Open AccessArticle

Inexact Iterates of Nonexpansive Mappings with Summable Errors in Metric Spaces with Graphs

Simeon Reich

and

Alexander J. Zaslavski

Department of Mathematics, The Technion—Israel Institute of Technology, Haifa 32000, Israel

Authors to whom correspondence should be addressed.

Symmetry 2023, 15(10), 1927; https://doi.org/10.3390/sym15101927

Submission received: 29 August 2023 / Revised: 25 September 2023 / Accepted: 30 September 2023 / Published: 17 October 2023

(This article belongs to the Special Issue Symmetry and Graph Theory)

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Abstract

In our joint work with Dan Butnariu (2006) we established the stability of the convergence of iterates of a nonexpansive mapping on a complete metric space in the presence of summable computational errors. In a recent paper of ours, we extended this result to inexact iterates of nonexpansive mappings on complete metric spaces with graphs under a certain assumption on the iterates. In the present paper we obtain an analogous result by removing that assumption on the iterates and replacing it with an additional assumption on the graph.

Keywords:

complete metric space; fixed point; graph; inexact iterate; nonexpansive mapping

MSC:

47H09; 47H10; 54E50

1. Introduction

During the last sixty years, many interesting developments have taken place in the fixed point theory of nonlinear mappings ([1,2,3,4,5,6,7,8,9,10,11,12,13]). The origin of these investigations is Banach’s classical work [14]. Since Banach’s celebrated theorem, many important and interesting results have been obtained in this field, including results on common fixed point problems, feasibility, iterative projection algorithms, and variational inequalities. In addition to their own importance, these results have numerous useful applications in various areas of research [13,15,16,17,18].

In [2], the authors considered a nonexpansive mapping acting on a complete metric space under the assumption that every sequence of its iterates converges to a fixed point. It was shown there that every sequence of inexact iterates of such a mapping with summable errors converges to a fixed point as well. This result is a generalization of a classical result of Ostrowski [8], which was obtained for strict contractions. In [19], we established an analog of this result for nonexpansive mappings acting on complete metric spaces with graphs under a certain assumption on the iterates. This result is an extension of an analogous theorem which was proved in [20] for strict contractions. In this connection, note that the investigation of mappings in metric spaces with graphs is now of great research interest [5,9,10,21,22,23,24]. In the present work, we obtain an analog of the main result of [19] by removing the assumption on the iterates and replacing it with an additional assumption on the graph.

At this point, it is worth recalling that the study of the behavior of inexact iterates is very important, as computational errors are always produced in calculations. Therefore, this has been and continues to be an important topic in analysis, beginning with the seminal paper [8], though see [2,13] as well.

Let

(X, ρ)

be a complete metric space and let G be a graph. We assume that the set of vertices

V (G)

of G is contained in the space X, its set of edges

E (G)

is a closed subset of the space

X \times X

endowed with the product metric, and that the following assumption holds.

(A1) For each pair of points

u, v \in X

, if

(u, v) \in E (G)

, then

(T (u), T (v)) \in E (G)

and the inequality

ρ (T (u), T (v)) \leq ρ (u, v)

is true.

Assumption (A1) appears in [21] (see [22,23]).

For each

ξ \in X

and every

E \subset X

, put

ρ (ξ, E) : = inf {ρ (ξ, η) : η \in E} .

For every

ξ \in X

and every

Δ > 0

, set

B (ξ, Δ) : = {η \in X : ρ (ξ, η) \leq Δ} .

For each map

G : X \to X

, let

G^{0} η = η

for all

η \in X

In [19], we established the following result.

Theorem 1.

Assume that a sequence

{x_{i}}_{i = 0}^{\infty} \subset X

satisfies

(T (x_{i}), x_{i + 1}) \in E (G), i = 0, 1, \dots

and

\sum_{i = 0}^{\infty} ρ (T (x_{i}), x_{i + 1}) < \infty,

and that a subsequence

{x_{i_{k}}}_{k = 1}^{\infty}

is given. Then, the following assertions hold.

1. Assume that for each integer k, the sequence

{T^{j} (x_{i_{k}})}_{j = 1}^{\infty}

converges. Then, there exists

x_{*} = lim_{i \to \infty} x_{i},

(x_{*}, T (x_{*})) \in E (G)

, and if T is continuous at the point

x_{*}

, then

T (x_{*}) = x_{*}

2. Assume that there exists a nonempty set F such that for each integer

k \geq 1

lim_{j \to \infty} ρ (T^{j} (x_{i_{k}}), F) = 0 .

