Abstract

In this paper, we discuss a class of mappings more general than -nonexpansive mapping defined on a modular space endowed with a graph. In our investigation, we prove the existence of fixed point results of these mappings. Then, we also introduce an iterative scheme for which proves the convergence to a fixed point of such mapping in a modular space with a graph.

1. Introduction

The abstract definition of modular spaces was introduced by Nakano [1] in 1950. Further, Musielak and Orlicz [2] redefined the modular spaces in such a way that a modular function space is seen as a vector space endowed with a modular function.

Fixed point theorems for monotone mappings in metric spaces endowed with partial ordering are considered firstly by Ran and Reurings [3] in 2004. They gave a generalization of Banach’s contraction theorem in partially ordered metric spaces. Their results were extended by Neito and Rodríguez-López [4]. Following this line of research, fixed point theory rapidly developed in the framework of spaces equipped with partial ordering. Some related works are [57].

Graphs are an ordered structure that contains and generalizes partial ordering as a particular case. The use of graphs, in the study of fixed points, was introduced by Jachymski [8]. The main idea of his work is that the contraction inequality satisfied only for some connected vertices of a graph. Therefore, he generalized Banach’s theorem in metric spaces with a graph. This work opened a new direction of research for the development of fixed point theory, in which many papers appeared [913].

In the last 10 years, many authors have considered the approach of Jachymski in the setting of modular spaces. Öztürk et al. [14] presented some fixed point results for mappings satisfying the integral contraction condition in a vector modular space equipped with a graph. Also, Alfuraidan [15] proved a generalization of Banach’s theorem in modular metric spaces endowed with a graph. Afterwards, Alfuraidan examined in [16] the existence of fixed points for multivalued monotone -contraction and -nonexpansive mappings in modular function spaces.

Over the past few decades, several iterative methods have been proposed by many mathematicians to find and approximate fixed points of nonlinear mappings. Some references to iterative processes are [1723].

In 2016, Ullah and Arshad [24] introduced a new iterative algorithm named AK iteration, defined by the following: for a given , where and are in . According to Ullah and Arshad, this scheme converges to fixed points of contractions and also has a better converge rate than all previous processes.

Subsequently, many authors extended and studied the convergence of iterative methods for nonlinear maps in various spaces endowed with a graph.

In 2012, Aleomraninejad et al. [25] proved the convergence of certain iterative processes for -contraction and -nonexpansive mapping acting on a graph. Alfuraidan [16] justified the existence and convergence for multivalued mapping in modular function spaces endowed with a graph. He also gave some results on the approximation of fixed points of Ćirić quasicontraction mappings in metric modular spaces equipped with a graph (see [15]).

In this paper, we start by introducing a new generalization of -nonexpansive mappings. Using Mann’s [20] iteration, we prove the existence of fixed points of this large class in modular spaces endowed with an oriented graph. Afterwards, we establish some convergence results for a new iterative process (see (55)) which can be considered an accelerated version of the AK iteration scheme.

2. Preliminaries

Throughout this work, stands for a linear vector space on the field .

Definition 1 (see [26]). A function is called a modular if the following holds: (1) if and only if (2)(3) for any for any and in .
If (3) is replaced by (3) for any and and in , then is called a convex modular.
A modular defines a corresponding modular space, i.e., the vector space given by Let be a convex modular, and the modular space is equipped with a norm called the Luxemburg norm, defined by

In the next definition, we give the basic definitions needed throughout this work.

Definition 2 (see [26]). Let be a modular defined on a vector space . (1)We say that a sequence is -convergent to if and only if converges to 0 when goes to infinity. Note that the limit is unique(2)A sequence is called -Cauchy if as (3)We say that is -complete if and only if any -Cauchy sequence is -convergent(4)A subset of is -closed if the -limit of a -convergent sequence of always belong to (5)A subset of is -bounded if we have (6)A subset of is -sequentially compact if any sequence of has a subsequence -convergent to a point (7)We say that satisfies the Fatou property if whenever -converges to for any , , and in Note that -convergence does not imply -Cauchy condition. Also, we have which does not imply in general for every

An important property associated with a modular which plays a powerful role in modular spaces is the properties -condition and -type condition.

