On the Equivalence between Differential and Integral Forms of Caputo-Type Fractional Problems on Hölder Spaces

In this paper, we investigate an absolutely fundamental problem arising from the application of the operator method (integral operators) to fractional-order differential problems, namely, the problem of equivalence of differential and integral problems. This problem depends not only on the problem under consideration or on the definition of the solution adopted, but also on the assumptions made about the functions and, therefore, especially about the function spaces in which integral operators operate. This problem has been extensively discussed and commented on in [1].

We will begin by introducing the function spaces on which we will study fractional-order integral operators. We will then discuss a very large class of such operators, examine their properties, and consider the equivalence problem of differential and integral problems, in which the very classes of operators discussed will be useful. We are particularly interested in problems involving operators that depend on two orders of the derivatives. These problems are analogous to those for integral equations involving multiple derivatives of unknown functions of different orders, but in this case, we have a much more complicated problem because successive differentiations and integrations are not the same operations. Of particular importance, then, is the study of how to formulate differential problems in equivalent integral form. When we study fractional differential problems, the key point is to find their equivalent integral forms.

This is an absolutely essential and fundamental step in the study of differential equations of fractional order. Surprisingly, it is sometimes overlooked.

The results of Hardy and Littlewood [2] are the origin of such studies for the Riemann–Liouville operator, and then this research continued in subspaces of the space of continuous functions (cf. [1]). Already the first results of this kind, based on the paper [2], showed clearly that the natural domains of the operators are Hölder spaces (cf. also [1,3,4,5]). This is because the values of fractional operators always lie in Hölder spaces. For instance, it is known that for $1/p<\alpha <1+1/p$ or $p=1$ and $1\le \alpha <2$, the fractional Riemann–Liouville integral operator $I^{\alpha }$ is bounded from $L^p[a,b]$ into ${\mathbb{H}}^{\alpha -\frac{1}{p}}[a,b]$; hence, for $u\in L^p$, $I^{\alpha }u$ is Hölder continuous with exponent $\alpha -1/p$, thus $I^{\alpha }u$ is continuous.

However, this is not sufficient to study the equivalence problem, since outside the space of absolutely continuous functions, differential and integral problems are not necessarily equivalent. We need to construct Hölder spaces suitable for the problem under consideration and study the properties of integral and differential operators on these spaces. Recall that Hölder spaces are fundamental to studying singular integral operators at all.

The equivalence problem of fractional-order differential and integral problems is much more complicated for operators depending on two different orders (fractional). Such operators, apart from achieving greater generality of considerations, have many natural applications and are, therefore, worth studying. Thus, in this paper, we will deal with the Langevin and Bagley–Torvik equations. Note, that the results for Langevin-type equations are particularly interesting because of the need to study two different orders of fractional derivatives [6].

The study of equivalence problems in this paper will, therefore, focus on the equations of the Langevin boundary-value problems with Caputo-type fractional derivatives [3,6]: for $t\in [a,b]$, $\lambda \in \mathbb{R}$, $f\in C[a,b]\times \mathbb{R}$ and $\beta ,\alpha \in (0,1),\beta +\alpha <1$, combined with an appropriate initial or boundary condition. It is worth noting that such questions have been studied in $C[a,b]$, an excessively large function space. We do not impose any ordering between $\alpha$ and $\beta$ (cf. [7,8,9] and references therein). We study the problems for several classes of tempered fractional-order derivatives obtaining a number of new results and each time establishing Hölder spaces suitable for the equivalence of differential and integral problems. The fractional Langevin equation was used for modeling of single-file diffusion and for a free particle driven by a power law type of noise. Also, the transformation of the Fokker–Planck equation, which corresponds to the Langevin equation with multiplicative white noise, into the Wiener process is made available for any prescription (cf. [10]). Interesting applications of this equation are discussed in detail in [11].

This problem is also related to the Bagley–Torvik equation (with derivatives of order 2 and $3/2$, respectively). This equation can be used to model the motion of a rigid plate immersed in a Newtonian fluid and connected to a fixed point by a massless spring. Moreover, in [12], they also showed applications of this fractional equation to the theory of viscoelasticity. At the same time, it is worth mentioning that the fractional Langevin equation has been used to discuss Brownian motion and anomalous diffusion, which is useful in the study of generalized elastic models and in protein dynamics (see [3] and references therein).