Then,

lim_{i \to \infty} ρ (x_{i}, F) = 0 .

3. Assume that for each integer

k \geq 1

there exists a nonempty compact set

E_{k} \subset X

such that

lim_{j \to \infty} ρ (T^{j} (x_{i_{k}}), E_{k}) = 0 .

Then, there exists a nonempty compact set

E \subset X

such that

{lim}_{i \to \infty} ρ (x_{i}, E) = 0 .

In our work, we obtain an extension of this result without assuming that

(T (x_{i}), x_{i + 1}) \in E (G), i = 0, 1, \dots .

Instead, we assume that a certain property of the graph is satisfied (see (A2) in the next section).

2. The Main Result

We use all the definitions and notations from Section 1 and assume that all the assumptions introduced in Section 1 hold. We then prove the following result.

Theorem 2.

Assume that a sequence

{x_{i}}_{i = 0}^{\infty} \subset X

satisfies

\sum_{i = 0}^{\infty} ρ (T (x_{i}), x_{i + 1}) < \infty,

(1)

c_{0} \geq 1

r_{0} \in (0, 1]

and that the following assumption holds.

(A2) for each integer

i \geq 0

and each point

z \in B (x_{i}, r_{0})

, there exists

ξ \in B (x_{i}, c_{0} ρ (z, x_{i}))

such that

(x_{i}, ξ), (z, ξ) \in E (G) .

Let a subsequence

{x_{i_{k}}}_{k = 1}^{\infty}

be given; then, the following assertions hold true:

1. Assume that the sequence

{T^{j} (x_{i_{k}})}_{j = 1}^{\infty}

converges for each integer k. Then, there exists

x_{*} = lim_{i \to \infty} x_{i},

(x_{*}, T (x_{*})) \in E (G)

, and if T is continuous at the point

x_{*}

, then

T (x_{*}) = x_{*}

2. Assume that there exists a nonempty set F such that, for each integer

k \geq 1

lim_{j \to \infty} ρ (T^{j} (x_{i_{k}}), F) = 0 .

Then,

lim_{i \to \infty} ρ (x_{i}, F) = 0 .

3. Assume that for each integer

k \geq 1

there exists a nonempty compact set

E_{k} \subset X

such that

lim_{j \to \infty} ρ (T^{j} (x_{i_{k}}), E_{k}) = 0 .

Then, there exists a nonempty compact set

E \subset X

such that

{lim}_{i \to \infty} ρ (x_{i}, E) = 0

Example 1.

Assume that

(Z, ∥ \cdot ∥)

is a Banach space ordered by a closed convex cone

Z_{+}

(

x \leq y

for

x, y \in Z

if and only if

y - x \in Z_{+}

) such that

Z_{+} - Z_{+} = Z .

Then, per the Krein–Shmulyan theorem [25], there exists

c_{0} > 0

such that for each

z \in Z

there exist points

z_{1}, z_{2} \in Z_{+}

such that

z = z_{1} - z_{2}, ∥ z_{i} ∥ \leq c_{0} ∥ z ∥, i = 1, 2 .

Let

V = X

be a nonempty closed subset of Z, with

(x, y) \in E (G)

if and only if

y \geq x

and

ρ (x, y) = ∥ x - y ∥

x, y \in X

.Assume that

r_{0} \in (0, 1]

c_{0} \geq 1

{x_{i}}_{i = 0}^{\infty} \subset X

and that

B (x_{i}, c_{0} r_{0}) \subset X, i = 0, 1, \dots .

Assume that

i \geq 0

is an integer and that

z \in X

satisfies

∥ z - x_{i} ∥ \leq r_{0} .

Then, there exist

v_{1}, v_{2} \in Z_{+}

such that

z - x_{2} = v_{2} - v_{1}, ∥ v_{j} ∥ \leq c_{0} ∥ z - x_{i} ∥, j = 1, 2 .

We now have

z = x_{i} + v_{2} - v_{1} \leq x_{i} + v_{2},

x_{i} \leq x_{i} + v_{2} .

Setting

ξ = x_{i} + v_{2},

it is easy to see that

ρ (x_{i}, ξ) = ∥ v_{2} ∥ \leq c_{0} ∥ z - x_{i} ∥ = ρ (z, x_{i}) c_{0},

and that

ρ (x_{i}, ξ) = ∥ v_{2} ∥ \leq c_{0} ∥ z - x_{i} ∥ \leq c_{0} r_{0} .

