Fixed Point Theorems for Noncyclic Monotone Relatively ρ-Nonexpansive Mappings in Modular Spaces

)e notion of modular spaces, as a generalization of Banach spaces, was firstly introduced by Nakano [1] in connection with the theory of ordered spaces. )ese spaces were developed and generalized by Orlicz and Musielak [2]. It is well known that fixed point theory is an active field of research, a powerful tool in solving integral and differential equations. Following the publications of Ran and Reuring [3] and Neito and Rrodriguez-Lopez [4], fixed point theory in partially ordered modular spaces has recently received a good attention from researchers. Alfuraidan et al. [5] gave a modular version of Ran and Reuring fixed point theorem and proved the existence of fixed point of monotone contraction mappings in modular function spaces. )ereafter, they gave an extension of their main results for pointwise monotone contractions. In [6], Gordji et al. have proved that any quasi-contraction mappings in partially ordered modular spaces without Δ2-condition have a fixed point. In 2016, Bin Dehaish and Khamsi proved in [7] the following theorems.


Introduction
e notion of modular spaces, as a generalization of Banach spaces, was firstly introduced by Nakano [1] in connection with the theory of ordered spaces. ese spaces were developed and generalized by Orlicz and Musielak [2].
It is well known that fixed point theory is an active field of research, a powerful tool in solving integral and differential equations. Following the publications of Ran and Reuring [3] and Neito and Rrodriguez-Lopez [4], fixed point theory in partially ordered modular spaces has recently received a good attention from researchers.
Alfuraidan et al. [5] gave a modular version of Ran and Reuring fixed point theorem and proved the existence of fixed point of monotone contraction mappings in modular function spaces. ereafter, they gave an extension of their main results for pointwise monotone contractions. In [6], Gordji et al. have proved that any quasi-contraction mappings in partially ordered modular spaces without Δ 2 -condition have a fixed point.
Theorem 1 (see [7]). Let ρ ∈ R be a (UUC1) and C be a nonempty convex ρ-closed ρ-bounded subset of L ρ not reduced to one point. Let T: C ⟶ C be a monotone ρ-nonexpansive mapping and ρ-continuous. Assume there exists f 0 ∈ C such that f 0 and Tf 0 are comparable. en, T has a fixed point.
Theorem 2 (see [7]). Assume that ρ ∈ R is UUCED and uniformly continuous. Assume that L ρ satisfies the property (R). Let C be a nonempty convex ρ-closed ρ-bounded subset of L ρ not reduced to one point. Let T: C ⟶ C be a monotone ρ-nonexpansive mapping and ρ-continuous. Assume there exists f 0 ∈ C such that f 0 and Tf 0 are comparable. en, T has a fixed point.
These theorems are a generalization to modular function spaces of Browder and Göhde fixed point theorem for monotone nonexpansive mappings in Banach spaces.
A mapping T: A ∪ B ⟶ A ∪ B is said to be noncyclic provided that T(A) ⊆ A and T(B) ⊆ B, where (A, B) is a nonempty pair in a modular space. When we consider this type of mappings, it is interesting to ask if it is possible to find a pair (x * , y * ) ∈ A × B such that that is, (x * , y * ) is a best proximity pair.
In this paper, we discuss some best proximity pair results for the class of monotone noncyclic relatively ρ-nonexpansive mappings in the framework of modular spaces equipped with a partial order defined by a ρ-closed convex cone P, that is, for any x and y in the modular space X ρ one has x ⪯ y if and only if y − x in P. We took this ordering because in the case of modular spaces the order intervals are not convex closed as in modular function spaces (see eorem 2.4 in [7]). Our results generalize eorems 1 and 2 and others for monotone ρ-nonexpansive mappings to the case of noncyclic monotone relatively ρ-nonexpansive mappings in modular spaces.
To illustrate the effectiveness of our main results, we give an application to an integral equation which involves noncyclic monotone ρ-nonexpansive mappings.

Materials and Methods
e materials used in this study are obtained from journal articles and books existing on the Internet. e main tools used in this manuscript are the uniform convexity and the uniform convexity in every direction of a modular ρ. e aim of this work is to extend some fixed point theorems for nonexansive mappings to best proximity pair for monotone noncyclic relatively ρ-nonexpansive mappings in modular spaces. e paper is organized as follows. In Section 3, we begin with recollection of some basic definitions and lemmas with corresponding references that will be used in the sequel. ereafter, we give the main results of the paper with some examples. To validate the utility of our results, we give in Section 4 an application to an integral equation.

