Best proximity points in modular function spaces

We generalize the notion of best proximity points in the context of modular function spaces. We have found sufficient conditions for the existence and uniqueness of best proximity points for cyclic maps in modular function spaces. We present an application of the main result for cyclic integral operators in Orlicz function spaces, endowed with an Orlicz function modular.


Introduction
A fundamental result in fixed point theory is the Banach contraction principle in Banach spaces or in complete metric spaces. Fixed point theory is an important tool for solving equations T x = x for mapping T defined on subsets of metric or normed spaces. It is widely applied to nonlinear integral equations and differential equations.
One kind of a generalization of the Banach contraction principle is the notion of cyclic maps [7]. Because a non-self mapping T : A → B does not necessarily have a fixed point, one often attempts to find an element x which is in some sense closest to T x. Best proximity point theorems are relevant in this perspective. The notion of best proximity point is introduced in [2]. This definition is more general than the notion of cyclic maps [7], in the sense that if the sets intersect then every best proximity point is a fixed point. A sufficient condition for the existence and uniqueness of the best proximity points in uniformly convex Banach spaces is given in [2].
Besides the idea of defining a norm and considering a Banach space, another direction of generalization of the Banach contraction principle is based on considering an abstractly given functional defined on a linear space, which controls the growth of the members of the space. This functional is usually called modular and defines a modular space. The theory of modular spaces was initiated by Nakano [15] in connection with the theory of ordered spaces, which was further generalized by Musielak and Orlicz [14]. Modular function spaces are a subclass of the modular spaces. The study of the geometry of modular function spaces was initiated by B. Zlatanov (B) Faculty of Mathematics and Informatics, Plovdiv University "Paisii Hilendarski", 24 "Tzar Assen" str., 4000 Plovdiv, Bulgaria E-mail: bzlatanov@gmail.com . Fixed point results in modular function spaces were obtained first by Khamsi et al. [5]. Further development of the theory of fixed points in modular function spaces can be found in the exhaustive references of the survey article [11] and in the book [4]. Kozlowski has contributed a lot towards the study of modular function spaces, both on his own and with his collaborators.
We have tried to generalize the idea of best proximity points in modular function spaces and to present an application for integral operators in Orlicz function spaces, endowed with an Orlicz function modular.

Modular function spaces
We will give a brief summary of basic definitions and results about modular function spaces. We will follow the survey paper [11], where the needed definitions and results for fixed point theory in modular function spaces are presented in the most concise manner. For an extensive study of fixed point theory in modular function spaces and the geometry of modular function spaces, we refer to [4] and for modular space to [14,15]. When we do not indicate a reference item in this section, it means that we are following [11].
Let Ω be a nonempty set and Σ be a nontrivial σ -algebra of subsets of Ω. Let P be a δ-ring of subsets of Ω, such that E ∩ A ∈ P for any E ∈ P and A ∈ Σ. Let us assume that there exists an increasing sequence of sets K n ∈ P, such that Ω = ∪K n . By E, we denote the linear space of all simple functions with supports from P. By M ∞ , we will denote the space of all extended measurable functions, i.e., all functions f : Ω → [−∞, ∞] such that there exists a sequence {g n } ⊂ E, |g n | ≤ | f | and g n (ω) → f (ω) for all ω ∈ Ω. By 1 A , we denote the characteristic function of the set A.
be a nontrivial convex and even function. We say that ρ is a regular convex function pseudomodular if: (v) ρ is order continuous in E, i.e., g n ∈ E and |g n (ω)| ↓ 0 implies ρ(g n ) ↓ 0.
Similarly to that in the case of measure spaces, we say that a set A ∈ Σ is ρ-null if ρ(g1 A ) = 0 for every g ∈ E. We say that a property holds ρ-almost everywhere if the exceptional set is ρ-null. As usual, we identify any pair of measurable sets whose symmetric difference is ρ-null as well as any pair of measurable functions differing only on a ρ-null set. With this in mind, we define where each f ∈ M(Ω, σ, P, ρ) is actually an equivalence class of functions equal to ρ a.e. rather than an individual function. Where no confusion exists, we will write M, instead of M(Ω, σ, P, ρ).

