Approximate Fixed Point Sequences of an Evolution Family on a Metric Space

Fuan, Si; Ullah, Rizwan; Rahmat, Gul; Numan, Muhammad; Butt, Saad Ihsan; Ge, Xun

doi:https://doi.org/10.1155/2020/1647193

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 1647193 | https://doi.org/10.1155/2020/1647193

Approximate Fixed Point Sequences of an Evolution Family on a Metric Space

Si Fuan,¹Rizwan Ullah,²Gul Rahmat,²Muhammad Numan,³Saad Ihsan Butt,⁴and Xun Ge⁵

Academic Editor: Ali Jaballah

Received26 Jan 2020

Accepted27 Mar 2020

Published14 Apr 2020

Abstract

In this article, we study the approximate fixed point sequence of an evolution family. A family of a bounded nonlinear operator acting on a metric space is said to be an evolution family if and for all . We prove that the common approximate fixed point sequence is equal to the intersection of the approximate fixed point sequence of two operators from the family. Furthermore, we apply the Ishikawa iteration process to construct an approximate fixed point sequence of an evolution family of nonlinear mapping.

1. Introduction

The study of an evolution equation is much more difficult but applicable for people working in this field of nonautonomous differential equation. The idea of an evolution equation arises from the solution of the nonautonomous Cauchy problem:

This article is about to explore the existence of approximate fixed point sequences for the subset of an evolution family of nonlinear operator acting on metric spaces. Fixed point theory is a muscular tool in various fields such as differential equations, economics, optimal control, game theory dynamical system [1–3]. In [4], Dehaish and Khamsi explore the existence of fixed points for the family of semigroups in metric spaces. In case of modular function spaces, various forms of fixed point conclusions for pointwise nonexpansive, asymptotic pointwise nonexpansive, pointwise contractions, and asymptotic pointwise contraction operators have been presented by Khamsi and Kozlowski [5, 6].

Since 1960, an extensive amount of work [7–9] has been devoted to fixed point theory, for the family of operator-like contractions and nonexpansive operators acting on Banach spaces as explored by Belluce and Kirk [10, 11], Bruck and Browder [12, 13], Lim [14], and DeMarr [15]. The asymptotic approach for the occurrence of common fixed points of semigroups of Lipschitzian operators has also been discussed, see the work done by Kirk and Xu [16]. Acting in Banach spaces, the existence of fixed points for asymptotic pointwise contraction operators and asymptotic pointwise nonexpansive operators is proved by Kirk and Xu in [16], and this result is extended to metric spaces by Hussain and Khamsi [17]. In [18], the author discussed, the intersection of fixed point sequence for nonexpansive semigroups of nonlinear operators , i.e., and on a metric, and the Ishikawa iteration is given for approximation of the common fixed point sequence of the semigroups. In the current article, we will define the common fixed point sequence for evolution families of bounded linear mappings on metric space and then demonstrate the collection of common fixed point sequence for a subset of the evolution family in terms of two operators in the family. For applications of fixed point theory, we refer to the readers [19–23] and the references therein.

2. Methodology

The idea of evolution family of bounded linear operators arises from the solutions of the nonautonomous differential system. It is more general than the semigroup of bounded linear operators. In this paper, we generalize the results from the semigroup to the evolution family.

First, we define the family of evolution equation of bounded linear operators . Then, for the subset of this family of evolution equation, we prove that for any two positive real numbers and such that is irrational, the common approximate fixed point sequence is equal to the intersection of the approximate fixed point sequence of two operators from the family, that is,

Finally, applying the Ishikawa iteration process defined as , we construct a sequence and we show that under certain restrictions, is an approximate fixed point sequence of both operators and .

3. Main Results

Definition 1. (see [18]). If is a nonempty subset of , an operator is called nonexpansive if for any , the following inequalityholds. If for any , then the point is said to be the fixed point of the operator , where is used to represent the collection of all fixed points of operator . A sequence in is called an approximate fixed point sequence of the operator if . The collection of approximate fixed point sequences of operator is given by .

Definition 2. A family of mappings is called a one-parameter continuous semigroup of nonexpansive mappings on a nonempty closed convex subset C of a Banach space X if the following conditions hold:(i)For each is a nonexpansive mapping on C;(ii), for all ;(iii)For each , the mapping from into C is continuous.In the case of Banach spaces, a nonexpansive operator defined on the subset, which is nonempty, convex, closed, and bounded, has an approximate fixed point sequence rather than a fixed point. Now, we discuss this concept in the family of evolution equations.

Definition 3. A family of bounded linear operator on a Banach space is called an evolution family if(i) for all .(ii) such that . Such a family is called nonexpansive, continuous, strongly continuous, and periodic if(iii), and .(iv).(v)For any nonempty and bounded subset , we have .(iv)There exists such that for all , exists, respectively.In the current article, we will assume the subset of an evolution family of bounded linear mapping on a Banach space , define by which is and periodic. Now, we define the collection of an approximate fixed point sequence of the family and as

Proposition 1 (see [24]). If is a nonempty and additive subgroup, then either or there exists a number , which is nonnegative such that . If there exists two real numbers and such that their fraction is irrational, then the collectionis dense in the set . Moreover, if then (D is dense in ).

Lemma 1. If such that is nonempty, and if is a nonexpansive operator, then for any , we have .