Preliminaries
roughout this work, X stands for a linear vector space on the field R. Let us start with some preliminaries and notations.
Definition 1 (see [8]). A function ρ: X ⟶ [0, +∞] is called a modular if the following holds: and for any x and y in X for any α ∈ [0, 1] and x and y in X, then ρ is called a convex modular. e modular space is defined as roughout this paper, we will assume that the modular ρ is convex. e Luxemburg norm in X ρ is defined as Associated to a modular, we introduce some basic notions needed throughout this work.
Definition 2 (see [8]). Let ρ be a modular defined on a vector space X: (1) We say that a sequence (x n ) n∈N ⊂ X ρ is ρ-convergent to x ∈ X ρ if and only if lim n⟶∞ ρ(x n − x) � 0. Note that the limit is unique.
We say that X ρ is ρ-complete if and only if any ρ-Cauchy sequence is ρ-convergent. (4) A subset C of X ρ is said ρ-closed if the ρ-limit of a ρ-convergent sequence of C always belong to C.
sequence (x n ) n of C has a subsequence ρ-convergent to a point x ∈ C. (7) We say that ρ satisfies the Fatou property if ρ(x − y) ≤ lim n⟶+∞ ρ(x − y n ) whenever (y n ) n ρ-converges to y for any x, y, and y n in X ρ .
Let us note that ρ-convergence does not imply ρ-Cauchy Definition 3. Let ρ be a modular and C be a nonempty subset of the modular space X ρ . A mapping T: C ⪯ C is said to be (a) Monotone, if T(x) ⪯ T(y) for any x, y ∈ C such that x ⪯ y.
whenever x, y ∈ X ρ and x ⪯ y. Recall that T: C ⟶ C is said to be ρ-continuous if (T(x n )) n ρ-converges to T(x) whenever (x n ) n ρ-converges to x. It is not true that a monotone ρ-nonexpansive mapping is ρ-continuous, since this result is not true in general when ρ is a norm.
Let A and B be nonempty subsets of a modular space X ρ . We adopt the notation A pair (A, B) is said to satisfy a property if both A and B satisfy that property. For instance, (A, B) is ρ-closed (resp. convex, ρ-bounded) if and only if A and B are ρ-closed (resp. convex, ρ-bounded). A pair (A, B) is not reduced to one point means that A and B are not singletons.
Recall the definition of the modular uniform convexity.
Definition 5 (see [9]). Let ρ be a modular. We say that the modular space X ρ satisfies the property (R) if and only if for every decreasing sequence (C n ) n∈N of nonempty ρ-closed convex and ρ-bounded subsets of X ρ has a nonempty intersection.
Lemma 1 (see [8]). Let ρ be a convex modular satisfying the Fatou property. Assume that X ρ is ρ-complete and ρ is (UUC 2 ). en, X ρ satisfies the property (R). Definition 6 (see [8]). Let (x n ) n be a sequence in X ρ and K be a nonempty subset of X ρ . e function τ: e following result, found in [8], plays a crucial role in the proof of many fixed point results in modular spaces. Lemma 2 (see [8]). Let ρ be a convex modular (UUC1) satisfying the Fatou property and X ρ a ρ-compete modular space. Let C be a nonempty ρ-closed convex subset of X ρ . Consider the ρ-type function τ: en, all the minimizing sequences of τ are ρ-convergent to the same limit.

Main Results
A subset P ⊂ X ρ is a pointed ρ-closed convex cone, if P is a nonempty ρ-closed subset of X ρ satisfying the following properties: Let ρ be a convex modular and T: C ⟶ C be a monotone mapping, where C is a nonempty convex subset of the modular space X ρ . Let x 0 ∈ C and λ ∈ (0, 1). Consider Let us introduce the class of mappings for which the problem of fixed point will be considered.
is said to be monotone noncyclic relatively ρ-nonexpansive if it satisfies the following conditions: Definition 8 (see [10]). Let (A, B) be a pair of a modular space X ρ and T: Definition 9. A space X ρ is said to satisfy the property (p), if x n ⟶ ρ x and y n ⟶ ρ y with x n ⪯ y n for all n ∈ N; then, x ⪯ y.
We use (A ⪯ 0 , B ⪯ 0 ) to denote the ordered ρ-proximal pair obtained from (A, B) by International Journal of Mathematics and Mathematical Sciences (A, B). us, for this y we take As the same we prove the converse.

