#### 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 [5–7].

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 [9–13].

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 [17–23].

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