Convergence for a Fixed Point of Evolution Families in Banach Space via Iterative Process

Abstract

In this paper, we present the convergence of Krasnoselskii Mann iterative process for common fixed points of pointwise Lipschitzian evolution family of bounded nonlinear operators on a Banach space .

1. Introduction

Let be a Banach space and be a nonempty, bounded, closed, and convex subset of . Let us consider a pointwise Lipschitzian evaluation family of nonlinear mappings, that is, a family of mappings satisfying the conditions , , is strongly continuous for every , and each is pointwise Lipschitzian. The latter means that there exists a family of functions such that . The existence of common fixed points for function of contraction and nonexpansive mappings have been investigated since the early 1960s, see Bruck[1, 2] Lim [3], Browder [4], Demarr [5], Belluce and Kirk [6, 7]. The asymptotic approach for finding common points of semigroups of Lipschitzian (but not pointwise Lipschitzian) mappings has been also investigated for some time, see Xu and Tan [8] and the references there in. Kirk and Xu [4] proved the existence of fixed points for asymptotic pointwise contraction and asymptotic pointwise nonexpansive mappings in Banach space, and latter on, Khamsi and Hussain extend these results to metric spaces [9], and Kozlowski and Khamsi to modular function spaces [10, 11].The generalization of known iterative fixed point construction process of pointwise asymptotically nonexpansive mapping has been studied by several authors, such as the Mann process [12, 13] or the Ishikawa process for detail see [14]. The iterative fixed point construction process for asymptotically nonexpansive mapping exists in Hilbert space, Banach space, and Metric spaces, see for further detail [8, 9, 15–28] and the works referred there in. The modified Mann iterative process for a fixed point of asymptotically nonexpansive mappings in uniformly convex Banach space which possess the Opial property of the weak convergence and the strong convergence for asymptotically nonexpansive mapping has been proved by Schu [20]. Xu and Tan [19] proved for weak convergence of the modified Ishikawa and modified Mann iterative process for asymptotically nonexpansive mapping on Banach space that satisfy the Opial property. The generalized Ishikawa and Mann process weakly converges to a common fixed point of a pointwise nonexpansive mapping ( is a nonempty, bounded, closed, and convex subset of a Banach space ) has been proved by Kozlowski in [29] which satisfying the Opial conditions.

In [30], Kozlowski and Braily Sims examine the convergence of generalized Mann and Ishikawa iteration process for . In this paper, we proved more general form of these results We extend this results to an evolution class of bounded nonlinear mappings on the set .

2. Definitions and Preliminary Results

Definition 1. A family on is pointwise Lipschitzian if there exists a family of mappings such thatIf for each , then the family is said to be a point-wise contraction family. If for all , then the family is called nonexpansive point-wise evolution family.

Definition 2. A pointwise Lipschitzian evolution family denoted by and defined as is said to be pointwise nonexpansive if for each . By letting and for , we say that is asymptotically nonexpansive ifwhere where , .
By defining , we have

Definition 3. By , we will represent the class of all asymptotic pointwise nonexpansive evolution families on such that

Lemma 1. (see [31]). Letbe a uniformly convex Banach space andbe an asymptotically nonexpansive pointwise Lipschitzian evolution family on. Then, the sethas a common fixed point, and the setof common fixed points is closed and convex.

Lemma 2. (see [15]). Assumebe a bounded sequence of real numbers andbe double index sequence of real number satisfyingand for each . Then, the sequence converged to , where .

Definition 4. A subset is said to be a generating set for if for each there exists , and such that .
The next lemmas will be used in the proof of our main result.

Lemma 3. (see [20, 32]). Ifis uniformly convex Banach space, letbe bounded away from 0 to 1, andbe such thatand , then .

3. Main Result

A sequence is called bounded away from 0 if exists a number such that , for all natural . Similarly, is called bounded away from 1 if exists a number such that for every natural number of n.

Lemma 4. Let be a subset of an evolution family on a Banach space . If for each , there exists such thatLet , then for all .

Proof 1. It follows from the fact that every is a bounded function that there exists a finite number , such thatThen, by using the inequality, we haveorThis completes the proof.

Lemma 5. If be a nonempty, bounded, closed, and convex subset of a Banach space and be an approximate fixed point sequence of for each , then the sequence is an approximate fixed point sequence of .

Proof 2. Considerimplies thatThe result follows by lemma 4 and the boundedness of .
Let us recall the definition of the Opial property which will play an important role in our paper.

Definition 5. A Banach space is said to have the Opial property if for every sequence weakly converging to a point (denoted by ) and for each such that there holdsor equivalentlyIn the next result, we will present the main work of this paper.