Thus, (A2) holds.

3. An Auxiliary Result

Lemma 1.

Assume that a sequence

{x_{i}}_{i = 0}^{\infty}

satisfies

\sum_{i = 0}^{\infty} ρ (T (x_{i}), x_{i + 1}) < \infty

and that there exist numbers

c_{0} \geq 1

and

r_{0} \in (0, 1]

such that (A2) holds. Let

q \geq 0

be an integer such that

\sum_{i = q}^{\infty} ρ (T (x_{i}), x_{i + 1}) < r_{0} .

(2)

Then, for each integer

n \geq 1

ρ (x_{n + q}, T^{n} (x_{q})) \leq (2 c_{0} + 1) \sum_{i = 0}^{n - 1} ρ (T (x_{q + i}), x_{q + i + 1}) .

Proof.

Setting

y_{q, 0} = x_{q},

(3)

y_{q + 1, 0} = T (y_{q, 0}), y_{q + 1, 1} = x_{q + 1},

(4)

per (2)–(4) we have

ρ (y_{q + 1, 0}, x_{q + 1}) \leq ρ (T (x_{q}), x_{q + 1}) < r_{0} .

(5)

Assumption (A2) and (5) imply that there exists

y_{q + 1, 1} \in X

such that

(x_{q + 1}, y_{q + 1, 1}) \in E (G), (y_{q + 1, 0}, y_{q + 1, 1}) \in E (G),

(6)

ρ (x_{q + 1}, y_{q + 1, 1}) \leq c_{0} ρ (x_{q + 1}, y_{q + 1, 0}) \leq c_{0} r_{0} .

(7)

Per (3), (4), and (7), we have

ρ (y_{q + 1, 1}, y_{q + 1, 0}) \leq ρ (x_{q + 1}, y_{q + 1, 1}) + ρ (x_{q + 1}, y_{q + 1, 0}),

\leq (c_{0} + 1) ρ (x_{q + 1}, y_{q + 1, 0}) = (c_{0} + 1) ρ (x_{q + 1}, T (x_{q})) .

(8)

Setting

y_{q + 1, 2} = x_{q + 1},

(9)

assume that

n \geq 1

is a natural number and that we have defined

y_{q + n, i} \in X, i = 0, \dots, 2 n,

such that

y_{q + n, 0} = T^{n} (x_{q}), y_{q + n, 2 n} = x_{q + n}

(10)

for each

i \in {0, \dots, 2 n - 1}

and at least one of the following relations holds:

(y_{q + n, i}, y_{q + n, i + 1}) \in E (G), (y_{q + n, i + 1}, y_{q + n, i}) \in E (G),

(11)

ρ (y_{q + n, 2 n}, y_{q + n, 2 n - 1}) \leq c_{0} ρ (x_{q + n}, T (x_{q + n - 1})),

(12)

ρ (y_{q + n, 2 n - 1}, y_{q + n, 2 n - 2}) \leq (c_{0} + 1) ρ (x_{q + n}, T (x_{q + n - 1})),

(13)

and

ρ (x_{q + n}, T^{n} (x_{q})) \leq \sum_{i = 0}^{2 n - 1} ρ (y_{q + n, i}, y_{q + n, i + 1}),

\leq (2 c_{0} + 1) \sum_{i = 0}^{n - 1} (x_{q + i + 1}, T (x_{q + i})) .

(14)

In view of (2)–(4) and (6)–(9), our assumption holds for

n = 1

. We now define

y_{q + n + 1, i} \in X

i = 0, \dots, 2 n + 2

. For

i = 0, \dots, 2 n - 1

, set

y_{q + n + 1, i} = T (y_{q + n, i}) .

(15)

Assumption (A1), (10), and (15) imply that for

i = 0, \dots, 2 n - 1

at least one of the following two inclusions holds:

(y_{q + n + 1, i}, y_{q + n + 1, i + 1}) \in E (G), (y_{q + n + 1, i + 1}, y_{q + n + 1, i}) \in E (G),

(16)

ρ (y_{q + n + 1, i}, y_{q + n + 1, i + 1}) \leq ρ (y_{q + n, i}, y_{q + n, i + 1}),

(17)

y_{q + n + 1, 0} = T^{n + 1} (x_{q}),

(18)

and

y_{q + n + 1, 2 n} = T (x_{q + n}) .