Proposition 2.
Let ρ be a convex modular satisfying the Fatou property and X ρ satisfies the property (P) . Let (A, B) be a nonempty convex pair of X ρ . Assume that A ⪯ 0 is nonempty: Proof. It is quite easy to verify (iii): (i) Let (x n ) n be a sequence in A ⪯ 0 . en, there exists a sequence (y n ) n in B such that x n ⪯ y n and ρ(x n − y n ) � d ρ (A, B) for all n ≥ 0. Since (A, B) is ρ-sequentially compact, there exists subsequences (x n k ) k and (y n k ) k of (x n ) n and (y n ) n , respectively, such that x n k ⪯ y n k for all k ≥ 0, and ρ-converge to x ∈ A and y ∈ B, respectively. From property (P), one has x ⪯ y. Since ρ satisfies the Fatou property, then (A, B). (A, B). Moreover, en, ρ(z − z ′ ) � d ρ (A, B). Hence, z ∈ A ⪯ 0 . erefore, A ⪯ 0 is convex. Using the same argument we prove that B ⪯ 0 is convex. (iii) Let x ∈ A ⪯ 0 , then there exists y ∈ B such that x ⪯ y and ρ(x − y) � d ρ (A, B). Since T is monotone noncyclic relatively ρ-nonexpansive, then Tx ⪯ Ty (A, B).
□ Theorem 3. Let ρ be a convex modular (UUC1) satisfying the Fatou property and X ρ be a ρ-complete modular space. Let (A, B) be a nonempty convex and ρ-bounded pair of X ρ such that the subset B ⪯ 0 is nonempty ρ-closed. Let T: A ∪ B ⟶ A ∪ B be a monotone noncyclic relatively ρ-nonexpansive mapping and ρ-continuous on B. Assume that there exists x 0 ∈ A ⪯ 0 such that x 0 ⪯ Tx 0 . en, T has a best proximity pair.
Proof. Let x 0 ∈ A ⪯ 0 such that x 0 ⪯ Tx 0 . Consider the sequence (x n ) n defined by x n+1 � λTx n + (1 − λ)x n for all n ≥ 0 and λ ∈ (0, 1). Set C n � y ∈ B ⪯ 0 : x n ⪯ y for any n ≥ 0. e sequence (C n ) n∈N is a decreasing sequence of nonempty ρ-closed convex ρ-bounded subsets. Since Hence, for all n ≥ 0, there exists y ∈ B such that x n ⪯ y and ρ(x n − y) � d ρ (A, B). us, there exists y ∈ B ⪯ 0 such that x n ⪯ y. erefore, C n is nonempty for all n ≥ 0. Moreover, since x n ⪯ x n+1 then (C n ) n∈N is a decreasing sequence.
Let (c p ) p ≥ 0 be a sequence in C n ρ-converges to b ∈ B ⪯ 0 . We have c p ∈ C n , then c p − x n ∈ P and c p ∈ B ⪯ 0 , for all p ≥ 0.
us, b − x n ∈ P since P is ρ-closed. Hence, b ∈ C n . erefore, C n is ρ-closed for all n ≥ 0.
As P and B ⪯ 0 are convex, it is easy to see that C n is convex for all n ≥ 0. Furthermore, C n is ρ-bounded for all n ≥ 0.
Moreover, T(C ∞ ) ⊆ C ∞ . In fact, let z ∈ C ∞ ; then, x n ⪯ z for all n ≥ 0. Since T is monotone and x n ⪯ Tx n , one has x n ⪯ Tz for all n ≥ 0. Moreover, Consider the type function τ: C ∞ ⟶ [0, +∞] generated by the sequence (x n ) n∈N , that is, τ(z) � lim sup n⟶+∞ ρ (x n − z) for all z ∈ C ∞ . Let (z p ) p∈N be a minimizing sequence of τ. By Lemma 2, (z p ) p ρ-converges to a point Hence, τ(Tz p ) ≤ τ(z p ). en, (Tz p ) p ≥ 0 is also a minimizing sequence of τ. By Lemma 2, (Tz p ) p ρ-converges to y. As T is ρ-continuous on B, then (Tz p ) p ρ-converges to Ty. erefore, Ty � y. en, there exists y ∈ B ⪯ 0 such that Ty �

International Journal of Mathematics and Mathematical Sciences
y and x n ⪯ y, for all n ∈ N. By the definition of B ⪯ 0 , there exists x ∈ A such that x ⪯ y and ρ(x − y) � d ρ (A, B). In order to finish the proof, we show that x is a fixed point of T on A. We have (A, B). (12) en, ρ(Tx − y) � d ρ (A, B). Moreover, (A, B). , (x, y) is a best proximity pair of the mapping T. □ To illustrate eorem 3, we consider the following example.
for all x � (x 1 , x 2 ) ∈ R 2 . e modular ρ is convex satisfying the Fatou property and (UUC1), and X ρ is a ρ-complete modular space. Consider the ρ-closed convex cone e mapping T is monotone noncyclic relatively ρ-nonexpansive ρ-continuous on A ∪ B, and for e following corollary is an immediate consequence of eorem 3, and it suffices to take A � B; therefore, A � A ⪯ 0 and B � B ⪯ 0 . It will be considered as a generalization of Browder and Göhde fixed point theorem for monotone ρ-nonexpansive mappings in modular spaces (see [7,11,12]). is corollary result has already been mentioned by Bin Dehaish and Khamsi (see eorem 1) [7] in the framework of modular function spaces.

