A + B) \leq μ (A) + μ (B)$ .
(vi): $μ (A ⋃ B) = max {μ (A), μ (B)}$ .
(vii): If $A_{n}$ is a sequence of nonempty, bounded, closed subsets of X such that $A_{n + 1} \subset A_{n},$ $n = 1, 2, 3, \dots$ , and ${lim}_{n \to \infty} μ (A_{n}) = 0$ , then the set $A_{\infty} = ⋂_{n = 1}^{\infty} A_{n}$ is nonempty.

Recall a classical example: the Hausdorff measure of noncompactness

β_{H} (X)

is defined as follows:

β_{H} (X) = i n f {r > 0 : there exists finite subset Y of X such that x \subset Y + B_{r}}

, where X is an arbitrary nonempty and bounded subset of X. To distinguish between measures of noncompactness

μ

in different spaces (if necessary), we will give an appropriate space as an index, i.e.,

μ_{X}

μ_{Y}

etc.

The main result used in the study of quadratic equations to date is the following:

Theorem 2

([17,25], Theorem 2.5). Let X be a Banach algebra with a measure of noncompactness

μ_{X}

. Assume that W is nonempty, bounded, closed, and convex subset of X, and the operators

T_{1} : W \to X

and

T_{2} : W \to X

are continuous with

T_{1} (W), T_{2} (W)

being bounded in X. Moreover, assume that

T = T_{1} \cdot T_{2}

transform the set W into itself. If

1.: There exists a constant $k_{1} > 0$ such that $T_{1}$ satisfies an inequality: $μ_{X} (T_{1} (U)) \leq k_{1} \cdot μ_{X} (U)$ for arbitrary bounded subset U of W;
2.: There exists a constant $k_{2} > 0$ such that $T_{2}$ satisfies an inequality: $μ_{X} (T_{2} (U)) \leq k_{2} \cdot μ_{X} (U)$ for arbitrary bounded subset U of W;
3.: The measure of noncompacntess $μ_{X}$ on X satisfies the condition

$μ_{X} (A \cdot B) \leq b \cdot μ_{X} (A) + c \cdot μ_{X} (B)$

for some positive constants $b, c$ ;
4.: The following inequality holds true

${∥ P (T) ∥}_{X} \cdot k_{2} + {∥ Q (T) ∥}_{X} \cdot k_{1} < 1,$

then there exists at least one fixed point for the operator H in the set

W \subset X

Applying this result to a given space requires choosing an suitable measure of noncompactness

μ_{X}

and checking Assumption 3 each time (e.g., [25]). It can be shown that the set

F i x T

of all fixed points of the operator T on the set W is a member of the so-called kernel of the measure

μ_{X}

(i.e., family of sets for which in the condition (i) in Definition 1 there is the equivalence; [17,18,25]) but is not generally unique. We should also mention another version of this theorem is applied directly to the product of operators ([26], Section 4) (cf. also [27]).

Remark 3.

Suppose we are able to prove directly that the operator T is nonexpansive (see [28] for some further study). If it is defined on a bounded subset W of the space X which is compactly embedded in

C [a, b]

, i.e., is compact in

C [a, b]

, then this weaker assumption implies the existence and uniqueness of a fixed point which can be easily found as a minimizer of

d (x, T (x))

. It then easily follows that the fixed point is the limit of any sequence of iterations of T. But, we still have to check if the product operator is nonexpansive, which seems to be difficult. Instead of doing this, we can follow the idea from our decomposition of a norm.

In the case of nonexpansive mappings with respect to the seminorm, we are able to prove the following version of the Edelstein fixed-point theorem (and Proposition 1):

Proposition 2.

Let

(X, ∥ \cdot ∥_{X})

X \subset C [a, b]

be closed under pointwise multiplication and equip this space with the norm

{∥ x ∥}_{X} = {∥ x ∥}_{\infty} + S (x),

where S is a given seminorm on X. Assume that for any

x, y \in X

and

l > 0

we have

S (x \cdot y) \leq l \cdot (∥ x ∥_{\infty} \cdot S (y) + S (x) \cdot ∥ y ∥_{\infty}) .

Let

T_{1}, T_{2}

be bounded mappings on X, with

∥ T_{1} ∥_{X} \leq M_{1}

∥ T_{2} ∥_{X} \leq M_{2}

Assume that there exist constants

M_{1}, M_{2}

such that

∥ T_{1} ∥_{\infty} < M_{1}

∥ T_{2} ∥_{\infty} < M_{2}

and positive constants

c_{1}, c_{2}, d_{1}, d_{2}

such that for any

x, y \in X

∥ T_{1} (x) - T_{1} {(y) ∥}_{\infty} \leq d_{1} \cdot {∥ x - y ∥}_{\infty}, S (T_{1} (x) - T_{1} (y)) \leq S (x - y),

and

∥ T_{2} (x) - T_{2} {(y) ∥}_{\infty} \leq d_{2} \cdot {∥ x - y ∥}_{\infty}, S (T_{2} (x) - T_{2} (y)) \leq S (x - y) .

Assume that

M_{2} \cdot d_{1} \cdot l + M_{1} \cdot d_{2} \cdot l < 1

and

l \cdot (M_{2} + M_{1}) \leq 1

. Then, the product

T = T_{1} \cdot T_{2}

is a nonexpansive mapping on

(X, ∥ \cdot ∥_{X})

Corollary 1.

Suppose that the assumptions of Proposition 2 are satisfied and that

T : W \to W \subset X

, where W is a compact subset of

C [a, b]

. Then, t has a unique fixed point in W and it can be found as a limit of iterations

x_{n + 1} = T (x_{n})

for an arbitrary choice of

x_{1} \in W

It is important to note that it is this version of the fixed-point theorem that is useful in the normed spaces considered in the paper. The bounded sets in the Hölder space will be compact in

C [a, b]

, and the condensing condition will hold when we check the contraction due to the supremum norm, and it is sufficient that the seminorm

S (x) = {[x]}_{γ}

does not grow. Instead of the condensing condition, it is also possible to consider the conditions from the Caristi or Browder theorems ([19]).

Remark 4.

Now, we can look for unique solutions for quadratic problems by applying our Theorem 1 instead of Theorem 2. This is based on the construction of the Darbo fixed-point theorem, and it is shown that T is contraction with respect to some measure of noncompactness of X. We must also assume that the space is a Banach algebra. Our Theorem 1 is based on a similar idea but allows for the uniqueness of the fixed point, and this is obtained by an iterative method, and the continuity of the multiplication operation is obtained independently. This theorem is the summary of our considerations and follows from Proposition 1, the generalized Maligranda–Orlicz lemma, and the Banach fixed-point theorem.

Of course, the contraction condition can be replaced in our research by some generalizations. We refer interested readers to the paper [29], where the ψ-contraction mappings in a complete cone metric space over Banach algebra are investigated. Although this case is considered in the context of Banach algebras, it involves generalized contractions over the whole space.

And now we will give a simple but illustrative example showing the differences between the classical Banach result and the version proved here, with an indication of the role of the assumptions.

Example 4.

Consider the operator

T (x) (t) = \frac{1}{4} x (t) + \frac{1}{4} x (t^{4})

, which is not a contraction with respect to the Hölder seminorm.

Let us calculate the difference between

T (x) (s)

and

T (x) (t)

. Since

T (x) (t) = \frac{1}{4} x (t) + \frac{1}{4} x (t^{4}), T (x) (s) = \frac{1}{4} x (s) + \frac{1}{4} x (s^{4}),

taking the difference, we obtain:

| T (x) (s) - T (x) (t) | \leq \frac{1}{4} (| x (s) - x (t) | + | x (s^{4}) - x (t^{4}) |) .

For simplicity, let

α = 1

(so, we consider the Lipschitz space

H^{1} [0, 1]

). We have:

\frac{| T (x) (s) - T (x) (t) |}{| s - t |} \leq \frac{1}{4} {[x]}_{α} + \frac{1}{4} \frac{| x (s^{4})) - x (t^{4}) |}{| s - t |},

and then

\frac{| T (x) (s) - T (x) (t) |}{{| s - t |}^{α}} \leq \frac{1}{4} {[x]}_{α} + \frac{1}{4} \frac{| x (s^{4})) - x (t^{4}) |}{| s^{4} - t^{4} |^{α}} \cdot ((s + t) (s^{2} + t^{2})) .

By taking the supremum over all

s \neq t

s, t \in [0, 1]

we obtain

{[T (x)]}_{α} \leq \frac{1}{4} {[x]}_{α} + \frac{1}{4} {[x]}_{α} \cdot 4 = \frac{5}{4} {[x]}_{α} .

The supremum is obtained for

x_{0} (t) = t

, i.e., with

{[x_{0}]}_{α} = 1

. Indeed,

T (x_{0}) (t) = \frac{1}{4} t + \frac{1}{4} t^{4}

and

{[T (x_{0})]}_{α} = \frac{5}{4}

. Thus,

{[T (x_{0})]}_{α} = \frac{5}{4} > 1 = {[x_{0}]}_{α}

, and T is not a contraction with respect to the Hölder seminorm on

H^{1} [0, 1]

. Obviously,

{∥ T ∥}_{\infty} \leq \frac{1}{4} {∥ x ∥}_{\infty} + \frac{1}{4} {∥ x ∥}_{\infty} = \frac{1}{2} {∥ x ∥}_{\infty},

so T transform bounded ball into itself and

{∥ T (x) - T (y) ∥}_{\infty} \leq \frac{1}{4} {∥ x - y ∥}_{\infty} + \frac{1}{4} {∥ x - y ∥}_{\infty} = \frac{1}{2} {∥ x - y ∥}_{\infty} .