Definition 3 (see [26, 27]). Let be a modular defined on a vector space . We say that satisfies (i)the -condition, if whenever as goes to (ii)the -type condition, if there exists such that for all If is assumed to be convex and to satisfy -type condition, we can define the growth function (see [28]): Then, . The arguments for this statement can be extracted from Lemma 2.6 in [10]. The minimal possible value of is usually denoted by .

In the sequel, we assume that is a convex modular.

Definition 4 (see [26]). Let be a modular and and . Define, for , that If , let If , we set : (i)We say that is uniformly convex (UC) if for every and , we have (ii)We say that is uniquely uniformly convex (UUC) if for all and , there exists such that , for (iii)We say that is strictly convex (SC), if for every such that and , we have

The following proposition characterizes the relationship between the above notions.

Proposition 5 (see [26]). (a)(UUC) implies (UC) for (b) for and (c)(UC1) implies (UC2) implies (SC)(d)(UUC1) implies (UUC2)

The following property plays a similar role as the reflexivity in Banach spaces for modular spaces.

Definition 6 (see [27]). Let be a modular. We say that the modular space satisfies the property if and only if every decreasing sequence of nonempty -closed convex and -bounded subsets of has a nonempty intersection.

Lemma 7 (see [26]. Let be a convex modular that satisfies the Fatou property. Assume that is -complete and is (UUC2). Then, satisfies the property .

Definition 8 [26]. Let be a sequence in and be a nonempty subset of . The function , defined by , is called a -type function.
Note that the -type function is convex since is convex. A sequence is called a minimizing sequence of if .

The following lemmas play an important role in the proof of the next fixed point theorem.

Lemma 9 (see [26]). Let be a convex modular (UUC1) satisfying the Fatou property and a -complete modular space. Let be a nonempty -closed convex subset of . Consider the -type function generated by a sequence in . Assume that . Then, all the minimizing sequences of are -convergent to the same limit.

Lemma 10 (see [29]). Suppose that is convex and satisfies the property (UUC1), and let be a sequence in , where . If there exists a positive real number such that , , and , then .

Lemma 11 (see [30]). Let and be two real bounded sequences. Then, (i)(ii)let , with convergent to a real . Then,

Let us finish this section with a few of graph theory terminology and basic definitions which will be needed throughout. Let be a nonempty subset of a modular space and be the loop set. Consider a directed graph such that the set of vertices coincides with and the set of its edges contains all loops, i.e., . Assume that has no parallel edges, so it can be identified to the pair .

Let and be vertices of a graph . A path from to of length is a finite sequence of elements for which , , and for .

A graph is said to be connected if there is a path between any two vertices of the graph . A directed graph is said to be transitive if for any , , and in such that and are in , then . Moreover, the conversion of a graph , denoted , is the graph obtained by reversing the direction of the edges of the graph . Thus, we have . For more details on graph theory, we refer readers to the book [31].

Definition 12. Let be a modular space. A graph is said to be convex if and only if for any , , , and in and , we have and which lead to .

Definition 13. Let be a nonempty subset of a modular space , and let be a directed graph such that . We say that have the property : if each sequence in -converges to and , then for all .

3. Results and Discussion

Throughout, we assume that is a modular space, where is a convex modular. Let be a nonempty -closed convex subset of . Let be a directed transitive graph such that is convex and . Moreover, we assume that -intervals are -closed. Recall that a -interval is any of the subsets , , and .

Definition 14. Let be a nonempty subset of a modular space . Let be a mapping and and such that . The mapping is said to (1)preserve edges if whenever for any (2)satisfy the condition if preserves edges which leads to , for all such that Note that if , then is -nonexpansive.

