On Asymptotic Pointwise Contractions in Modular Metric Spaces

Abstract

In this paper we study and prove some new fixed points theorems for pointwise and asymptotic pointwise contraction mappings in modular metric spaces.

1. Introduction

The notion of modulars on linear spaces and the corresponding theory of modular linear spaces were founded by Nakano [1] and were extensively developed by his mathematical school. In many cases, particularly in applications to integral operators, approximation and fixed point results, modular type conditions are much more natural as modular type assumptions can be more easily verified than their metric or norm counterparts. In present time the theory of modulars and modular spaces is extensively investigated. Chistyakov in [2, 3] introduced the notion of a Modular metric on arbitrary set and their corresponding modular spaces. The main idea behind this new concept according to Chistyakov is the physical interpretation of the modular. Informally speaking, whereas a metric on a set represents nonnegative finite distances between any two points of the set, a modular on a set attributes a nonnegative (possibly, infinite valued) “field of (generalized) velocities”: to each “time” 𝜆>0 (the absolute value of) an average velocity 𝑤_𝜆(𝑥,𝑦) is associated in such a way that in order to cover the “distance” between points 𝑥,𝑦∈𝑋 it takes time 𝜆 to move from 𝑥 to 𝑦 with velocity 𝑤_𝜆(𝑥,𝑦). But our approach to modular metric spaces is different. Indeed we look at these spaces as the nonlinear version of the classical modular spaces as introduced by Nakano [1] on vector spaces and modular function spaces introduced by Musielak [4] and Orlicz [5].

In recent years, there was a strong interest to study the fixed point property in modular function spaces, which are natural generalization of both function and sequence spaces, after the first paper [6] was published by Khamsi et al. in 1990. More recently, the authors presented some fixed point results for pointwise contractions and asymptotic pointwise contractions acting in modular functions spaces [7, 8]. The theory of contractions and nonexpansive mappings defined on convex subsets of Banach spaces has been well developed since the 1960s (see, e.g., Belluce and Kirk [9, 10], Browder [11], Bruck [12], DeMarr [13], and Lim [14]) and generalized to other metric spaces (see, e.g., [15–17]) and modular function spaces (see, e.g., [6]). The corresponding fixed point results were then extended to larger classes of mappings like pointwise contractions [18, 19] and asymptotic pointwise contractions and nonexpansive mappings [20, 21].

In this paper we prove the existence of fixed point theorems for pointwise mappings without the use of ultrapower technique. Our results extend and improve several known results including the corresponding recent fixed point results of [7, 8, 20].

For more on metric fixed point theory, the reader may consult the book [22].

2. Basic Definitions and Properties

Let 𝑋 be a nonempty set. Throughout this paper for a function 𝑤∶(0,∞)×𝑋×𝑋→(0,∞], we will write 𝑤_𝜆𝑥,𝑦=𝑤𝜆,𝑥,𝑦, (1) for all 𝜆>0 and 𝑥,𝑦∈𝑋.

Definition 1 (see [2, 3]). A function 𝑤∶(0,∞)×𝑋×𝑋→[0,∞] is said to be modular metric on 𝑋 if it satisfies the following axioms: (i)𝑥=𝑦 if and only if 𝑤_𝜆(𝑥,𝑦)=0, for all 𝜆>0; (ii)𝑤_𝜆(𝑥,𝑦)=𝑤_𝜆(𝑦,𝑥), for all 𝜆>0, and 𝑥,𝑦∈𝑀; (iii)𝑤_𝜆+𝜇(𝑥,𝑦)≤𝑤_𝜆(𝑥,𝑧)+𝑤_𝜇(𝑧,𝑦), for all 𝜆,𝜇>0 and 𝑥,𝑦,𝑧∈𝑋.If instead of (i), we have only the condition (i′) 𝑤_𝜆(𝑥,𝑥)=0,∀𝜆>0,𝑥∈𝑋, (2) then 𝑤 is said to be a pseudomodular (metric) on 𝑋. A modular metric 𝑤 on 𝑋 is said to be regular if the following weaker version of (i) is satisfied: 𝑥=𝑦iff𝑤_𝜆𝑥,𝑦=0,forsome𝜆>0. (3) Finally 𝑤 is said to be convex if for 𝜆,𝜇>0 and 𝑥,𝑦,𝑧∈𝑋, it satisfies the inequality 𝑤_𝜆+𝜇≤𝜆𝑥,𝑦𝑤𝜆+𝜇^𝜆()+𝜇𝑥,𝑧𝑤𝜆+𝜇^𝜇𝑧,𝑦. (4)