This operator is a contraction with respect to the supremum norm.

Let us now find the fixed points of the operator T and its nonlinear counterpart

H (x) (t) = T (x) (t) \cdot T (x) (t)

. A fixed point

x^{*} (t)

satisfies

T (x^{*}) (t) = x^{*} (t)

, which gives us the equation:

x^{*} (t) = \frac{1}{4} x^{*} (t) + \frac{1}{4} x^{*} (t^{4}) .

Rearranging this equation, we obtain

\begin{matrix} x^{*} (t) - \frac{1}{4} x^{*} (t) & = & \frac{1}{4} x^{*} (t^{4}) \\ x^{*} (t) & = & \frac{1}{3} x^{*} (t^{4}) \end{matrix}

This is the functional equation we need to solve to find the fixed points of T. Let us examine possible solutions. A candidate for a fixed point is a constant function, say

x^{*} (t) = c

, where c is some constant. Substituting this into the equation:

c = \frac{1}{3} c .

This equation is only satisfied when

c = 0

. Therefore, the only constant fixed point is

x^{*} (t) = 0

. Moreover, by observing the iterative nature of the functional equation, we can investigate the behavior at

t = 0

and

t = 1

. Since we are looking for continuous solutions, we have a unique fixed point

x^{*} (t) \equiv 0

. For non-constant solutions, the functional equation

x^{*} (t) = \frac{1}{3} x^{*} (t^{2})

leads to a recursive structure, which becomes progressively smaller for

t \in (0, 1)

, suggesting that

x^{*} (t) = 0

is the only solution.

Fixed points of

H (x) (t) = T (x) (t) \cdot T (x) (t)

. Recall, that

H (x) (t) = T (x) (t) \cdot T (x) (t) .

Substituting

T (x) (t) = \frac{1}{4} x (t) + \frac{1}{4} x (t^{4})

into this definition:

H (x) (t) = (\frac{1}{4} x (t) + \frac{1}{4} x (t^{4})) \cdot (\frac{1}{4} x (t) + \frac{1}{4} x (t^{4})) .

Expanding the product:

H (x) (t) = \frac{1}{16} x {(t)}^{2} + \frac{1}{16} x (t) x (t^{4}) + \frac{1}{16} x (t^{4}) x (t) + \frac{1}{16} x {(t^{4})}^{2},

which can be simplified to

H (x) (t) = \frac{1}{16} (x {(t)}^{2} + 2 x (t) x (t^{4}) + x {(t^{4})}^{2}) .

If we are looking for a fixed point

x^{*} (t)

such that

H (x^{*}) (t) = x^{*} (t)

, i.e.,

x^{*} (t) = \frac{1}{16} (x^{*} {(t)}^{2} + 2 x^{*} (t) x^{*} (t^{4}) + x^{*} {(t^{4})}^{2}) .

Let us examine possible fixed points, paying particular attention to whether a fixed point exists and especially whether it is the only one. We first check whether a constant function is a fixed point. Let

x^{*} (t) = c

, where c is a constant. Substituting into the equation:

c = \frac{1}{16} (c^{2} + 2 c \cdot c + c^{2}),

and then

\begin{matrix} c = \frac{1}{16} (2 c^{2} + 2 c^{2}) & = & \frac{1}{16} \cdot 4 c^{2} = \frac{c^{2}}{4} \\ c (c - 4) & = & 0 \end{matrix}

Thus,

c = 0

c = 4

. If

c = 0

, then

x^{*} (t) = 0

is a fixed point. The same is true for

c = 4

. Then,

x^{*} (t) = 4

is also a fixed point. As we can see, we certainly do not have the uniqueness of a fixed point.

Suppose now that we consider a ball in

C [0, 1]

with radius M, i.e.,

{∥ T ∥}_{\infty} \leq M

. In the considered case of Lipschitz functions, the constant

l = 1

(in Theorem 1).

Thus, the contraction condition from our fixed-point theorem became

M \cdot \frac{1}{4} + M \cdot \frac{1}{4} = \frac{M}{2} < 1,

whenever

M < 2

. Note that in this ball we have a unique fixed point (as

x^{*} (t) = 4

is not in this ball). As we can see, this very condition in the theorem still ensures the uniqueness of the fixed point (even for an operator that is not a contraction in the Hölder space).

5.1. Fractional Operators

As we have already pointed out, an excellent example of applications of our theorem are quadratic integral equations, and in particular those for fractional order operators. We will investigate such questions for a very general class of fractional order operators on suitable Hölder spaces. To this end, we will study their properties and those of the spaces, including showing that they are Banach algebras. In view of the results we have obtained so far, we will recall two classical operators of fractional order.

Since the results presented in this paper are motivated by and concern a generalization of those known for the classical fractional order Riemann–Liouville integral operator, we will first recall the classical ones. Let us now collect the known results for the Riemann–Liouville operator and the spaces explored so far (cf. [11]).

Definition 2

([30]). The Riemann–Liouville fractional integral of order τ of a well-defined function x on D is given by

I^{τ} x (t) = \frac{1}{Γ (τ)} \int_{0}^{t} \frac{x (s)}{{(t - s)}^{1 - τ}} d s, τ > 0, t > 0,

where

Γ (τ) = \int_{0}^{\infty} e^{- t} t^{τ - 1} d t

Proposition 3

([31], Theorem 3.1, [13]). Let

α > 0

and

0 \leq γ < 1

1.: $I^{α}$ is a linear operator defined on $L^{1}$ . For $1 \leq p \leq \infty$ , $I^{α}$ is a bounded operator from $L^{p}$ into $L^{p}$ and

$∥ I^{α} {u ∥}_{L^{p}} \leq \frac{T^{α}}{Γ (α + 1)} {∥ u ∥}_{L^{p}} .$
2.: For $1 \leq p < 1 / α$ , $I^{α}$ is a bounded operator from $L^{p} [0, T]$ into $L^{r} [0, T]$ for $1 \leq r < p / (1 - α p)$ . If $1 < p < 1 / α$ , then $I^{α}$ is a bounded operator from $L^{p} [0, T]$ into $L^{r} [0, T]$ for $r = p / (1 - α p)$ .
3.: For $1 / p < α < 1 + 1 / p$ or $p = 1$ and $1 \leq α < 2$ , the fractional integral operator $I^{α}$ is bounded from $L^{p}$ into a Hölder space $H^{α - \frac{1}{p}} ([a, b])$ ; hence, for $u \in L^{p}$ , $I^{α} u$ is Hölder continuous with exponent $α - 1 / p$ , thus $I^{α} u$ is continuous.

Various modifications and generalizations of the classical fractional integration operators are known and widely used in both theory and application. Here is another definition of an operator that is often used in applications:

Definition 3.

The Hadamard fractional integral of order

τ > 0

of a function

x \in L^{1} ([1, e], R)

, is defined as

{}^{H}I_{1}^{τ} x (t) = \frac{1}{Γ (τ)} \int_{1}^{t} {(log \frac{t}{s})}^{τ - 1} \frac{x (s)}{s} d s,

where

Γ (\cdot)

is the Euler gamma function. For completeness, we define

{}^{H}I^{τ} x (1) = 0

For the sake of our general considerations, we will only mention that a result analogous to Proposition 3 and concerning the value of the Hadamard operator in spaces of Hölder can be found in [32], Theorem 4.4.

We will analyze the following general class of operators (introduced on the basis of the so-called tempered calculus [33] and studied in [32,34]):

Definition 4

([15,32], Definition 1, [11]). Let

g \in C^{1} [a, b]

be a positive increasing function such that

g^{'} (t) \neq 0

, for all

t \in (a, b)

. The generalized fractional integral of a function

x : [a, b] \to E

of order

α > 0

and parameter

μ \in R^{+}

is that defined by

ℑ_{a, g}^{α, μ} x (t) = \frac{1}{Γ (α)} \int_{a}^{t} {(g (t) - g (s))}^{α - 1} e^{- μ (g (t) - g (s))} x (s) g^{'} (s) d s,

where

- \infty \leq a < b \leq \infty

. To ensure completeness, we define

ℑ_{a, g}^{α, μ} x (a) = 0

The generalized fractional operator defined above generalizes existing fractional integral operators including the two mentioned above:

(H): $ℑ_{0, g}^{α, μ}$ , $t \in [0, 1]$ , with

$g (t) = ln (1 + t) and μ \in R^{+}$

where $α > 0$ is the generalized version of the Hadamard model of fractional operators (the classical Hadamard operators under the additional condition $μ = 0$ ).
(RL): $ℑ_{0, t}^{α, 0}$ , $t \in [0, 1]$ is the classical fractional operator of the Riemann–Liouville type. We can, for example, refer the reader to the book [30].

The next result shows the fundamental role of Hölder spaces for generalized fractional operators ([32]). It is a starting point for studying the maximum regularity of the values of such operators.

Lemma 4.

α \in (0, 1)

μ > 0

and

p > max {\frac{1}{α}, 1}

, then the map

ℑ_{a, g}^{α, μ} : L^{p} [a, b] \to H_{0}^{α - \frac{1}{p}} [a, b]

is bounded.