Lemma 15. Let be a convex modular and be a modular space endowed with a directed transitive graph such that is convex and . Let satisfy the condition on . If is a fixed point of , then for all such that .

Proof. Let be a fixed point of and such that . We have By condition , Hence, , which implies Since , then

It is easy to obtain the following consequence of condition (3), which we need in the sequel, by using the convexity of the modular , and we prove by induction on the inequality of the property.

Property 16. Let be a convex modular. For in such that and in , we have

Proposition 17. Let be a convex modular and be a modular space endowed with a directed transitive graph such that is convex and . Let be a self-mapping on monotone and satisfy the condition on , where . Then, (i)for all such that , we have (ii)if satisfies the -type condition, then for all and in such that and are in and for , we have (a) or (b): (a)(b)

Proof. (i)Let such that . Then, we have . By the condition for , we have which implies . Since , then . (ii)Assume that satisfies the -type condition, that is, for all . Let such that and are in . Assume on the contrary that and . We have Then, which is impossible. Therefore, (a) or (b) holds.
Let be a mapping preserving edges. Let be a sequence . Let such that (resp., ). By induction, we construct a sequence in , called the Mann iteration, defined by Recall that , , and (resp., , , and ) are in , for any .

In the following result, we prove the existence of fixed points of mappings using the Mann iteration defined by (14) such that the sequence converges to .

Theorem 18. Let be a (UUC1) convex modular that satisfies the Fatou property and the -type condition, and is a -complete modular space endowed with a directed transitive graph, where is a nonempty -closed convex and -bounded, is convex, and . be a mapping satisfying the condition with and -continuous on . Assume that there exists such that (resp., ). Then, has a fixed point in such that (resp., ) if and only if .

Proof. Assume that has a fixed point such that . Using the convexity of the modular , we have It follows that is a decreasing nonnegative sequence. Since is convex, then for all . Moreover, -bounded implies that for all . These lead to the conclusion that is convergent. Let . We have . Then, Moreover, Thus, by Lemma 10, we get . In the following, let us prove the converse statement. Let such that . Set . The subset is a nonempty -closed convex. In fact, set for all . We have which is a decreasing sequence of nonempty -closed convex -bounded subsets of . Indeed, it is easy to verify that is nonempty and -bounded for all . Let and . Then, . Hence, ; that is, is convex for all . Moreover, is -closed for all . In fact, let -converge to . By the definition of , we have . Since and are supposed to be -closed, then for all . Since , then ; that is, is a decreasing sequence. By Lemma 7, the modular space satisfies the property . Then, is a nonempty -closed convex.
Let be two -type functions generated by the sequences and , respectively; that is, and . Let us prove that for all such that and , one has

Case 1. If satisfies (ii, a) of Proposition 17 for some , that is, , then by condition for , we have for . Then, for all . Hence, for all , since .

Case 2. If satisfies (ii, b) of Proposition 17 for some , that is, where . By the condition for , we have for all . It follows that for all . Since , then for all . Moreover, for all , By Proposition 17, we have . Then, for all . Thus, for all . Therefore, by (21) and (27), we get By the as goes to infinity, we have since . Otherwise, for all . By as goes to infinity, we get Thus, by (29) and (30), one has for all . Let be a minimizing sequence of the type function . By Lemma 9, -converges to a point . By (31), the sequence is also a minimizing sequence of . Hence, converges also to . Since is -continuous, then -converges to . Thus, the uniqueness of the limit implies that .

Example 1. For the function , consider the modular space where the modular is given by such that for all , where and .
We have which is a convex modular satisfying the Fatou property and (UUC2) (for more details, see [32]). Moreover, satisfies the -type condition. In fact, let . Since , we have . Thus, . Therefore, for all . Taking , one has Recall that if satisfies the -type condition, then convergence is equivalent to -convergence (see [33]). Since is a Banach space (see [32]), then is a -complete modular space.
Recall the following two inequalities: (i)For all , we havefor all such that . (ii)For all , we havefor all .
The modular is (UUC1). In fact, let , , and such that , , and . Let us partition the set into two subsets and given as follows: and . Using the definition of the modular , we will denote , because the two sums on and converge, with for all . Thus, leads to or .