Definition 2.2
Let ρ be a regular convex function pseudomodular.
The class of all nonzero regular convex function modular defined on Ω will be denoted by R.
It is easy to prove that ρ( f, E) is a function pseudomodular in the sense of Definition 2.1.1 in [10] (more precisely, it is a function pseudomodular with the Fatou property). Therefore, we can use all results of the standard theory of modular function spaces as per the framework defined by Kozlowski [8][9][10]; see also Musielak and Orlicz [14] for the basics of the general modular theory.
For the rest of the article, if we state something about a norm we will mean Luxemburg norm · ρ , which is generated by the modular ρ.
In this way, Lebesgue, Orlicz, Musielak-Orlicz, Lorentz and Orlicz-Lorentz are examples of modular function spaces.
In the following theorem, we recall some of the basic properties of modular function spaces.
(1) (L ρ , f ρ ) is complete and the norm · ρ is a monotone w.r.t the natural order in M. ( There exists a nondecreasing sequence of sets H k ∈ P such that H k ↑ Ω and f n converge uniformly to f on every H k (Egoroff Theorem).
(note that this property is equivalent to the Fatou property).
The next definition gives generalizations of the classical notions for normed spaces in the context of modular function spaces.
converges to some f, then we must have f ∈ C. (g) Let A, B ⊂ L ρ . We define the ρ-distance between the sets A and B by If A consists of a single element f, (h) We say that ρ has the 2 -property if sup n∈N ρ(2 f n , D k ) → 0, whenever D k ↓ ∅ and sup n∈N ρ( f n , D k ) → 0. (i) [4, p.116] A function modular ρ ∈ R is called uniformly continuous if for any L > 0 and ε > 0 there exists δ = δ(L , ε) > 0, such that if ρ(x) ≤ L and ρ(y) < δ there holds the inequality |ρ(x + y) − ρ(x)| < ε.
The following conditions are equivalent , the modular convergence is equivalent to the norm convergence.
Let us mention that the ρ-convergence does not imply ρ-Cauchy, since ρ does not satisfy the triangle inequality. If ρ has 2 -property, then ρ-convergence implies ρ-Cauchy.
The generalization of convexity properties for Banach spaces is investigated for modular function spaces in [6]. As demonstrated in [11], one concept of uniform convexity for Banach spaces generates several different types of uniform convexity in modular function spaces. This is due primarily to the fact that in general, the modular function is not homogeneous.

Proposition 2.9
The following conditions characterize the relationship between the notions that are defined in Definition 2.8 We recall that M is called an Orlicz function, provided M is even, convex and continuous nondecreasing in The function M is a regular convex function modular and it is called Orlicz function modular. An extensive study of Orlicz spaces can be found in [1,12,13,16,17].
We say that M satisfies the 2 -condition if there exist constants It is easy to observe that if M satisfies the 2 -condition, then the Orlicz function modular M has the 2 property.
If we restrict to the Orlicz space L M (0, 1), then the Orlicz function modular is defined by We will denote the corresponding modular function space by L M (0, 1). When M = |t| p , we will denote L M (0, 1) by L p (0, 1). [7] Let A, B be two sets. The map T :

Definition 2.11
We will generalize the notion of best proximity point in a metric spaces [2] for modular function spaces.

Definition 2.12
Let ρ ∈ R, A 1 , A 2 ⊂ L ρ be two subsets and T : We will prove a general result about ρ-best proximity points for modular function spaces. An application of this general result for Orlicz function spaces, endowed with an Orlicz function modular, will be presented.

Main result
Definition 3.1 Let ρ ∈ R, A, B ⊆ L ρ be subsets. The map T : A∪ B → A∪ B is called a cyclic ρ-contraction if it is a cyclic map and there exists k ∈ (0, 1), such that the inequality holds for every x ∈ A, y ∈ B.
Assume that ρ satisfies (UC1), has the 2 -property and is uniformly continuous.

Auxiliary results
For the rest of the article, k will be the constant from (2), and for simplicity of the notations we denote d ρ (A, B) by d ρ .
After applying n-times (2), we obtain the chain of inequalities Thus from k ∈ (0, 1), it follows that lim n→∞ ρ(T n+1 x − T n x) = d ρ . Proof Without loss of generality, we may assume that n < m. By assumption, m − n is an odd number. Let us take M = sup{ρ(x − T 2 j−1 x) : j ∈ N}. It is easy to observe that M > d ρ . From the ρ-boundedness of the set A ∪ B, it follows that M < +∞. Let ε > 0 be arbitrary. There exists N 0 ∈ N, such that the inequality k n (M − d ρ ) < ε holds true for every n ≥ N 0 . From (4) for every m > n ≥ N 0 , we get the inequalities Then for every ε > 0, there exists N 1 ∈ N, such that for every m > n ≥ N 1 , there holds the inequality ρ(x m − z n ) < ε.
Using the convexity of the set A, (1) and (6), we get the inequalities which is a contradiction. Therefore for every ε > 0, there exists N 1 ∈ N, such that for every m > n ≥ N 1 there holds the inequality ρ(x m − z n ) < ε.
In a similar fashion, we can prove the following lemma.

Lemma 4.4
Let ρ ∈ R. Assume that ρ satisfies (UC1), and has the 2 -property, and let A be a ρ-closed and convex subset of L ρ , B be ρ-closed subset of L ρ and A∪B be ρ-bounded. If the sequences {x n } ∞ n=1 , {z n } ∞ n=1 ⊂ A and {y n } ∞ n=1 ⊂ B be such that: then lim n→∞ ρ(x n − z n ) = 0. We would like to mention that if ρ satisfies the triangle inequality, the proof is trivial and we do not need the assumption that ρ is uniform continuous.