Our aim will also be to make the results as general as possible. Therefore, we will replace the above lemma by defining suitable generalized spaces of Hölder type about which we will show that they satisfy the conditions for the normalization discussed so far. We will show that the generalized fractional order operators are invariant on the newly introduced spaces and that they are Banach algebras (cf. [2,18]). This will allow us to study quadratic integral equations, obtaining the uniqueness of their solutions using our fixed-point theorem.

5.2. Generalized Hölder Spaces

For simplicity, in this subsection we will consider the case

[a, b] = [0, 1]

. In addition to Hölder spaces

H^{λ} [0, 1]

, we are interested in another general class of function spaces.

For a continuously increasing function

ϑ : R^{+} \to R^{+}

with

ϑ (0 +) = 0

{lim}_{t \to 0 +} \frac{ϑ (t)}{t} = + \infty

, we define the (generalized) Hölder space

H^{ϑ} [0, 1]

as the class of functions satisfying the following condition:

| x (t) - x (s) | \leq L \cdot ϑ (| t - s |), L > 0, x \in C [0, 1] .

Given the formula above, it is sometimes called a

ϑ

-Lipschitz (or

ϑ

-Hölder) space. See also [2] for a special case of this type of space. Using the standard proof, we conclude that this space is of the type investigated in our paper.

Lemma 5.

Let

ϑ : R^{+} \to R^{+}

be continuous increasing function with

ϑ (0 +) = 0

{lim}_{t \to 0 +} \frac{ϑ (t)}{t} = + \infty

. Then, the space

H^{ϑ} [0, 1]

equipped with the norm

{∥x∥}_{ϑ} = {∥ x ∥}_{\infty} + {[x]}_{ϑ}, where {[x]}_{ϑ} = sup_{t \neq s} \frac{| x (t) - x (s) |}{ϑ (| t - s |)},

becomes a Banach space.

Obviously this seminorm vanishes for any constant function. The special choice

g (t) = t

ϑ (t) = t^{p}

p \in (0, 1)

leads to the classical Hölder spaces

H^{p} [0, 1]

. Note that the role of this g function is to test differentiability with respect to that function, and the

ϑ

function is to control the growth rate of the continuity modulus and so is “tempered” by

ϑ

Now, we define one more function space suitable for fractional operators. Let

g \in C^{1} [0, 1]

be a positive increasing function such that

g^{'} (t) \neq 0

, for all

t \in [0, 1]

. For a continuous increasing function

ϑ : R^{+} \to R^{+}

with

ϑ (0) = 0

{lim}_{t \to 0 +} \frac{ϑ (t)}{t} = + \infty

we define the (generalized) Hölder space

H_{g}^{ϑ} [0, 1]

as consisting of functions for which

| x (t) - x (s) | \leq L ϑ (| g (t) - g (s) |), for some L > 0, x \in C [0, 1] .

Put (for any

σ > 0

)

ω_{ϑ}^{g} (x, σ) = sup_{t \neq s} \{\frac{| x (t) - x (s) |}{ϑ (| g (t) - g (s) |)} : | t - s | < σ\} .

As in the classical case of spaces tempered by a modulus of continuity, the following is obtained:

Lemma 6

([2], cf. also [35]). Let

g \in C^{1} [0, 1]

be a positive increasing function such that

g^{'} (t) \neq 0

, for all

t \in [0, 1]

. Assume that

ϑ : R^{+} \to R^{+}

is a continuous increasing function with

ϑ (0) = 0

and

{lim}_{t \to 0 +} \frac{ϑ (t)}{t} = + \infty

. Then, the functional

ω_{ϑ}^{g} : C [0, 1] \to R^{+}

ω_{ϑ}^{g} (x) = lim_{σ \to 0^{+}} ω_{ϑ}^{g} (x, σ)

is a seminorm on

C [0, 1]

We will now extend some ideas from [2,18,36]. Consider a new class of subspaces of

C [0, 1]

that we will associate with fractional type operators. If we define a seminorm as an integral-type variation

j_{α, β}^{g, ϑ} (x, [0, 1]) = \int_{0}^{1} σ^{- (β + 1)} ω_{ϑ}^{g} {(x, σ)}^{\frac{β}{α}} d σ,

then we obtain a Banach space of the type we are interested in. We have

Proposition 4.

Let

0 < α \leq 1, α < β < \infty

and let

g \in C^{1} [0, 1]

be a positive increasing function such that

g^{'} (t) \neq 0

, for all

t \in [0, 1]

. Suppose

ϑ : R^{+} \to R^{+}

continuous and increasing function with

ϑ (0) = 0

and

{lim}_{t \to 0 +} \frac{ϑ (t)}{t} = + \infty

. Consider the space

J_{α, β}^{g, ϑ} [0, 1]

consisting of all of functions x with finite values of the functional

{∥ | x | ∥}_{α, β}^{g, ϑ} = {∥ x ∥}_{\infty} + (j_{α, β}^{g, ϑ} (x, [0, 1]))^{\frac{α}{β}} .

Then,

∥ | x | ∥_{α, β}^{g, ϑ}

is a norm on that space and

(J_{α, β}^{g, ϑ} [0, 1], ∥ | x | ∥_{α, β}^{g, ϑ})

is a Banach space.

It is straightforward to show that

∥ | x | ∥_{α, β}^{g, ϑ}

is a norm, and completeness is proved as in the case of Hölder spaces (see, for example, [14]) or in spaces tempered by some kind of modulus of continuity (see [37]). Furthermore, similar to classical Hölder spaces, by definition, if

1 < β_{1} < β_{2}

, then

J_{α, β_{1}}^{g, ϑ} [0, 1] \subset J_{α, β_{2}}^{g, ϑ} [0, 1]

It is worthwhile to note that, as claimed in [38], Section 5, contraction conditions are much easier to obtain in the integral-type Hölder space than in the classical Hölder spaces. However, in this paper, we show that we do not even need the conditions for contraction.

With the proposed assumptions about

α, β

ϑ

and g in mind as in the above Proposition, the boundedness of

ω_{ϑ}^{g} {(x, σ)}^{\frac{β}{α}}

always implies that

J_{α, β}^{g, ϑ} [0, 1] \subset H_{g}^{ϑ} [0, 1] .

First, note that

H_{g}^{{\tilde{ϑ}}_{α}} [a, b]

is compactly embedded in

C [a, b]

, but since the later space has a weaker norm than Hölder spaces, we prefer such a property of Hölder spaces rather than embeddings in

C [a, b]

(cf. assumptions of Theorem 1).

Let us present a compact embedding theorem with respect to the parameter

ϑ

for generalized Hölder spaces

J_{α, β}^{g, ϑ} [0, 1]

. Denote by

H^{*}

the class of non-decreasing functions

ϑ

from

R^{+}

R^{+}

, with right limit at zero equals zero and

ϑ (t) / t \to C > 0

t \to 0^{+}

. It is also an extension of [15], Lemma 1 for

H_{g}^{ϑ} [0, 1]

Lemma 7.

Let

g \in C^{1} [0, 1]

be a positive increasing function such that

g^{'} (t) \neq 0

, for all

t \in [0, 1]

. Suppose that

ϕ, ψ \in H^{*}

. If

\underset{t \to 0^{+}}{lim inf} \frac{ψ (t)}{ϕ (t)} = C < \infty,

then

J_{α, β}^{g, ψ} [0, 1]

is continuously embedded in

J_{α, β}^{g, ϕ} [0, 1]

. If, moreover,

lim_{t \to 0^{+}} \frac{ψ (t)}{ϕ (t)} = 0,

(3)

then the embedding is also compact.

Proof.

Recall that

j_{α, β}^{g, ϑ} (x, [0, 1]) = \int_{0}^{1} σ^{- (β + 1)} ω_{ϑ}^{g} {(x, σ)}^{\frac{β}{α}} d σ .

I. Consider the first case when

{lim inf}_{t \to 0^{+}} \frac{ψ (t)}{ϕ (t)} = C < \infty

. It implies that there exists

δ_{1} > 0

such that

ψ (t) \leq C \cdot ϕ (t)

for

t \in (0, δ_{1})

. Clearly,

\frac{ψ (t)}{ϕ (t)}

is bounded on

[δ_{1}, 1]

. Due to the monotonicity of these functions

\frac{ψ (δ_{1})}{ϕ (1)} \leq \frac{ψ (t)}{ϕ (t)}

for

t \in [δ_{1}, 1]

. Let

x \in J_{α, β}^{g, ψ} [0, 1]

. So for

| g (t) - g (s) | < δ_{1}

, we obtain

| x (t) - x (s) | \leq ψ (| g (t) - g (s) |) \cdot ω_{ψ}^{g} (x) \leq C \cdot ϕ (| g (t) - g (s) |) \cdot ω_{ψ}^{g} (x) .

and then

sup_{t \neq s, 0 < | t - s | < δ_{1}} \frac{| x (t) - x (s) |}{ϕ (| g (t) - g (s) |)} \leq C \cdot sup_{t \neq s, 0 < | t - s | < δ_{1}} \frac{| x (t) - x (s) |}{ψ (| g (t) - g (s) |)} .

Moreover,

sup_{t \neq s, 0 < | t - s | < δ_{1}} \frac{| x (t) - x (s) |}{ϕ (| g (t) - g (s) |)} = sup_{t \neq s, 0 < | t - s | < δ_{1}} \frac{ψ (| g (t) - g (s) |)}{ϕ (| g (t) - g (s) |)} \cdot \frac{| x (t) - x (s) |}{ψ (| g (t) - g (s) |)} .