A more complete description of the two types of problems and their interrelationships can be found in [3]. It is worth noting that in the classical fractional-order problem, we have $\alpha +\beta <1$, and so far, only the operators that are not weakly singular (e.g., for $\alpha =3$ and $\beta =3/2$) have been studied. We will cover this case. A study of such problems with a purely mathematical motivation can be found in [13,14], for instance.

The main goal of this paper is to establish equivalence between appropriately defined solutions of a fractional differential boundary value problem and solutions of the corresponding integral equation. This type of equivalence is well known for functions from the space $AC[a,b]$ [3]. However, it is known that solutions of integral equations are Hölder continuous, so this space does not seem to be the best choice. With the results obtained, the paper was able not only to investigate the existence of continuous solutions of the Langevin equation (see [5,15]), but also to demonstrate their existence in Hölder spaces. In this paper, we will identify corresponding differential and integral problems with tempered fractional-order derivatives and study the operators generating such equations on Hölder-type spaces. Due to the completely different norms and their properties, this requires a separate and deeper analysis. In the cases of interest, we will show the equivalence of differential and integral problems by bridging the gap in the current studies.

As an example of the application of the results obtained, the paper will be complemented by results on inhomogeneous Langevin-type problems.

We begin by introducing the function spaces on which we will study fractional-order integral operators. Fractional integrals can improve local properties of functions, but fractional derivatives can have the opposite effect. Therefore, it is necessary to study the mapping properties of fractional integrals in different function spaces. As we have already pointed out, Hölder continuity of solutions is strongly related to fractional-order problems (for example, [1,2,4,5]). This type of space is, therefore, a natural candidate for equivalence studies of differential and integral problems of fractional order. We should start with the Hölder-type spaces. For the convenience of the reader, we will recall the basic concepts and facts.

For $\gamma \in (0,1)$, the Hölder space ${\mathcal{H}}^{\gamma }[a,b]$ consists of all functions $f:[a,b]\to \mathbb{R}$ such that for $t,s\in [a,b]$, there exists a constant $M>0$ (independent on $t,s$) for which $|f\left(t\right)-f\left(s\right)|\le {M|t-s|}^{\gamma }$. In addition, if $f\left(a\right)=0$, we consider the so-called little Hölder spaces and we write $f\in {\mathcal{H}}_0^{\gamma }[a,b]$. Obviously, $f\in {\mathcal{H}}^{\gamma }[a,b]$ if the following seminorm is finite. Also, when ${\mathcal{H}}^{\gamma }[a,b]$ equipped with the norm ${∥f∥}_{\gamma }:={∥f∥}_{\infty }+{\left[f\right]}_{\gamma }$, it becomes a Banach space (cf. [4,5,15,16]).

It is important to note that there is a relationship between the Hölder space and nowhere differentiable functions. There are functions that are Hölder continuous but not absolutely continuous, such as the Weierstrass function, and functions that are absolutely continuous but not Hölder continuous. For example, the function $f:[0,1]\to \mathbb{R}$ defined by ${\displaystyle f\left(t\right):=t^{\alpha },0<\alpha <\gamma <1,}$ is absolutely continuous on $[0,1]$, but $f\notin {\mathcal{H}}^{\gamma }[0,1]$. However, for $0<{\gamma }_1<{\gamma }_2<1$, it is not difficult to show that

Obviously all embeddings are strict, e.g., the function $f\left(x\right)=1/\log (x-2)$ for $x\in (0,1]$ and $f\left(0\right)=0$ is continuous, but not in any Hölder space of exponent $\alpha >0$. It is worth noting at this point that the embeddings between Hölder spaces and their embeddings in $C[a,b]$ are compact (as a consequence of the Arzelá–Ascoli theorem), and so using compactness when studying operators acting between them will be a natural method, where otherwise either the norm contraction property or additional assumptions about the functions under study are needed. The embeddings of different Hölder spaces are not dense.

There are various modifications and generalizations of classical fractional integration operators that are widely used in both theory and applications. The following definition unifies various fractional integrals for integrable functions, allowing for the solution of initial and/or boundary value problems with different types of fractional integrals and derivatives to be solved in a unified way.