Corollary 4. Let ρ be a convex modular (UUC1) satisfying the Fatou property and X ρ be a ρ-complete modular space. Let
C be a nonempty ρ-closed convex ρ-bounded subset of X ρ and T: C ⟶ C be a monotone ρ-nonexpansive mapping and ρ-continuous. If there exists x 0 ∈ C such that x 0 ⪯ T(x 0 ), then T has a fixed point.
Particular case of normed vector spaces:  Second case: assume that (A ⪯ 0 , B ⪯ 0 ) is nonempty. Let (x n ) n be a sequence in A ⪯ 0 such that x n ρ-converges to x ∈ A. ere exists a sequence (y n ) n in B such that x n ⪯ y n and ‖x n − y n ‖ � d (A, B), for all n ≥ 0. Since X is reflexive and (y n ) n is bounded, then there exists a subsequence (y φ(n) ) n in B which converges weakly to y ∈ B. Since (x φ(n) ) n converges weakly to x, then us, ‖x − y‖ � d(A, B) and x ⪯ y. Hence, x ∈ A ⪯ 0 . erefore, A ⪯ 0 is closed. Using the same argument, we prove that B ⪯ 0 is also closed. In the context of uniformly convex normed vector spaces, eorem 3 is given as follows. (A, B) be a nonempty convex and bounded pair of a partially ordered uniformly convex Banach space X such that B ⪯ 0 nonempty and closed. Let T: A ∪ B ⟶ A ∪ B be a noncyclic monotone relatively nonexpansive mapping. Assume that there exists x 0 ∈ A ⪯ 0 such that x 0 ⪯ Tx 0 . en, T has a best proximity pair.

Theorem 5. Let
In order to weaken the assumptions of Theorem 3, we introduce the notions of uniform continuity and uniform convexity in every direction.
Definition 11. Let ρ be a modular. We say that ρ is uniformly convex in every direction (UCED) if for any r > 0 and a non null z ∈ X ρ , we have

International Journal of Mathematics and Mathematical Sciences
We say that ρ is unique uniform convexity in every direction (UUCED) if there exists η(s, z) > 0, for s ≥ 0 and z nonnull in X ρ , such that δ(r, z) > η(s, z) , for r > s.
(17) e following proposition characterizes relationships between uniform convexity, uniform convexity in every direction, and strict convexity of a modular.

Lemma 3. Let ρ be a convex modular uniformly continuous and (UUCED). Assume that X ρ satisfies the property (R). Let
C be a nonempty ρ-closed convex and ρ-bounded subset of X ρ and K be a nonempty ρ-closed convex subset of C. Let (x k ) k∈N be a sequence in C and consider the ρ-type function τ: K ⟶ [0, +∞] defined by en, τ has a unique minimum point in K. such that x 0 ⪯ Tx 0 . en, T has a best proximity pair.
Consider the type function τ: B ∞ ⟶ [0, +∞] generated by the sequence (x n ) n∈N , that is, τ(z) � lim sup n⟶+∞ ρ(x n − z) for all z ∈ B ∞ . By Lemma 3, we know that τ has a unique minimum point y ∈ B ∞ . Since T is monotone noncyclic relatively ρ-nonexpansive, we have which implies that τ(T(y)) ≤ τ(y), since λ ∈ (0, 1). erefore, T(y) is also a minimum point of τ in B ∞ . e uniqueness of the minimum point of τ implies that T(y) � y.

□
Since y ∈ B ∞ , then y ∈ B ⪯ 0 . us, there exists x ∈ A such that x ⪯ y and ρ(x − y) � d ρ (A, B). Using the same idea as in the proof of eorem 3, we show that x is a fixed point of (A, B), that is, (x, y) is a best proximity pair.
In the case A � B, therefore A � A ⪯ 0 and B � B ⪯ 0 , we obtain the following corollary. e result of this corollary has already been mentioned by Khamsi and Bin Dehaish (see eorem 2) [7] in modular function spaces.

Corollary 7.
Let ρ be a convex modular (UUCED) and uniformly continuous. Assume that X ρ satisfies the property (R). Let C be a nonempty ρ-closed convex and ρ-bounded subset of X ρ . Let T: C ⟶ C be a monotone ρ-nonexpansive mapping. If there exists x 0 ∈ C such that x 0 ⪯ T(x 0 ), then T has a fixed point.