\frac{ψ (δ_{1})}{ϕ (1)} \cdot sup_{t \neq s, 0 < | t - s | < δ_{1}} \frac{| x (t) - x (s) |}{ϕ (| g (t) - g (s) |)} \leq sup_{t \neq s, 0 < | t - s | < δ_{1}} \frac{| x (t) - x (s) |}{ψ (| g (t) - g (s) |)} .

Hence, for any

σ > 0

ω_{ϕ}^{g} (x, σ) \leq \frac{ω_{ψ}^{g} (x, σ)}{min {C, \frac{ϕ (δ_{1})}{ϕ (1)}}} .

Thus, by passing to the limit with

σ

and using the definition of

j_{α, β}^{g, ϕ} (x, [0, 1])

we obtain

min \{C, \frac{ϕ (δ_{1})}{ϕ (1)}\} \cdot j_{α, β}^{g, ϕ} (x, [0, 1]) \leq j_{α, β}^{g, ψ} (x, [0, 1]) .

Finally,

∥ | x | ∥_{α, β}^{g, ϕ} \leq const . \cdot ∥ | x | ∥_{α, β}^{g, ψ}

and the embedding is continuous.

II. Now, the case of compact embeddings. Let

(u_{n}) \subset J_{α, β}^{g, ψ} [0, 1]

be arbitrary sequence taken from the unit ball in this space, so, in particular, its supremum norm is less than 1, i.e.,

∥ u_{n} ∥_{\infty} \leq 1

and

| u_{n} (t) - u_{n} (s) | \leq A \cdot ψ (| g (t) - g (s) |)

for some

A > 0

and all

(t, s) \in [0, 1]

. By the monotonicity and continuity properties of g and

ψ

we can apply the Arzelà–Ascoli theorem, and consequently we can subtract a subsequence

({\tilde{u}}_{n})

convergent in the supremum norm in

C [0, 1]

to some

u \in C [0, 1]

. By applying the uniform convergence of this sequence we get

| u (t) - u (s) | = lim_{n \to \infty} | {\tilde{u}}_{n} (t) - {\tilde{u}}_{n} (s) | \leq A \cdot ψ (| g (t) - g (s) |),

and again using the properties of

ψ

and g, we obtain that

ω_{ψ}^{g} (u, σ) \leq A < \infty

and then

j_{α, β}^{g, ψ} (u, [0, 1]) < \infty

, so

u \in J_{α, β}^{g, ψ} [0, 1]

as well.

Now, we are in a position to prove that

(u_{n} - {\tilde{u}}_{n}) \subset J_{α, β}^{g, ψ} [0, 1]

is convergent in

J_{α, β}^{g, ϕ} [0, 1]

. Obviously, it converges in

C [a, b]

, so it remains to prove its convergence in the seminorm

j_{α, β}^{g, ψ}

. Fix arbitrary

σ > 0

. Let us consider first the seminorm

ω_{ψ}^{g} (x, σ)

Given

δ > 0

, by the properties of g, let

t, s \in [a, b]

be such that

t \neq s

and

| g (t) - g (s) | < δ

. Place

v_{n} = u - {\tilde{u}}_{n}

. Since

ψ

is increasing, we obtain

\begin{matrix} ω_{ϕ}^{g} (x, σ) & = & sup_{t \neq s} \frac{| v_{n} (t) - v_{n} (s) |}{ϕ (| g (t) - g (s) |)} \\ = & max \{sup_{t \neq s, 0 < | t - s | < δ} \frac{| v_{n} (t) - v_{n} (s) |}{ϕ (| g (t) - g (s) |)}, sup_{t \neq s, δ \leq | t - s | \leq b - a} \frac{| v_{n} (t) - v_{n} (s) |}{ϕ (| g (t) - g (s) |)}\} \\ \leq & max \{sup_{t \neq s, 0 < | t - s | < δ} \frac{ψ (| g (t) - g (s) |)}{ϕ (| g (t) - g (s) |)} \cdot \frac{| v_{n} (t) - v_{n} (s) |}{ψ (| g (t) - g (s) |)}, \\ sup_{t \neq s, δ \leq | t - s | \leq b - a} \frac{| v_{n} (t) - v_{n} (s) |}{ϕ (| g (t) - g (s) |)}\} . \end{matrix}

Recall, that we consider functions

ψ, ϕ

from the class

H^{*}

. If

{lim}_{δ \to 0^{+}} \frac{ψ (δ)}{ϕ (δ)} = 0

, then as

{lim}_{t \to 0^{+}} \frac{ψ (t)}{ϕ (t)} = 0

, for sufficiently small t, i.e.,

t < δ_{C}

, we have:

\frac{ψ (t)}{ϕ (t)} < ϵ .

Thus, for

| g (t) - g (s) | < δ_{C}

| v_{n} (t) - v_{n} (s) | \leq ϕ (| g (t) - g (s) |) \cdot ω_{ϕ}^{g} (v_{n}) \leq ϵ ϕ (| g (t) - g (s) |) \cdot ω_{ψ}^{g} (v_{n}) .

It implies, that

{v_{n}}

is equicontinuous in the

ϕ

–seminorm. Therefore,

sup_{t \neq s, 0 < | t - s | < δ_{C}} \frac{ϕ (| g (t) - g (s) |)}{ψ (| g (t) - g (s) |)} \cdot \frac{| v_{n} (t) - v_{n} (s) |}{ϕ (| g (t) - g (s) |)} \leq ε \cdot ω_{ψ}^{g} (v_{n}),

and arguing as above we obtain

sup_{t \neq s, 0 < | t - s | < δ_{C}} \frac{ϕ (| g (t) - g (s) |)}{ψ (| g (t) - g (s) |)} \cdot j_{α, β}^{g, ϕ} (v_{n}, [0, 1]) \leq ε \cdot j_{α, β}^{g, ψ} (v_{n}, [0, 1]) .

(4)

Clearly,

sup_{t \neq s, δ_{C} \leq | t - s | \leq b} \frac{| v_{n} (t) - v_{n} (s) |}{ψ (| g (t) - g (s) |)} \leq \frac{2}{ψ (δ_{C})} \cdot {∥ v_{n} ∥}_{\infty} .

(5)

Hence,

{v_{n}}

is uniformly bounded and equicontinuous in

J_{α, β}^{g, ϕ} [0, 1]

, the Arzelà–Ascoli theorem guarantees that there is a subsequence

{v_{n_{k}}}

convergent in

J_{α, β}^{g, ϕ} [0, 1]

. Indeed, since both

j_{α, β}^{g, ϕ} (v_{n}, [0, 1])

and

∥ v_{n} ∥_{\infty}

converge to zero as

n \to \infty

, so does

{∥ | v_{n} | ∥}_{α, β}^{g, ϕ}

, and we are finished. □

This class of spaces extends some well-known ideas for bounded variation spaces, in particular those of [38] for the Riemann–Liouville integral operator

I^{α}

and the special case of

J_{α, β}

with

ϑ (t) = t

([2]). As claimed in [35], such spaces are useful when studying generalized fractional integral operators and the special choice of g and

ϑ

allow us to consider such spaces as natural domains and the range of these operators ([35], Lemma 4.4, Theorem 4.2).

For completeness, we recall that the space consisting of functions with bounded variation

B V [a, b]

endowed with the norm

{∥ x ∥}_{B V} = {∥ x ∥}_{\infty} + V a r (x, [a, b])

is a Banach space (even a Banach algebra), where

V a r (x, [a, b]) = sup_{Π \in P} \sum_{i = 0}^{n_{P} - 1} | f (x_{i + 1}) - f (x_{i}) | .

Here, the supremum is taken over the set

P

of all partitions

\{x_{0}, \dots, x_{n_{P}}\}

of the interval

[a, b]

satisfying

x_{i} \leq x_{i + 1}

for

0 \leq i \leq n_{P} - 1

of all partitions of the interval considered.

However, in order to show the uniqueness of the fixed point (as solutions of the quadratic fractional integral equation) with our approach, we need to prove that such spaces are Banach algebras. But, for this class of spaces, there is no reason to study them separately (as in many previous papers). Instead, we will use the generalized Maligranda–Orlicz lemma [18], Theorem 1, which is a universal tool in such studies.

Proposition 5.

Let

0 < α \leq 1, α < β < \infty

and let

g \in C^{1} [0, 1]

be a positive increasing function such that

g^{'} (t) \neq 0

, for all

t \in [0, 1]

. Suppose

ϑ : R^{+} \to R^{+}

is a continuous and increasing function with

ϑ (0) = 0

and

{lim}_{t \to 0 +} \frac{ϑ (t)}{t} = + \infty

. Then, the space

J_{α, β}^{g, ϑ} [0, 1]

consisting of all of functions x with finite values of the norm

{∥ | x | ∥}_{α, β}^{g, ϑ} = {∥ x ∥}_{\infty} + (j_{α, β}^{g, ϑ} (x, [0, 1]))^{\frac{α}{β}}

is is a commutative Banach algebra with unity when equipped with this norm. For arbitrary

x, y \in J_{α, β}^{g, ϑ} [0, 1]

, we obtain

{∥ | x \cdot y | ∥}_{α, β}^{g, ϑ} \leq (1 + 2^{2 - \frac{α}{β}}) \cdot {∥ | x | ∥}_{α, β}^{g, ϑ} {∥ | y | ∥}_{α, β}^{g, ϑ}

Proof.