(19)

In view of (2),

ρ (T (x_{q + n}), x_{q + n + 1}) < r_{0} .

(20)

Assumption (A2), (19), and (20) imply that there exists

y_{q + n + 1, 2 n + 1} \in X

such that

(y_{q + n + 1, 2 n}, y_{q + n + 1, 2 n + 1}) \in E (G),

(21)

(T (x_{q + n}), y_{q + n + 1, 2 n + 1}) \in E (G),

(22)

(x_{q + n + 1}, y_{q + n + 1, 2 n + 1}) \in E (G),

(23)

and

ρ (x_{q + n + 1}, y_{q + n + 1, 2 n + 1}) \leq c_{0} ρ (x_{q + n + 1}, T (x_{q + n})) .

(24)

Setting

y_{q + n + 1, 2 n + 2} = x_{q + n + 1},

(25)

per (24) and (25) we have

ρ (y_{q + n + 1, 2 n + 2}, y_{q + n + 1, 2 n + 1}) \leq c_{0} ρ (x_{q + n + 1}, T (x_{q + n})) .

(26)

It follows from (19), (25), and (26) that

ρ (y_{q + n + 1, 2 n + 1}, y_{q + n + 1, 2 n}),

\leq ρ (y_{q + n + 1, 2 n + 1}, x_{q + n + 1}) + ρ (x_{q + n + 1}, T (x_{q + n})),

\leq (c_{0} + 1) ρ (x_{q + n + 1}, T (x_{q + n})) .

(27)

It now follows from (14), (18), and (25)–(27) that

ρ (T^{n + 1} (x_{q}), x_{q + n + 1}) = ρ (y_{q + n + 1, 0}, y_{q + n + 1, 2 n + 2}),

\leq \sum_{i = 0}^{2 n + 1} ρ (y_{q + n + 1, i}, y_{q + n + 1, i + 1}),

= \sum_{i = 0}^{2 n - 1} ρ (y_{q + n + 1, i}, y_{q + n + 1, i + 1}),

+ ρ (y_{q + n + 1, 2 n}, y_{q + n + 1, 2 n + 1}) + ρ (y_{q + n + 1, 2 n + 1}, y_{q + n + 1, 2 n + 2}),

\leq \sum_{i = 0}^{2 n - 1} ρ (y_{q + n, i}, y_{q + n, i + 1}),

+ (2 c_{0} + 1) ρ (T (x_{q + n}), x_{q + n + 1}),

\leq (2 c_{0} + 1) \sum_{i = 0}^{n} ρ (x_{q + i + 1}, T (x_{q + i})) .

(28)

Per (18), (23), and (25)–(28), the assumption made for n holds for

n + 1

as well. Thus, using induction, for each integer

n \geq 1

we have defined

y_{q + n, i} \in X

i = 0, \dots, 2 n

, such that (10)–(14) hold. This completes the proof of Lemma 1. □

4. Proof of Theorem 2

Let

ϵ \in (0, r_{0} / 2) .

(29)

Per (1), there exists an integer

k \geq 1

such that

\sum_{j = i_{k} - 1}^{\infty} ρ (x_{j + 1}, T (x_{j})) < {(2 c_{0} + 1)}^{- 1} ϵ / 4 .

(30)

Lemma 1 and inequality (30) imply that for each integer

n \geq 1

, we have

ρ (x_{n + i_{k}}, T^{n} (x_{i_{k}})) \leq (2 c_{0} + 1) \sum_{j = i_{k}}^{i_{k} + n - 1} ρ (T (x_{j}), x_{j + 1}) < ϵ / 4 .

(31)

We first prove Assertion 1. There exists

y_{k} = lim_{j \to \infty} T^{j} (x_{i_{k}}) .

Per (31), for all sufficiently large natural numbers n,

ρ (x_{n + i_{k}}, y_{k}) \leq ρ (y_{k}, T^{n} (x_{i_{k}})) + ρ (T^{n} (x_{i_{k}}), x_{i_{k} + n}) < ϵ .

Because

ϵ

is an arbitrary sufficiently small positive number, we conclude that

{x_{n}}_{n = 0}^{\infty}

is a Cauchy sequence and that there exists the limit

x_{*} = lim_{n \to \infty} x_{n} .

It follows from (1) that

x_{*} = lim_{i \to \infty} T (x_{i}) .

Because the set

E (G)

is closed, Assumption (A2) implies that

(x_{*}, x_{*}) \in E (G) .