For the convenience of the reader, we will now recall all the definitions related to the class of integrals and fractional derivatives of interest. Let $\psi \in C^1[a,b]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Throughout the paper, we will make these assumptions about the function $\psi$.

For completeness, we define ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(a\right):=0$. Let us recall that

If ${\mathcal{E}}_{\alpha ,\beta }(·)$ denotes the Mittag–Leffler function (see [19], then see [20])

We can now define the concept of fractional differentiation [17,21] and list the properties needed to prove the results (see also [22,23,24]).

Let us recall that

It can easily be seen (see, e.g., [25]) that

In this paper, we do not study classical Caputo, as it is not an inverse operator of the generalized integral operators under study, and its definition—although modeled on the Caputo idea—is nevertheless different.

The above definition of the derivative is very general and includes many other fractional integral operators which will be considered later. In order not to stray too far from the purpose of the paper, we will limit ourselves to citing the work of [5,15,22], where the interested reader will find a discussion of the special cases covered by our definition.

Let us describe now the acting condition for little Hölder spaces.

Also, according to ([5], Lemma 3), there exists a constant $D_A>0$ such that whenever $\alpha +\zeta <1$.

Recall that for any $f\in C[a,b]$, we have

However, we have

Interestingly,

Let us now describe problems that take advantage of defining equivalent problems. We now turn to the Caputo-type fractional Langevin differential equation. In this section, we will consider the linear case. It is clear that our focus is on a purely fractional problem, i.e., a situation where the sum of the two fractional parameters, $\alpha$ and $\beta$, is less than one. with $t\in [a,b]$, $\lambda \in \mathbb{R}$, $f\in C[a,b]$, $\beta ,\alpha \in (0,1),\beta +\alpha <1$, combined with an appropriate initial or boundary condition.

As a direct consequence of our previous discussion of tempered derivatives of fractional order, it follows that we will consider the following integral form:

We will first study the formal form of the integral problem that is equivalent to the one we are studying. We will then verify the equivalence of these two problems in the spaces studied above. It is important to note that this equivalence depends strongly on the domain of the operators considered. Therefore, for some constant $C_1$. Similarly,

Then, the formal integral form (of any solution) for (2) has the following form: for $t\in [a,b]$ and $C_1\in \mathbb{R}$.

According to ([15], Lemma 5) and in view of Lemma 1, we see that if $f\in {\mathcal{H}}_0^{\zeta }[a,b]$ with $\zeta +2\alpha +\beta <1$, then (cf. Lemma 1) ${\mathbb{I}}_{\psi }^{\alpha +\beta ,\mu }f\in {\mathcal{H}}_0^{\zeta +\alpha +\beta }[a,b]$ and (3) admits a Hölder continuous solution $x\in {\mathcal{H}}_0^{\zeta +\alpha +\beta }[a,b]$.

It is unfortunate that, as shown in [26], the relationship between the left inverse of the function $d_{\psi }^{\alpha ,\mu }/d{t}^{\alpha }$ and ${\mathbb{I}}_{\psi }^{\alpha ,\mu }$ is not as straightforward as one might expect, if not in the space of absolutely continuous functions, then in Hölder spaces.

Therefore, the existence of a Hölder continuous solution to (3) does not guarantee the existence of solutions to (2). Many authors avoided the problem of showing the equivalence between the Caputo-type fractional differential problems and the corresponding integral forms by defining so-called mild-type solutions. Recall, that this type of solution is exactly defined to be a solution to an integral form of a Caputo-type fractional differential problem. Also, some authors avoided the problem of showing the equivalence by redefining the Caputo fractional derivative as for $\alpha \in (0,1)$. Note that the definition (4) of the Caputo fractional derivative coincides with the standard definition (3) on the space of absolutely continuous functions.

Considering spaces other than $AC[a,b]$ leads to mistakes about the equivalence of problems. We will explain this.

In what follows, we have avoided the problem by showing the equivalence in a different way, using the following definition:

Since the integrals improve the continuity properties of functions, similarly as in Lemma 1, we can prove the following lemma for little Hölder spaces ${\mathcal{H}}_0$.