In the light of [18], Theorem 1, in order to prove that the space

J_{α, β}^{g, ϑ} [0, 1]

with the proposed norm is a Banach algebra, it is sufficient to check its assumptions. In particular, it is necessary to study the properties of the seminorm

S (x) = (j_{α, β}^{g, ϑ} (x, [0, 1]))^{\frac{α}{β}}

Let

x, y \in J_{α, β}^{g, ϑ} [0, 1]

. For arbitrary

t, s \in [0, 1]

, we have

\begin{matrix} | x (t) y (t) - x (s) y (s) | & \leq & | x (t) y (t) - x (t) y (s) + x (t) y (s) - x (s) y (s) | \\ \leq & | x (t) | | y (t) - y (s) | + | y (s) | | x (t) - x (s) | . \end{matrix}

Since

sup (f \cdot g) \leq sup f \cdot sup g,

for any

σ \in [0, 1]

, we obtain

ω_{ϑ}^{g} (x y, σ) = sup_{t \neq s} \{\frac{| x y (t) - x y (s) |}{ϑ (| g (t) - g (s) |)} : | t - s | < σ\}

and then

ω_{ϑ}^{g} (x y, σ) \leq {∥ x ∥}_{\infty} \cdot ω_{ϑ}^{g} (y, σ) + {∥ y ∥}_{\infty} \cdot ω_{ϑ}^{g} (x, σ) .

Now, we can estimate the seminorm

(j_{α, β}^{g, ϑ} (x, [0, 1]))^{\frac{α}{β}}

. We have

\begin{matrix} j_{α, β}^{g, ϑ} (x y, [0, 1]) & = & \int_{0}^{1} σ^{- (β + 1)} ω_{ϑ}^{g} {(x y, σ)}^{\frac{β}{α}} d σ \\ \leq & \int_{0}^{1} σ^{- (β + 1)} \cdot {(∥ x ∥}_{\infty} \cdot ω_{ϑ}^{g} (y, σ) + {∥ y ∥}_{\infty} \cdot ω_{ϑ}^{g} {(x, σ))}^{\frac{β}{α}} d σ . \end{matrix}

Since we have

β \geq α

, we can apply the Minkowski inequality, and then we have

\begin{matrix} j_{α, β}^{g, ϑ} (x y, [0, 1]) & \leq & 2^{\frac{β}{α} - 1} \cdot {(∥ x ∥}_{\infty})^{\frac{β}{α}} \cdot \int_{0}^{1} σ^{- (β + 1)} ω_{ϑ}^{g} {(y, σ)}^{\frac{β}{α}} d σ \\ + 2^{\frac{β}{α} - 1} \cdot {(∥ y ∥}_{\infty})^{\frac{β}{α}} \cdot \int_{0}^{1} σ^{- (β + 1)} ω_{ϑ}^{g} {(x, σ)}^{\frac{β}{α}} d σ . \end{matrix}

Then,

\begin{matrix} (j_{α, β}^{g, ϑ} (x y, [0, 1]))^{\frac{α}{β}} & \leq & {(2^{\frac{β}{α} - 1} \cdot {(∥ x ∥}_{\infty})^{\frac{β}{α}} \cdot \int_{0}^{1} σ^{- (β + 1)} ω_{ϑ}^{g} {(y, σ)}^{\frac{β}{α}} d σ)}^{\frac{α}{β}} \\ + {(2^{\frac{β}{α} - 1} \cdot {(∥ y ∥}_{\infty})^{\frac{β}{α}} \cdot \int_{0}^{1} σ^{- (β + 1)} ω_{ϑ}^{g} {(x, σ)}^{\frac{β}{α}} d σ)}^{\frac{α}{β}} \\ = & 2^{1 - \frac{α}{β}} \cdot {∥ x ∥}_{\infty} \cdot (j_{α, β}^{g, ϑ} (y, [0, 1]))^{\frac{α}{β}} \\ + 2^{1 - \frac{α}{β}} \cdot {∥ y ∥}_{\infty} \cdot (j_{α, β}^{g, ϑ} (x, [0, 1]))^{\frac{α}{β}} . \end{matrix}

As we can see, the seminorm satisfies the Maligranda–Orlicz lemma with the constant

l = 2^{1 - \frac{α}{β}}

. The space under consideration is therefore a Banach algebra.

To obtain the second part of the thesis and to calculate the constant in the estimation, let us use the inequalities obtained. By the definition of the norm, for any

x \in J_{α, β}^{g, ϑ} [0, 1]

we have

(j_{α, β}^{g, ϑ} (x y, [0, 1]))^{\frac{α}{β}} \leq ∥ | x \cdot y | ∥_{α, β}^{g, ϑ}

and

{∥ x y ∥}_{\infty} \leq ∥ | x \cdot y | ∥_{α, β}^{g, ϑ}

, so we can obtain the estimation

\begin{matrix} ∥ | x y | ∥_{α, β}^{g, ϑ} & = & {∥ x y ∥}_{\infty} + (j_{α, β}^{g, ϑ} (x y, [0, 1]))^{\frac{α}{β}} \\ \leq & {∥ x y ∥}_{\infty} + 2^{1 - \frac{α}{β}} \cdot {∥ x ∥}_{\infty} ∥ | y | ∥_{α, β}^{g, ϑ} + 2^{1 - \frac{α}{β}} \cdot {∥ y ∥}_{\infty} ∥ | x | ∥_{α, β}^{g, ϑ} \\ \leq & {∥ x ∥}_{\infty} {∥ y ∥}_{\infty} + 2^{1 - \frac{α}{β}} \cdot {∥ x ∥}_{\infty} ∥ | y | ∥_{α, β}^{g, ϑ} + 2^{1 - \frac{α}{β}} \cdot {∥ y ∥}_{\infty} ∥ | x | ∥_{α, β}^{g, ϑ} \\ \leq & ∥ | x | ∥_{α, β}^{g, ϑ} ∥ | y | ∥_{α, β}^{g, ϑ} + 2^{1 - \frac{α}{β}} \cdot ∥ | x | ∥_{α, β}^{g, ϑ} ∥ | y | ∥_{α, β}^{g, ϑ} + 2^{1 - \frac{α}{β}} \cdot ∥ | x | ∥_{α, β}^{g, ϑ} ∥ | y | ∥_{α, β}^{g, ϑ} \\ = & (1 + 2^{2 - \frac{α}{β}}) \cdot ∥ | x | ∥_{α, β}^{g, ϑ} \cdot ∥ | y | ∥_{α, β}^{g, ϑ} . \end{matrix}

Obviously, this is commutative algebra with unity

1 (t) \equiv 1

(clearly

1 \in J_{α, β}^{g, ϑ} [0, 1]

). Since

∥ | 1 | ∥_{α, β}^{g, ϑ} \neq 1

, it is not normalized algebra. □

5.3. Quadratic Fractional Equations

We will now present a sample result on the uniqueness of solutions of quadratic integral equations, which is an example of the application of the proposed version of the fixed-point theorem. To the best of our knowledge, all previous uniqueness theorems for such (and other) problems have been considered under two important restrictions. Namely, the regularity of the solutions was lost, and the investigations were concerned only with the space

C [a, b]

, and consequently the operators were simply contractions on their domain (cf. [9]). Surprisingly, this always amounted to an application of the Banach fixed-point theorem to subsets of

C [a, b]

and had no mathematically interesting aspects. Meanwhile, as we have shown in the paper, the quadratic integral operator does not need to be a contraction in the solution space at all to have a unique fixed point. As we have shown in this paper, this is too strong an assumption (cf. Lemma 3), and worse, we lose natural classes of solutions.

On the other hand, by applying Theorem 2, one can obtain non-emptiness (and compactness) of the set of solutions but generally without their uniqueness and the iterative method of their construction (cf. [1,2,3,18]).

Let us also note an interesting mathematical aspect of the situation: we are studying a whole class of subspaces

C [a, b]

related to the regularity of solutions, so there are other interesting function spaces to study, e.g., properties of nonlinear operators on such spaces. The necessary and sufficient conditions for their good definition, boundedness, or continuity are still not fully explored.

By selecting certain fractional integral operators (a very general class of operators) to study, we can specify which g and

ϑ

functions involve these operators and study the properties of the operators on such function spaces.

Definition 5

([39]). Suppose that a function

f (t, x) : [a, b] \times R \to R

; we will assign the superposition (Nemytskii) operator F, as follows:

F (x) (t) = f (t, x (t)), t \in [a, b] .

For an interesting discussion of the properties of this operator on function spaces, see [40] or [38], Section 4.

Let us consider the following quadratic integral equations with fractional order operators:

x (t) = ℑ_{a, g}^{τ_{1}, μ} F_{1} (x) (t) \cdot ℑ_{a, g}^{τ_{2}, μ} F_{2} (x) (t)

(6)

where

0 < α < τ_{i} < 1

i = 1, 2

in the Banach algebra

J_{α, β}^{g, ϑ} [0, 1]

, where

β \geq max {τ_{1}, τ_{2}}

. Here

F_{k}

denotes Nemytskii superposition generated by functions

f_{k} (\cdot, \cdot)

(

k = 1, 2

Since the result now presented is intended to be an illustration of the usefulness of the fixed-point theorem obtained, we will not focus on the necessary and sufficient conditions for the properties of the operators in the spaces under consideration (this part is still an open problem) and leave it as an interesting mathematical problem for separate research.

We will start by studying the generalized fractional operator in the space we introduced, with its boundedness.