Proof of main result
Proof Let d ρ > 0. Let x 0 ∈ A be arbitrary. From Lemma 4.8, we have that the sequence {T 2n x 0 } ∞ n=1 is a ρ-Cauchy sequence. From Theorem 2.6, the space L ρ is ρ-complete and therefore the sequence {T 2n x 0 } ∞ n=1 is ρ-convergent. Let us denote by x ∈ A the ρ limit of the sequence {T 2n x 0 } ∞ n=1 . From Lemma 4.7, we get that x is a ρ-best proximity point of T in A.
From the inequality ρ( Suppose that there exists another ρ-best proximity point y = x. Then, T 2 y = y. From the system of inequalities From the equalities ρ(T x − y) = ρ(T x − x) = d ρ and Lemma 4.4, it follows that x = y.
If d ρ = 0, we get that {T 2n x} ∞ n=1 is a ρ-Cauchy sequence. From the ρ-completeness of L ρ , it follows that there exists x = lim n→∞ T 2n x ∈ A. From Lemma 4.7, we get that ρ(x − T x) = d ρ = 0, i.e., x is a fixed point for the map T and x ∈ A ∩ B.
Suppose that there exists another fixed point y = x. From the inequality we get a contradiction. Consequently, there exists a unique fixed point x ∈ A ∩ B.
We would like to state a corollary which covers a wide class of modular function spaces.

Corollary 5.1 Let ρ ∈ R. Assume that ρ satisfies (UC1), and is uniformly continuous and there exists C > 0 such that ρ(2 f ) ≤ Cρ( f ) for any f ∈ L ρ . Let A, B ⊆ Lρ be ρ-closed, convex subsets, A ∪ B be ρ-bounded and T : A ∪ B → A ∪ B be a cyclic ρ-contraction. Then there there exists a unique x ∈ A such that x is a ρ-best proximity point of T in A, T 2 x = x and for any x 0 ∈ A the point x is a ρ-limit of the sequence
{T 2n x 0 } ∞ n=1 .

Best proximity points for integral equations in Orlicz function spaces
We will apply Theorem 3.2 to study the existence and uniqueness of the best proximity points to a class of integral operators in Orlicz modular space L M (0, 1). For simplicity of the notation, we will denote L M (0, 1) by L M .
The uniform convexity of the Orlicz function implies that (UC1) [3]. It is known [6,11] that for Orlicz spaces over a finite, atomless measure space, the Orlicz modular M is (UC2) if and only if M is very convex. From Proposition 2.9, (UC2) is equivalent to (UC1), provided that the Orlicz function M satisfies the 2condition. Thus, we will replace in the following examples the assumption that ρ is (UC1), to be very convex and to satisfy the 2 -condition.
It can be proved [4, p.116] that in Orlicz spaces over a finite atomless measure, the uniform continuity of the Orlicz function modular is equivalent to the 2 -condition.
Let us point out a well-known fact that the set A = {u : Example 6.3 Let M be an Oricz function, which is very convex and satisfies the 2 -condition. Let us consider the modular function space L M . We assume that α ∈ L M and let n ∈ N be a fixed number. Denote the Let u, v ∈ A ∪ B. Then using the fact that M satisfies the 2 -condition, we get the inequality Using the convexity of the function M, we get the inequality and consequently T is a cyclic M-contraction. Thus from Theorem 3.2, it follows that there exists a unique M-best proximity point u ∈ A, such that for any For simplicity of the notations, let us denote for a function u : Let the following conditions take place: (14) and for any u ∈ A, v ∈ B; (c4) for any u ∈ A and v ∈ B there hold the inclusions T u ∈ B and T v ∈ A.
Then T is a cyclic M-contraction and there exists a unique u ∈ A, such that u is a M-best proximity point of T in A, T 2 u = u and for any u 0 ∈ A the sequence {T 2n u 0 } ∞ n=1 is M-convergent to u. Proof The condition (c4) implies that T is a cyclic map.
Let u 1 , u 2 ∈ A and v 1 , v 2 ∈ B. Then using the fact that M satisfies the 2 -condition, we get the inequalities Let us consider the functions f (x) = |x|, K (x, s) = xs 2 and g(x) = We will show that the map T satisfies the conditions of Theorem 6.4. The function M(t) = |t| 2 is an Orlicz function, from the class |t| p , p ∈ (1, +∞) and therefore A ∪ B is M-bounded and |t| 2 is uniformly convex. The sets A and B are convex.
For any t ∈ [t 0 , +∞), t 0 > 0, p > 1 there holds the inequality Indeed, let us define the function From the fact that the function 1+t t is a decreasing function on the interval (0, +∞), it follows that F (t) ≥ 0 for every t ∈ [t 0 , +∞). From the equality F(t 0 ) = 0, we get that the inequality (15) holds true.
We will check that the map T satisfies the conditions (c1)-(c4). As p = 2, we get that q = 2, which is the solution of the equation 1 p + 1 q = 1.

Concluding remarks
We would like to pose some open questions. Is it possible to replace the condition for uniform continuity of the function modular ρ in Theorem 3.2 with just a continuity? Is it possible to replace (UC1) with (UC2) in Theorem 3.2?
We expect that all generalizations for best proximity points for different kinds of maps will be possible to be done for modular spaces.