Proof. Without the loss of generality, we consider that . Let . Then, . Let fix , thenfor any greater or equal to 1. As is fixed, therefore, , and we gethence .

Lemma 2. If such that is nonempty. Let be a subset of an evolution family acting on . If and are two positive real numbers, which are also the periods of this family. Then,

Proof. Without the loss of generality, we consider . Let .
Recall that if , and by definition of , where . Then,It means thatfor any greater or equal to 1. According to Lemma 1, we get , which implies . Next, consider such that or is negative. Without the loss of generality, consider . Then, we haveSince and are nonexpansive, we getfor any greater or equal to 1. Again by Lemma 1, we have which implies . Hence,

Theorem 1. is a subset of the metric space which is nonempty and bounded. Let be a subset of an evolution family of nonexpansive operators from into . Consider that is strongly continuous. If and are two numbers, which are positive and real and also periods of this family with the condition that is irrational. Then,

Proof. Sinceit is enough to prove thatWithout the loss of generality consider that is not empty. Let , and Lemma 2 implies that .
From Proposition 1, is dense in . Then, there exists and such that . Then,Let , and the family is continuous in strong sense, hence there exists such that for any , we haveSince , from Lemma 2, there exists , such thatfor any . Hence,for any . Since was positive, hence we havee.g., .

Corollary 1. If is a subset of the metric space which is nonempty and bounded. Let be an evolution family of a nonexpansive operator from into . Consider is strongly continuous. Then,

In the next section, we discuss a technique for constructing an approximate fixed point sequence of two operators acting in a metric space.

4. Common Approximate Fixed Point Sequence of Two Nonexpansive Mappings

Reich and Shafrir in [25] discuss hyperbolic metric space and intersection of an approximate fixed point sequence of two operators define on this space, see also [26]. Now, we will discuss the construction of such an approximate fixed point sequence.

Definition 4. (see [27]). If is a metric space, it is called convex, if for any in there exist a unique metric segment , and for a unique point of and , denoted by , the following inequality holds:where is a family of segments of the points of .

Definition 5 (see [25]). Let a metric space be convex, and it is called a hyperbolic metric space if for all in and , the inequalityholds.
The subset of a hyperbolic metric space is convex if for any , in , the segment is also in .

Definition 6. (see [18]). If a metric space is hyperbolic, then for any and every positive real number and , if the inequalityholds then it is uniformly convex.
The idea of uniform convexity was initially studied in Banach spaces [28]. This idea was generalized to metric spaces in [29]. One can also see [25, 26, 30].

Lemma 3. If is a hyperbolic space which is uniformly convex,then for , , and positive numbers , we havefor all , such that , , and .

Lemma 4. If a hyperbolic metric space is uniformly convex. Consider we have such thatwhere such that . Then, .

Let be a subset of a hyperbolic metric space which is nonempty and convex. Let be two operators. Fix . The strong convergence of Ishikawa iterates is defined byas given by Das and Debata, where . Under certain restrictions, we will show that is an approximate fixed point sequence of both operators and . Consider and as nonexpansive, and is their common fixed point. Then,where which shows that the sequence is decreasing, hence exists. Base on the above inequalities, we have

Theorem 2. where it is a hyperbolic metric space, which is also completely uniformly convex, and is nonempty, convex, and closed. Let be two mappings, such that Fix and generate in by (1) such that and belong to with , and then we have and .

Proof. Let . Then, is a decreasing sequence. Fix . Now, in case for , then all results are obvious. Therefore, we assume . Then,where . By the above two inequalities, we havewhich means . Therefore, we getBy Lemma 4, we conclude that . Next, from (2) and (3), we getwhich implies that . Since , we have . Since , we get similarly. We haveUsing Lemma 4, we get . Finally, sincewe conclude that .

Remark 1. It is difficult to find a common fixed point of operators and . If , it means that the operators and commute, i.e., by Theorem 2 and is which is bounded, the operators and have a common fixed point. Indeed, fix and definefor and . Considerwhere belong to . Which show that, is a contraction. Then, in is the unique fixed point of by the Banach contraction principle. As is bounded andwe have . Introduce the mappingAs is uniformly convex, is a minimum point of which is also unique:and . For any , since the operator has an approximate fixed point sequence , we getTherefore, this gives . As the collection is strictly convex, is a subset of which is nonempty and convex. Also, the operators and commute, hence . From above, it is clear that has a fixed point in , which means that operators and have a fixed point in common.

Remark 2. If an evolution family is periodic of every period , then becomes a semigroup. Hence, paper [18] become a special case of our work.

5. Conclusions

The study of an evolution equation is much more difficult but applicable for people working in this field of nonautonomous differential equation. The idea of evolution family of bounded linear operators arises from the solutions of nonautonomous differential. In this paper, we introduce a sequence of an evolution family, which is more general than the semigroup of bounded linear operators. We proved that the common approximate fixed point sequence is equal to the intersection of an approximate fixed point sequence of two operators from the family. Furthermore, we apply the Ishikawa iteration process to construct an approximate fixed point sequence of an evolution family of nonlinear mapping. Interesting readers can use other iterations instead of Ishikawa iteration and prove the results of this paper.

Data Availability

All data required for this paper are included within this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this paper.

Acknowledgments

This research was supported by the National Key R&D Program of China (2016YFC0401801).

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Copyright © 2020 Si Fuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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