Lemma 8

([35], Lemma 3.4). Let

g \in C^{1} [0, 1]

be a positive increasing function such that

g^{'} (t) \neq 0

, for all

t \in [a, b]

. Let

0 < α < 1

μ \geq 0

and

β > α

. Assume that

ϑ : R^{+} \to R^{+}

is an arbitrary continuous increasing function with

ϑ (0) = 0

with

{lim}_{t \to 0 +} \frac{ϑ (t)}{t} = + \infty

. Then the operator

ℑ_{0, g}^{α, μ}

is bounded from

J_{α, β}^{g, ϑ} [0, 1]

C (([0, 1]), ∥ \cdot ∥_{\infty})

Lemma 9

([35], Theorem 3.2). Let

ϑ (x) = x^{ρ}

and let

g \in C^{1} [0, 1]

be a positive increasing function such that

g^{'} (t) \neq 0

, for all

t \in [0, 1]

. Suppose

0 < τ < 1

0 < α < τ

ρ < τ - α

, and

β > α

. Then

ℑ_{0, g}^{τ, μ} x

maps the space

J_{α, β}^{g, ϑ} [0, 1]

into itself and is bounded.

Furthermore, its generalized modulus of continuity can be estimated as follows:

ω_{ϑ}^{g} (ℑ_{0, g}^{τ, μ} x, δ) \leq c (τ, μ) \cdot [{∥ g ∥}_{\infty}^{τ} \cdot ω_{ϑ}^{g} (x, δ) + {∥ x ∥}_{\infty} \cdot ω_{ϑ}^{g} (g^{τ} \cdot e^{- μ g}, δ)]

for some constant

c (α, μ)

The above estimate is important for checking the condition (2), so we need to calculate this constant

c (α, μ)

(which is not performed in [35]). In fact, in [35], we have the following estimate:

c (τ, μ) = \int_{0}^{1} {(1 - u)}^{τ} e^{- μ (1 - u)} d u / Γ (τ)

. So,

c (τ, μ) = \frac{\int_{0}^{1} {(1 - u)}^{τ} e^{- μ (1 - u)} d u}{Γ (τ)} \leq \frac{Γ (τ + 1) - γ (τ + 1, μ)}{Γ (τ) \cdot μ^{τ + 1}},

where

γ (\cdot, \cdot)

is the incomplete Gamma function. Since this expression (especially for large

μ

) can be less than one, we put in (2) the constant

l = max {1, c (α, μ)}

. This seminorm is good enough for our approach.

Finally, we will illustrate the usefulness of the result obtained, which allows us to obtain the uniqueness of the solutions of the equation and of the scheme for its calculation in conjunction with an examination of the corresponding regularity. Let us note that the operator under consideration need not be a contraction in the solution space and that, in spite of rather technical assumptions, Theorem 2 cannot be applied.

Let

0 < α < τ_{i} < 1

, (

i = 1, 2

β > max (τ_{1}, τ_{2})

ϑ (x) = x^{ρ}

with

ρ < τ_{1} - α

and

ρ < τ_{2} - α

. Assume the following:

(i): Assume that for $i = 1, 2$ , the superposition operators be such that $F_{i} : J_{α, β}^{g, ϑ} [0, 1] \to J_{α, β}^{g, ϑ} [0, 1]$ are bounded and continuous. Assume that there exist constants $c_{1}, c_{2}, N_{2}$ such that $∥ F_{2} ∥_{\infty} \leq c_{2} {∥ x ∥}_{\infty} + N_{2}$ , $∥ F_{1} ∥_{\infty} \leq c_{1} {∥ x ∥}_{\infty}$ .
(ii): For arbitrary $σ \in [0, 1]$ , assume that $ω_{ϑ}^{g} (F_{i} (x), σ) \leq {[a_{i} (σ) σ^{β + 1} + b_{i} \cdot ω_{ϑ}^{g} (x, σ)]}^{\frac{α}{β}}$ for $i = 1, 2$ , where $a_{i}$ are positive and continuous functions and $b_{i} > 0$ ( $i = 1, 2$ ).
(iii): Define the following constants:

$\begin{matrix} K_{i} & = & \frac{(2^{\frac{β}{α}} - 1) \cdot {∥ g ∥}_{\infty}^{\frac{τ_{i} \cdot β}{α}} \cdot c {(τ_{i}, μ)}^{\frac{β}{α}}}{Γ {(τ_{i} + 1)}^{\frac{β}{α}}}, \\ c (τ_{i}, μ) & = & \frac{\int_{0}^{1} {(1 - u)}^{τ_{2}} e^{- μ (1 - u)} d u}{Γ (τ_{i})} \\ L_{i} = & K_{i} \cdot 2^{\frac{β}{α} - 1} \int_{0}^{1} σ^{[- (β + 1) + \frac{β τ_{i}}{α}]} c_{i}^{\frac{β}{α}} d σ \\ M_{i} & = & K_{i} \cdot \int_{0}^{1} σ^{[- (β + 1) + \frac{β τ_{2}}{α}]} N_{i}^{\frac{β}{α}} \cdot ω_{\tilde{ϑ}}^{g} (g^{τ_{2}} \cdot e^{- μ g}, σ) d σ . \end{matrix}$

for $i = 1, 2$ . Assume that, under our assumptions, the constants $c_{1}, c_{2}, N_{2}$ are such that the above constants are positive and finite.
(iv): Define the following functions:

$I_{1} (x, τ_{1}, μ, g) = (c_{i} {∥ x ∥}_{\infty} + N_{i}) \cdot \frac{{(g (1) - g (0))}^{τ_{i}}}{Γ (τ_{i})} \cdot \frac{1}{μ^{τ_{i}}} γ (τ_{i}, μ (g (1) - g (0)))$

for $i = 1, 2$ .
Assume that for defined constants there exists $r > 0$ such that

$(1 + 2^{2 - \frac{α}{β}}) \cdot [I_{1} (r, α, μ, g) + S_{1} \cdot r + R_{1}] \cdot [I_{2} (r, α, μ, g) + S_{2} \cdot r + R_{2}] \leq r,$

where $R_{i} = max {K_{i} \cdot b_{i}, L_{i}}$ , $S_{i} = K_{i} {∥ a_{i} ∥}_{\infty} + M_{i}$ for $i = 1, 2$ .
(v): The following inequality holds:

$(C_{τ_{1}, μ} \cdot | ℑ_{0, g}^{τ_{2}, μ} | \cdot (c_{2} \cdot r) + | ℑ_{0, g}^{τ_{1}, μ} | \cdot (c_{1} \cdot r + N_{1}) \cdot C_{τ_{2}, μ}) < 1,$

where $C_{τ_{k}, μ} (t) = ℑ_{0, g}^{τ_{k}, μ} 1 (t) = \frac{1}{Γ (τ_{1})} \int_{0}^{1} {(g (t) - g (s))}^{τ_{k} - 1} e^{- μ (g (t) - g (s))} g^{'} (s) d s$ ( $k = 1, 2$ ).

Theorem 3.

Let

g \in C^{1} [0, 1]

be a positive increasing function such that

g^{'} (t) \neq 0

, for all

t \in (0, 1)

. Suppose that

0 < α < τ_{i} < 1

(i = 1, 2)

β > max (τ_{1}, τ_{2})

ϑ (x) = x^{ρ}

. If assumptions (i)–(v) are satisfied, then the equation (6) has a unique solution

z \in J_{α, β}^{g, ϑ} [0, 1]

, which can be obtained as a limit of the sequence of iterations

x_{n + 1} = T (x_{n})

, for any

x_{1} \in J_{α, β}^{g, ϑ} [0, 1]

Proof.

The proof can be divided into steps according to the testability of the assumptions of the Theorem 1. Let us rewrite the quadratic integral equation (6) in the operator form:

x = H (x) = P (x) \cdot Q (x)

with

P (x) = [ℑ_{0, g}^{τ_{1}, μ} \circ F_{1}] (x) a n d Q (x) (t) = [ℑ_{0, g}^{τ_{2}, μ} \circ F_{2}] (x) .

From Assumption (ii), it follows that the nonlinear superposition operators

F_{i}

(

i = 1, 2

) are bounded and continuous from

J_{α, β}^{g, ϑ} [0, 1]

into itself. Then, due to Lemma 8, the operators

ℑ_{0, g}^{τ_{1}, μ}

and

ℑ_{0, g}^{τ_{2}, μ}

are well defined and bounded from

J_{α, β}^{g, ϑ} [0, 1]

into

C [0, 1]

Lemma 9 implies that P and Q are bounded as acting from

J_{α, β}^{g, ϑ} [0, 1]

into itself. Continuity of

ℑ_{0, g}^{τ_{i}, μ}

(

i = 1, 2

), due to its linearity, is equivalent to boundedness, so we have composition of continuous operators, and, finally, operators P and Q map continuously

J_{α, β}^{g, ϑ} [0, 1]

into

C [0, 1]

into itself. Since in Proposition 5 we proved that this space is a Banach algebra, we conclude that H has the same property.