Note that for a metric pseudomodular 𝑤 on a set 𝑋, and any 𝑥,𝑦∈𝑋, the function 𝜆→𝑤_𝜆(𝑥,𝑦) is nonincreasing on (0,∞). Indeed if 0<𝜇<𝜆, then 𝑤_𝜆𝑥,𝑦≤𝑤_𝜆−𝜇(𝑥,𝑥)+𝑤_𝜇𝑥,𝑦=𝑤_𝜇𝑥,𝑦. (5)

Definition 2 (see [2, 3]). Let 𝑤 be a pseudomodular on 𝑋. Fix 𝑥₀∈𝑋. The two sets 𝑋_𝑤=𝑋_𝑤𝑥₀=𝑥∈𝑋∶𝑤_𝜆𝑥,𝑥₀,𝑋⟶0as𝜆⟶∞_∗𝑤=𝑋^∗_𝑤𝑥₀=𝑥∈𝑋∶∃𝜆=𝜆(𝑥)>0suchthat𝑤_𝜆𝑥,𝑥₀<∞ (6) are said to be modular spaces (around 𝑥₀).

It is clear that 𝑋_𝑤⊂𝑋^∗_𝑤, but this inclusion may be proper in general. It follows from [2, 3] that if 𝑤 is a modular on 𝑋, then the modular space 𝑋_𝑤 can be equipped with a (nontrivial) metric, generated by 𝑤 and given by 𝑑_𝑤𝑥,𝑦=inf𝜆>0∶𝑤_𝜆𝑥,𝑦≤𝜆, (7) for any 𝑥,𝑦∈𝑋_𝑤. If 𝑤 is convex modular on 𝑋, according to [2, 3] the two modular spaces coincide, 𝑋^∗_𝑤=𝑋_𝑤, and this common set can be endowed with the metric 𝑑^∗_𝑤 given by 𝑑^∗_𝑤𝑥,𝑦=inf𝜆>0∶𝑤_𝜆𝑥,𝑦≤1, (8) for any 𝑥,𝑦∈𝑋_𝑤. These distances will be called Luxemburg distances (see example below for the justification).

Next we give an example of a modular metric space.