Theorem 1. Let be a uniformly convex Banach space with the Opial property. Let be a nonempty, convex, and bounded subset of . Assume that there exists and such that . Assume there exists an such that as . Then, for every .

Proof 3. By defining a function for . For fixed , , we haveSince all functions are bounded and hence , it follows thatBy Lemma 4, we haveHence,Since is asymptotic pointwise nonexpansive, therefore,for each . By applying this to and passing with , we haveSince , for with , then by Opial property of , we have comparing this with give us
.
By Theorem 2, [33] and Proposition 3.4, [4] for every , there exists a continuous mapsuch that if and only if . This yieldsfor each , for every such that , .
Put ,
and , we getBy applying on both sides and , we getLet , we get thatDue to the property of , for each fix using the same argument, we haveor

Definition 6. Let , and , and . In the generalized iterative process generated by the evolution family , the iterative formula is defined by the following:where is chosen arbitrarily, (i) is bounded in 0 and 1. (ii) . (iii) for all .

Definition 7. If , then the above processThe following lemmas are important for the Mann convergence theorems.

Lemma 6. Let is a closed, bounded, and convex subset of a Banach space . Let , and be a sequence generated by . Then, an element such that .

Proof 4. Let . Then,This implies thatIt follows for each natural number ,Now, for any Let for every natural number , we haveWe observe that . By lemma 2, there exists an element , such that

Lemma 7. Let is a closed, convex, and bounded subset of a uniform convex Banach space . Let be a sequence generated by the generalized Krasnoselskii-Mann process . Then,and

Proof 5. Let . For a fix element, . By lemma 6, an element such that where , then it is trueWe observe thatBy Lemma 3 apply for and Now, the construction of sequence is equivalent toThis completes the proof.
Now, we are to show the consequential lemma which proves that under the appropriate assumption of sequence, generated by the following:The iterative process to become an approximate fixed point sequence will give the final determination of proving the process for convergence.

Lemma 8. Let be a convex and bounded subset of the Banach space . Let the is well defined, and is a subset of such that for all a strictly increasing sequence with the following conditions:(i) as. (ii), where Then, is an approximate fixed points sequence for each bounded linear operators for each and , that is,If , then

Proof 6. By Lemma 4, it is sufficient to show that for . To this end, let , and noted that as . We haveIn view of Lemma 7, we observe that
as .
Indeed,which approaches to zero as k tends to because of Lemma 7 and the fact that the process is well defined, and by boundedness of .
Also,which approaches to zero where by Lemma 7 the facts that the process is well defined because the evolution family is asymptotic pointwise nonexpansive. Also, , so by Lemma 5, is a fixed point approximate sequence of .
Now, we prove the weak convergence of the sequence and generate the iteration process.

Theorem 2. If which is closed, convex, and bounded, with Opial property. Let the sequence , iteration process is well defined and is an approximate fixed point sequence for each , then a sequence weakly converges to the common fixed point .

Proof 7. Let , be two weak limit points of the sequence . Then, two subsequences and of , such that and . For any fix , and since is approximate fixed point sequence for , it satisfiesBy the Demiclosedness principle theorem, we have and . Now, by Lemma 6, the following limits and exist. We claim that . On the contrary base let , then by Opial propertywhich is an absurd hence . This shows that the sequence has mostly one weak limit point . Since is weakly sequentially compact, therefore, has only one weak limit point, that is, . Again, using the Demiclosedness principle, we get that Since is chosen to be arbitrary and by the construction of does not depend on , where , we conclude that for each .
Now, we can use the result for some more distinct situations. First, we start the discrete case.

Theorem 3. If which is nonempty, convex, closed, and bounded with Opial property. Let be an evolution family with discrete generating set , and iterative process with . If for each , a quasiperiodic strictly increasing sequence of natural numbers with quasiperiod , and for each natural number . Then, the sequence generated by is the approximate fixed point sequence for each , and weakly converges to a common fixed point .

Proof 8. To prove the result, we will verify both the conditions of Lemma 8. The condition of Lemma 8 is clearly satisfied because . To show we noted by quasiperiodicity of for each natural number , such that . Assuming that . Fix Noted that by Lemma 7, for . Therefore, for natural number sufficiently large, it is true thatThis shows that (i) is also satisfied; hence, the sequence generated by is the approximate fixed point sequence for each . By Theorem 2, the sequence is weakly converges to the common fixed point .

Remark 1. We noted that Theorem 1 in [29] is a special case of Theorem3with.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding this work.