We are looking for an invariant ball (compact in a target space, cf. Theorem 1). To accomplish this, we consider a subspace of the same type, for which balls are compact in

J_{α, β}^{g, ϑ} [0, 1])

. By Lemma 7, it is sufficient to consider

ϑ (t) = t^{ρ}

\tilde{ϑ} (t) = t^{ζ}

with

ρ < ζ < τ_{k} - α

(

k = 1, 2

). The balls will be considered in this space

J_{α, β}^{g, \tilde{ϑ}} [0, 1]

, so they will be compact as subsets of

J_{α, β}^{g, ϑ} [0, 1]

(as expected in Theorem 1). Since the operator H is bounded, when restricted to the ball

B_{r} (J_{α, β}^{g, \tilde{ϑ}} [0, 1]) = {x \in J_{α, β}^{g, \tilde{ϑ}} [0, 1] : ∥ | x | ∥_{α, β}^{g, \tilde{ϑ}} \leq r}

(

r > 0

), we obtain

∥ | H (B_{r} (J_{α, β}^{g, \tilde{ϑ}} [0, 1])) | ∥_{α, β}^{g, \tilde{ϑ}} \leq R

, for some

R > 0

Due to the goals of this paper and the illustrative nature of the theorem on the uniqueness of solutions of a quadratic fractional integral equation to be proved and the very technical, although standard, proof of this part of the theorem, we will limit the details of the proof. The chosen generalized fractional order operators require technical changes in the calculation (see [35]). We will give a detailed justification for one of the operators and then show the existence of the radius of such a ball, without determining it precisely. This will allow us to concentrate on the main problem of the theorem while keeping the proof complete. Let us examine the operator Q.

Let

x \in J_{α, β}^{g, \tilde{ϑ}} [0, 1]

. Then

\begin{matrix} ∥ | Q (x) | ∥_{α, β}^{g, \tilde{ϑ}} & = & ∥ | ℑ_{0, g}^{τ_{2}, μ} \circ F_{2} (x) | ∥_{α, β}^{g, \tilde{ϑ}} \\ = & ∥ ℑ_{0, g}^{τ_{2}, μ} \circ F_{2} (x) ∥_{\infty} + j_{α, β}^{g, \tilde{ϑ}} {(ℑ_{0, g}^{τ_{2}, μ} \circ F_{2}, [0, 1])}^{\frac{α}{β}}, \end{matrix}

(7)

and

\begin{matrix} ∥ ℑ_{0, g}^{τ_{2}, μ} \circ F_{2} (x) ∥_{\infty} \leq ∥ F_{2} (x) ∥_{\infty} \cdot sup_{t \in [0, 1]} | \frac{1}{Γ (τ_{2})} \int_{0}^{t} {(g (t) - g (s))}^{τ_{2} - 1} e^{- μ (g (t) - g (s))} g^{'} (s) d s | \\ \leq & (c_{2} {∥ x ∥}_{\infty} + N_{2}) \cdot \frac{{(g (1) - g (0))}^{τ_{2}}}{Γ (τ_{2})} \cdot \frac{1}{μ^{τ_{2}}} γ (τ_{2}, μ (g (1) - g (0))) = : I_{2} (x, τ_{2}, μ, g), \end{matrix}

(8)

where

γ (\cdot, \cdot)

denotes the incomplete Gamma function. Moreover, similarly, as in [35], by applying our assumptions together with the Minkowski inequality, we can now estimate the second term of the norm

\begin{matrix} j_{α, β}^{g, \tilde{ϑ}} (ℑ_{0, g}^{τ_{2}, μ} \circ F_{2} (x), [0, 1]) = \int_{0}^{1} σ^{- (β + 1)} ω_{\tilde{ϑ}}^{g} {(ℑ_{0, g}^{τ_{2}, μ} \circ F_{2} (x), σ)}^{\frac{β}{α}} d σ \\ \leq \int_{0}^{1} σ^{- (β + 1)} \cdot c {(α, μ)}^{\frac{β}{α}} \cdot {[{∥ g ∥}_{\infty}^{τ_{2}} \cdot ω_{\tilde{ϑ}}^{g} (F_{2} (x), σ) + {∥ F_{2} (x) ∥}_{\infty} \cdot ω_{\tilde{ϑ}}^{g} (g_{2}^{τ} \cdot e^{- μ g}, σ)]}^{\frac{β}{α}} d σ \\ \leq K_{2} \cdot \int_{0}^{1} σ^{- (β + 1)} \cdot [ω_{\tilde{ϑ}}^{g} {(F_{2} (x), σ)}^{\frac{β}{α}} + (c_{2} {∥ x ∥}_{\infty} + N_{2})^{\frac{β}{α}} \cdot ω_{\tilde{ϑ}}^{g} {(g^{τ_{2}} \cdot e^{- μ g}, σ)}^{\frac{β}{α}}] d σ \\ \leq K_{2} \cdot \int_{0}^{1} σ^{- (β + 1)} [ω_{\tilde{ϑ}}^{g} {(F_{2} (x), σ)}^{\frac{β}{α}} + σ^{\frac{β τ_{2}}{α}} (c_{2} {∥ x ∥}_{\infty} + N_{2})^{\frac{β}{α}} \cdot ω_{\tilde{ϑ}}^{g} (g^{τ_{2}} \cdot e^{- μ g}, σ)] d σ \\ \leq K_{2} \cdot [\int_{0}^{1} σ^{- (β + 1)} ω_{\tilde{ϑ}}^{g} {(F_{2} (x), σ)}^{\frac{β}{α}} d σ \\ + \int_{0}^{1} σ^{- (β + 1)} σ^{\frac{β τ_{2}}{α}} (c_{2} {∥ x ∥}_{\infty} + N_{2})^{\frac{β}{α}} \cdot ω_{\tilde{ϑ}}^{g} (g^{τ_{2}} \cdot e^{- μ g}, σ) d σ] \\ \leq K_{2} \cdot [\int_{0}^{1} (a_{2} (σ) + b_{2} \cdot σ^{- (β + 1)} \cdot ω_{\tilde{ϑ}}^{g} {(x, σ)}^{\frac{β}{α}}) d σ \\ + 2^{\frac{β}{α} - 1} \int_{0}^{1} σ^{[- (β + 1) + \frac{β τ_{2}}{α}]} (c_{2}^{\frac{β}{α}} {∥ x ∥}_{\infty}^{\frac{β}{α}} + N_{2}^{\frac{β}{α}} \cdot ω_{\tilde{ϑ}}^{g} (g^{τ_{2}} \cdot e^{- μ g}, σ)) d σ], \end{matrix}

(9)

where

K_{2} = \frac{(2^{\frac{β}{α}} - 1) \cdot {∥ g ∥}_{\infty}^{\frac{τ_{2} \cdot β}{α}} \cdot c {(τ_{2}, μ)}^{\frac{β}{α}}}{Γ {(τ_{2} + 1)}^{\frac{β}{α}}}, c (τ_{2}, μ) = \frac{\int_{0}^{1} {(1 - u)}^{τ_{2}} e^{- μ (1 - u)} d u}{Γ (τ_{2})} .

Now, let us estimate

ω_{\tilde{ϑ}}^{g} {(g_{2}^{τ} e^{- μ g}, σ)}^{\frac{β}{α}}

. Fix arbitrary

σ \in (0, 1]

. For

| t - s | < σ

we have

\frac{{| (g (t))}^{τ_{2}} \cdot e^{- μ g (t)} - {(g (s))}^{τ_{2}} \cdot e^{- μ g (s)} |}{(| g (t) - {g (s) |)}^{ζ}} \leq \frac{{| g (t)}^{τ_{2}} - g {(s)}^{τ_{2}} |}{(| g (t) - {g (s) |)}^{ζ}} \leq {| g (t) - g (s) |}^{τ_{2} - ζ} .

Then,

ω_{\tilde{ϑ}}^{g} {(g^{τ_{2}} \cdot e^{- μ g}, σ)}^{\frac{β}{α}} \leq σ^{\frac{(τ_{2} - ζ) \cdot α}{β}} .

Since

- (β + 1) + \frac{(τ_{2} - ζ) \cdot α}{β} < 1

, the integral

\int_{0}^{1} σ^{[- (β + 1)]} \cdot σ^{\frac{(τ_{2} - ζ) \cdot α}{β}} d σ

in the estimation (9) is convergent. Therefore,

\begin{matrix} {(j_{α, β}^{g, ϑ} (ℑ_{0, g}^{τ_{2}, μ} x, [0, 1]))}^{\frac{α}{β}} & \leq & c_{2} (τ_{2}, μ) \cdot {∥ g ∥}_{\infty}^{\frac{τ_{2} β}{α}} \cdot {(j_{α, β}^{g, ϑ} (x, [0, 1])))}^{\frac{α}{β}} \\ + {∥ x ∥}_{\infty} \cdot \int_{0}^{1} σ^{- 1 - β + \frac{(τ_{2} - ζ) \cdot β}{α}} d σ . \end{matrix}

Thus,

j_{α, β}^{g, ϑ} {(ℑ_{0, g}^{τ_{2}, μ} \circ F_{2} (x), [0, 1])}^{\frac{α}{β}} \leq K_{2} \cdot ∥ a_{2} ∥_{\infty} + K_{2} \cdot b_{2} \cdot {(j_{α, β}^{g, ϑ} (x, [0, 1]))}^{\frac{α}{β}} + {∥ x ∥}_{\infty} \cdot L_{2} + M_{2},

with

L_{2} = K_{2} \cdot 2^{\frac{β}{α} - 1} \int_{0}^{1} σ^{[- (β + 1) + \frac{β τ_{2}}{α}]} c_{2}^{\frac{β}{α}} d σ

and

M_{2} = K_{2} \cdot \int_{0}^{1} σ^{[- (β + 1) + \frac{β τ_{2}}{α}]} N_{2}^{\frac{β}{α}} \cdot ω_{\tilde{ϑ}}^{g} (g^{τ_{2}} \cdot e^{- μ g}, σ) d σ .