If the mapping T is continuous at the point

x_{*}

, then we have

T (x_{*}) = x_{*}

. Assertion 1 is proved.

Next, we prove Assertion 2. Based on our assumptions,

lim_{j \to \infty} ρ (T^{j} (x_{i_{k}}), F) = 0 .

When combined with (31), this implies that for each sufficiently large natural number n we have

ρ (x_{n + i_{k}}, F) \leq ρ (x_{n + i_{k}}, T^{n} (x_{i_{k}})) + ρ (T^{n} (x_{i_{k}}), F) < ϵ .

Because

ϵ

is an arbitrary sufficiently small positive number, we can conclude that

lim_{i \to \infty} ρ (x_{i}, F) = 0 .

Assertion 2 is proved.

Finally, we prove Assertion 3. There exists a compact set

E_{0} \subset X

such that

lim_{n \to \infty} ρ (T^{n} (x_{k}), E_{0}) = 0 .

Per (31), for each sufficiently large natural number n we have

ρ (x_{n + i_{k}}, E_{0}) \leq ρ (x_{n + i_{k}}, T^{n} (x_{i_{k}})) + ρ (T^{n} (x_{i_{k}}), E_{0}) < ϵ .

Thus, we have shown that there exists a compact set

E_{0}

such that

ρ (x_{n + i_{k}}, E_{0}) < ϵ

for every sufficiently large natural number n. We may assume that

E_{0}

is finite. Because

ϵ

is any element of the interval

(0, r_{0} / 2)

, this implies that each subsequence of

{x_{i}}_{i = 0}^{\infty}

has a convergent subsequence. Denoting the set of all limit points of the sequence

{x_{i}}_{i = 0}^{\infty}

by E, it is not difficult to see that E is compact and that

lim_{i \to \infty} ρ (x_{i}, E) = 0 .

This completes the proof of Assertion 3 and of Theorem 2 itself.

5. Conclusions

In the present paper, we have shown that if all exact orbits of a nonexpansive self-mapping of a complete metric space with a graph converge, then this convergence property holds for all its inexact orbits with summable errors as well. This is an analog of the result of [2] for inexact iterates of nonexpansive mappings defined on complete metric spaces. In this connection, we recall that the study of the behavior of inexact iterates is very important, as computational errors are always present in calculations. Therefore, this is a rapidly growing area of research, starting with the seminal paper [8] (although see [2,13] and references mentioned therein as well). Our results show that if all exact iterates converge, then inexact iterates with summable errors converge as well.

Author Contributions

Conceptualization, S.R. and A.J.Z.; Methodology, S.R. and A.J.Z.; Validation, S.R. and A.J.Z.; Formal analysis, S.R. and A.J.Z. All authors have read and agreed to the published version of the manuscript.

Funding

Simeon Reich was partially supported by the Israel Science Foundation (Grant 820/17), by the Fund for the Promotion of Research at the Technion (Grant 2001893), and by the Technion General Research Fund (Grant 2016723).

Data Availability Statement

Not applicable.

Acknowledgments

Both authors are grateful to the editor and the three anonymous referees for their useful comments and helpful suggestions.

Conflicts of Interest

The authors declare no conflict of interest.

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Reich, S.; Zaslavski, A.J. Inexact Iterates of Nonexpansive Mappings with Summable Errors in Metric Spaces with Graphs. Symmetry 2023, 15, 1927. https://doi.org/10.3390/sym15101927

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Reich S, Zaslavski AJ. Inexact Iterates of Nonexpansive Mappings with Summable Errors in Metric Spaces with Graphs. Symmetry. 2023; 15(10):1927. https://doi.org/10.3390/sym15101927

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Reich, Simeon, and Alexander J. Zaslavski. 2023. "Inexact Iterates of Nonexpansive Mappings with Summable Errors in Metric Spaces with Graphs" Symmetry 15, no. 10: 1927. https://doi.org/10.3390/sym15101927

APA Style

Reich, S., & Zaslavski, A. J. (2023). Inexact Iterates of Nonexpansive Mappings with Summable Errors in Metric Spaces with Graphs. Symmetry, 15(10), 1927. https://doi.org/10.3390/sym15101927

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Inexact Iterates of Nonexpansive Mappings with Summable Errors in Metric Spaces with Graphs

Abstract

1. Introduction

2. The Main Result

3. An Auxiliary Result

4. Proof of Theorem 2

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

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