Case 1. Assume that . Since , then Since , then by the inequality (ii), we get By the convexity of , we have Thus,

Case 2. Assume that . Let be a subset of defined by , where . Let . Then, for all . Thus, Thus, since . Hence, since . Otherwise, for all . Thus, by the inequality (i), for all . Thus, Hence, Thus, Since , then Therefore, It follows since . Let . Since is convex, we have Therefore, by (40) and (50), we put . Thus, is (UUC1).
Consider the nonempty set . It is clear that is convex and -bounded, and it is -closed. Indeed, let be a sequence in which -converges to a point , where , for all . We have for all , where . Since -converges to , then by (51), , i.e., . Thus, because for all .
Let the convex directed graph such that and . The graph is transitive such that is convex, , and -intervals are -closed convex. Let be a mapping defined by . The mapping satisfies the condition for and . In fact, let and in such that . It is quite easy to check that preserves edges and is -continuous. Moreover, For the sequel, we consider . We have and . Let , and we have . Set , for all , where . By induction on , we construct the sequence given as follows: for all . Moreover, we have Thus, . Therefore, has a fixed point such that .

In the sequel, we will use the fixed point set with the partial orders given by

Now, we discuss the convergence of a modified version of the AK iteration scheme in order to get an accelerated iteration. The modified version is given by the following: for a given , where , , and are in such that there exists with , a convex subset of a modular space , and satisfying the condition on .

In the proof of the following lemma, we use the notations , , and for all .

Lemma 19. Let be a convex modular and be a modular space endowed with a directed transitive graph, where a nonempty -closed convex and -bounded, is convex, and . Also, let be a mapping preserving edges. Assume that the sequence is defined by iteration (55) and . Then, , , and are in for all .

Proof. Let such that . We have the -interval which is convex. Then, ; that is, and are in . Since preserves edges, then and are in . Thus, and are in . Since and are in , we have and since preserves edges. Using the transitivity of for and , one finds . Since is convex, then and are in . Again, preserves edges, then and are in ; that is, and are in . By transitivity of for and , one has . From and since preserves edges, we get . Combining and , we obtain . Again, the -interval is convex. Then, and are in . Since preserves edges, one has and which are in . That is, and are in . Using the transitivity of the graph for and , one finds . Again, preserves edges; then, implies that . Thus, combining with and the transitivity of , we get . Again, for the transitivity of for the edges , , and , one has . Moreover, from , , and , we get . Therefore, we have , , and which are in .
Next, we assume that . Using the same argument as before, we deduce that and are in since . Again, preserves edges which implies that and are in . Applying transitivity of for and , we obtain . Since preserves edges, then from , we have . Again, transitivity of implies that for and , we have .
From the convexity of , we have and which are in . Since preserves edges, then and are in . Thus, and are in . Applying transitivity of for and , we get . Since preserves edges, then . Again, by transitivity for and , one has . Since is convex, then ; that is, and are in . Since the mapping preserves edges, then and are in ; that is, and are in . Again, the transitivity of for and implies that .
Since preserves edges, then implies that . Transitivity of for and leads to . Moreover, from , , and , we deduce that . And from , , and , one finds . By and and since preserves edges, we obtain .
Therefore, , , , and are in . Finally, we conclude, by induction, that , , and are in for all .

Lemma 20. Let be a convex modular and be a modular space endowed with a directed transitive graph , where is a nonempty -closed convex subset, is convex, and . Let be a mapping satisfying on . Assume that there exists such that and is nonempty. Then, exists for all , where is defined by (55).

Proof. Assume that there exists such that and is nonempty. Fix . Using the convexity of , one can prove by induction that , , and are in for all .
Moreover, we have for all . Hence, is a decreasing sequence and bounded in . Thus, exists.