The above inequality therefore gives an inequality for the estimation of the norm, of the form

j_{α, β}^{g, \tilde{ϑ}} {(ℑ_{0, g}^{τ_{2}, μ} \circ F_{2} (x), [0, 1])}^{\frac{α}{β}} \leq S_{2} \cdot ∥ | x | ∥_{α, β}^{g, \tilde{ϑ}} + R_{2},

for some constants calculated on the basis of the above estimation. In order to focus the attention of the reader on the essential steps of the proof, we will take the liberty of leaving the calculations to those who are further interested. By substituting from (8) and (9) to (7), we have

∥ | Q (x) | ∥_{α, β}^{g, \tilde{ϑ}} \leq I_{2} (x, α, μ, g) + S_{2} \cdot ∥ | x | ∥_{α, β}^{g, \tilde{ϑ}} + R_{2} .

(10)

Similarly, we can show that

P (x)

is bounded on

[0, 1]

with

\begin{matrix} ∥ ℑ_{0, g}^{τ_{1}, μ} \circ F_{1} (x) ∥_{\infty} & \leq & ∥ F_{1} (x) ∥_{\infty} \cdot sup_{t \in [0, 1]} | \frac{1}{Γ (τ_{1})} \int_{0}^{t} {(g (t) - g (s))}^{τ_{1} - 1} e^{- μ (g (t) - g (s))} g^{'} (s) d s | \\ \leq & (c_{1} {∥ x ∥}_{\infty} + N_{1}) \cdot sup_{t \in [0, 1]} \frac{{(g (t) - g (0))}^{τ_{1}}}{Γ (τ_{1})} γ (τ_{1}, μ (g (t) - g (0))) \\ \leq & (c_{1} {∥ x ∥}_{\infty} + N_{1}) \cdot \frac{{(g (1) - g (0))}^{τ_{1}}}{Γ (τ_{1})} \cdot \frac{1}{μ^{τ_{1}}} γ (τ_{1}, μ (g (1) - g (0))) \\ = : I_{1} (x, τ_{1}, μ, g), \end{matrix}

(11)

and by taking similar calculation as for the operator Q, we obtain

∥ | P (x) | ∥_{α, β}^{g, \tilde{ϑ}} \leq I_{1} (x, τ_{1}, μ, g) + S_{1} \cdot ∥ | x | ∥_{α, β}^{g, \tilde{ϑ}} + R_{1}

(12)

for constants, again calculated on the basis of the above estimation. This means that the operator P is bounded from the Banach algebra

J_{α, β}^{g, ϑ} [0, 1]

into itself.

Now, with the help of the algebra property, we have that

∥ | H (x) | ∥_{α, β}^{g, ϑ} \leq (1 + 2^{2 - \frac{α}{β}}) \cdot ∥ | P (x) | ∥_{α, β}^{g, \tilde{ϑ}} \cdot ∥ | Q (x) | ∥_{α, β}^{g, \tilde{ϑ}} .

Moreover, by virtue of (10) and (12), the above inequality implies the following estimates:

∥ | P (B_{r}) | ∥_{α, β}^{g, \tilde{ϑ}} \leq I_{1} (r, τ_{1}, μ, g) + S_{1} \cdot r + R_{1}

∥ | Q (B_{r}) | ∥_{α, β}^{g, \tilde{ϑ}} \leq I_{1} (r, τ_{1}, μ, g) + S_{2} \cdot r + R_{2} .

For

x \in B_{r} (J_{α, β}^{g, \tilde{ϑ}} [0, 1])

, we obtain

\begin{matrix} ∥ | H (x) | ∥_{α, β}^{g, \tilde{ϑ}} & \leq & (1 + 2^{2 - \frac{α}{β}}) \cdot (I_{1} (x, τ_{1}, μ, g) + N_{1} \cdot ∥ | x | ∥_{α, β}^{g, \tilde{ϑ}} + R_{1}) \\ \cdot (I_{2} (x, τ_{2}, μ, g) + N_{2} \cdot ∥ | x | ∥_{α, β}^{g, \tilde{ϑ}} + R_{2}) \\ \leq & (1 + 2^{2 - \frac{α}{β}}) \cdot (I_{1} (x, τ_{1}, μ, g) + N_{1} \cdot r + R_{1}) \\ \cdot (I_{2} (x, τ_{2}, μ, g) + N_{2} \cdot r + R_{2}) \leq r . \end{matrix}

Then, due to our assumptions, we infer that

H : B_{r} (J_{α, β}^{g, \tilde{ϑ}} [0, 1]) \to B_{r} (J_{α, β}^{g, \tilde{ϑ}} [0, 1])

and is continuous. Recall that the seminorm satisfies the condition (2). Due to Lemma 7, it suffices to check that P and Q are Lipschitz-continuous with respect to the supremum norm (with some constants

d_{1}, d_{2}

, respectively). Thus, for

x, y \in B_{r} (J_{α, β}^{g, \tilde{ϑ}} [0, 1])

we can repeat our argumentation from Lemma 2 to obtain the Lipschitz condition for H:

\begin{matrix} {∥ H (x) - H (y) ∥}_{\infty} & \leq & ∥ [ℑ_{0, g}^{τ_{1}, μ} \circ F_{1}] (x)] \cdot [ℑ_{0, g}^{τ_{2}, μ} \circ F_{2}] (x) \\ - [ℑ_{0, g}^{τ_{1}, μ} \circ F_{1}] (y) {] \cdot [ℑ_{0, g}^{τ_{2}, μ} \circ F_{2}] (y) ∥}_{\infty} \\ \leq & {∥ y - x ∥}_{\infty} (C_{τ_{1}, μ} \cdot | ℑ_{0, g}^{τ_{2}, μ} | \cdot (c_{2} \cdot r) + | ℑ_{0, g}^{τ_{1}, μ} | \cdot (c_{1} \cdot r + N_{1}) \cdot C_{τ_{2}, μ}), \end{matrix}

where

C_{τ_{k}, μ} (t) = ℑ_{0, g}^{τ_{k}, μ} 1 (t) = \frac{1}{Γ (τ_{1})} \int_{0}^{1} {(g (t) - g (s))}^{τ_{k} - 1} e^{- μ (g (t) - g (s))} g^{'} (s) d s

(

k = 1, 2

Combining all the above properties, we see, that H satisfies all the assumptions of Theorem 1. This completes the proof. □

6. Conclusions

In this paper, we proved a variant of the Banach fixed-point theorem for operators which are not necessarily contractions on subspaces of the space of continuous functions

C [a, b]

with norms stronger than the supremum norm. Then the uniqueness of fixed points is obtained under very mild conditions which are not restrictive in applications. For many differential or integral problems, it is important to prove not only the existence of their solutions, but also their uniqueness and the method of finding them. The classical method, based on the Banach contraction theorem and the iterative formula

x_{n + 1} = T (x_{n})

, is the basic method. Of course, not only contractions have unique fixed points.

Our results allow us to keep the uniqueness of the solutions, as well as the iterative method of their construction, but at the same time to investigate the regularity of the solutions (generalized Hölder continuity instead of just continuity).

The results of this paper are illustrated by proving the singularity of the solutions of a quadratic integral equation with generalized fractional order operators, for which the regularity of the solutions is well known as satisfying a certain Hölder condition. It is noteworthy that this is the first result on the singularity of solutions without assuming an operator contraction condition. For this purpose, a suitable function space (generalized Hölder space) is constructed and it is proved that it is a Banach algebra.

Author Contributions

All authors contributed equally to obtaining the results and writing the paper. All authors have read and approved of the final manuscript.

Funding

The author Maciej Ciesielski was funded by the Poznan University of Technology under Grant no. 0213/SBAD/0119.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

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Cichoń, K.; Cichoń, M.; Ciesielski, M. On the Problem of the Uniqueness of Fixed Points and Solutions for Quadratic Fractional-Integral Equations on Banach Algebras. Symmetry 2024, 16, 1535. https://doi.org/10.3390/sym16111535

AMA Style

Cichoń K, Cichoń M, Ciesielski M. On the Problem of the Uniqueness of Fixed Points and Solutions for Quadratic Fractional-Integral Equations on Banach Algebras. Symmetry. 2024; 16(11):1535. https://doi.org/10.3390/sym16111535

Chicago/Turabian Style

Cichoń, Kinga, Mieczysław Cichoń, and Maciej Ciesielski. 2024. "On the Problem of the Uniqueness of Fixed Points and Solutions for Quadratic Fractional-Integral Equations on Banach Algebras" Symmetry 16, no. 11: 1535. https://doi.org/10.3390/sym16111535

APA Style

Cichoń, K., Cichoń, M., & Ciesielski, M. (2024). On the Problem of the Uniqueness of Fixed Points and Solutions for Quadratic Fractional-Integral Equations on Banach Algebras. Symmetry, 16(11), 1535. https://doi.org/10.3390/sym16111535

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On the Problem of the Uniqueness of Fixed Points and Solutions for Quadratic Fractional-Integral Equations on Banach Algebras

Abstract

1. Introduction

2. Preliminaries

3. Uniqueness, Iterative Methods, and Fixed Point Theorems

4. Contractive Product Operators

5. Quadratic Problem Applications

5.1. Fractional Operators

5.2. Generalized Hölder Spaces

5.3. Quadratic Fractional Equations

6. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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