Theorem 21. Let be a convex modular (UUC1) satisfying -type condition and be a modular space endowed with a directed transitive graph , where is a nonempty -closed convex subset, is convex, and . Let be a mapping satisfying on . Assume that there exists such that and is nonempty. Then, .

Proof. Let such that and is nonempty. Fix . By Lemma 20, exists. We assume that , because if , there is nothing to prove. We have Otherwise, by the inequalities (56), we have for all . Hence, By Lemma 10 and Equations (57) and (59), we obtain .

Theorem 22. Let be a convex modular (UUC1) satisfying -type condition and be a modular space endowed with a directed transitive graph , where is a nonempty -sequentially compact convex and -bounded subset having property , is convex, and . Also, let be a mapping satisfying on with . Assume that there exists such that and is nonempty. Then, the sequence given by (55) -converges to a fixed point of .

Proof. Assume that there exists such that . Since is -sequentially compact, then has a subsequence which -converges to , where is an increasing function on . Moreover, for all . Thus, the property implies that for all . Next, let us prove that is a fixed point of .

Case 1. If there exists a subsequence such that for all , that is, satisfies (ii, a) of Proposition 17 for the subsequence and . By the condition , we obtain For all , Hence, for all . It follows since . Taking , we obtain . Otherwise, for all . Therefore, again by and by Theorem 21, .

Case 2. If there exists a subsequence of such that for all , that is, satisfies (ii, b) of Proposition 17 for the subsequence and . By the condition , we obtain Hence, for all . This implies for all . Hence, since for all . Taking the limit as goes to infinity, we get . Moreover, for all , we have By , we get . Thus, from both cases, we conclude that , since -converges to . Hence, is a fixed point of the mapping .
In order to complete the proof, let us show that -converges to . By Lemma 20, exists, for all . Since is a -decreasing sequence, then for all . Moreover, we have for all . Thus, . Hence, exists. Since -converges to , then . Thus, . Therefore, -converges to a fixed point of .

4. Numerical Example

Example 2. Let , where the modular is defined by , for all , with , and let the graph such that and . Consider the mapping defined by The mapping preserves edges and satisfies the condition for and . Indeed, if and are in or in such that , we have which is -nonexpansive and preserves edges. Then, it is . Otherwise, if and , then , and we have . Otherwise, If , we have and all the hypotheses of Theorem 18 are fulfilled. Then, the mapping has a fixed point .

To illustrate the convergence of the proposed algorithm, we provide some numerical results and a comparison with other recent iterations: AK iteration [24], M iteration [34], K iteration [35], and the iteration of Piri [22]. Firstly, we show the convergence behavior of the iterative process (55) with different initial points. To do this, we take , , , and as a stop criterion. From the given initial points , , , and , the convergence behaviors of algorithm (55) are displayed in Table 1 and graphicaly in Figures 1 and 2. The numerical results of Table 1 show that the decrease of the initial point affects the convergence speed. That is, the sequence generated by algorithm (55) will converge in less number of iterations to a fixed point when is decreased.

We show, in Table 2, the stability of the iterative process (55) based on different iteration parameters , , and . Moreover, for more precision of the convergence to the fixed point , we take also as a stop criterion. Table 2 shows that different parameters , , and which have a small effect on the number of iterations needed to achieve the fixed point . Finally, we compare the iteration numbers of the proposed scheme (55) with the other processes. The numerical results of Table 2 prove that our scheme is advantageous because it required less number of iterations for the convergence than the other schemes.

5. Conclusions

Throughout the paper, we have established an existence result and some convergence theorems of a new iterative scheme for the class of mappings satisfying the condition in the framework of modular spaces endowed with a directed graph . Further, we gave an example of this class of mapping to space endowed with a directed graph . Our results are generalization of several results as in relevant items from the reference section of this paper, as well as in the literature in general.

Abbreviations

UC:Uniform convexity
UUC:Unique uniform convexity
UCED:Uniform convexity in every direction
UUCED:Unique uniform convexity in every direction
SC:Strict convexity.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.