Example 3. 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, that is 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. Let 𝜌∶ℳ_∞→[0,∞] be a nontrivial, convex, and even function. We say that 𝜌 is a regular convex function pseudomodular if(i)𝜌(0)=0; (ii)𝜌 is monotone; that is, |𝑓(𝜔)|≤|𝑔(𝜔)| for all 𝜔∈Ω implies 𝜌(𝑓)≤𝜌(𝑔), where 𝑓,𝑔∈ℳ_∞; (iii)𝜌 is orthogonally subadditive; that is, 𝜌(𝑓1_𝐴∪𝐵)≤𝜌(𝑓1_𝐴)+𝜌(𝑓1_𝐵) for any 𝐴,𝐵∈Σ such that 𝐴∩𝐵≠∅,𝑓∈ℳ; (iv)𝜌 has the Fatou property; that is, |𝑓_𝑛(𝜔)|↑|𝑓(𝜔)| for all 𝜔∈Ω implies 𝜌(𝑓_𝑛)↑𝜌(𝑓), where 𝑓∈ℳ_∞; (v)𝜌 is order continuous in ℰ; that is, 𝑔_𝑛∈ℰ and |𝑔_𝑛(𝜔)|↓0 implies 𝜌(𝑔_𝑛)↓0. Similarly as 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 ℳΩ,Σ,𝒫,𝜌=𝑓∈ℳ_∞;𝑓(𝜔)<∞𝜌−a.e, (9) where each 𝑓∈ℳ(Ω,Σ,𝒫,𝜌) is actually an equivalence class of functions equal 𝜌-a.e. rather than an individual function. Where no confusion exists we will write ℳ instead of ℳ(Ω,Σ,𝒫,𝜌). Let 𝜌 be a regular function pseudomodular. (a)We say that 𝜌 is a regular function semimodular if 𝜌(𝛼𝑓)=0 for every 𝛼>0 implies 𝑓=0𝜌−a.e.(b)We say that 𝜌 is a regular function modular if 𝜌(𝑓)=0 implies 𝑓=0𝜌−a.e.The class of all nonzero regular convex function modulars defined on Ω will be denoted by ℜ. Let us denote 𝜌(𝑓,𝐸)=𝜌(𝑓1_𝐸) for 𝑓∈ℳ,𝐸∈Σ. It is easy to prove that 𝜌(𝑓,𝐸) is a function pseudomodular in the sense of Definition  2.1.1 in [23] (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 in [23–25], and see also Musielak [4] for the basics of the general modular theory. Let 𝜌 be a convex function modular. (a)The associated modular function space is the vector space 𝐿_𝜌(Ω,Σ), or briefly 𝐿_𝜌, defined by 𝐿_𝜌=𝑓∈ℳ;𝜌𝜆𝑓⟶0as𝜆⟶0. (10)(b)The following formula defines a norm in 𝐿_𝜌 (frequently called Luxemburg norm): 𝑓_𝜌𝑓=inf𝛼>0;𝜌𝛼≤1. (11) Modular function space furnishes a wonderful example of a modular metric space. Indeed let 𝐿_𝜌 be modular function space. Define the function 𝑤 by 𝑤_𝜆𝑓,𝑔=𝜌𝑓−𝑔𝜆, (12) for all 𝜆>0, and 𝑓,𝑔∈𝐿_𝜌; then 𝑤 is a modular metric on 𝐿_𝜌. Note that 𝑤 is convex if and only if 𝜌 is convex. Moreover we have 𝑓−𝑔_𝜌=𝑑^∗_𝑤𝑓,𝑔, (13) for any 𝑓,𝑔∈𝐿_𝜌.

Definition 4. Let 𝑋_𝑤 be a modular metric space. (1)The sequence (𝑥_𝑛)_𝑛∈ℕ in 𝑋_𝑤 is said to be 𝑤-convergent to 𝑥∈𝑋_𝑤 if and only if 𝑤₁(𝑥_𝑛,𝑥)→0, as 𝑛→∞. 𝑥 will be called the 𝑤-limit of (𝑥_𝑛). (2)The sequence (𝑥_𝑛)_𝑛∈𝑁 in 𝑋_𝑤 is said to be 𝑤-Cauchy if 𝑤₁(𝑥_𝑚,𝑥_𝑛)→0, as 𝑚,𝑛→∞. (3)A subset 𝑀 of 𝑋_𝑤 is said to be 𝑤-closed if the 𝑤-limit of a 𝑤-convergent sequence of 𝑀 always belong to 𝑀. (4)A subset 𝑀 of 𝑋_𝑤 is said to be 𝑤-complete if any 𝑤-Cauchy sequence in 𝑀 is a 𝑤-convergent sequence and its 𝑤-limit is in 𝑀.(5)A subset 𝑀 of 𝑋_𝑤 is said to be 𝑤-bounded one has 𝛿_𝑤(𝑀)=sup𝑤₁𝑥,𝑦;𝑥,𝑦∈𝑀<∞. (14)Note that if 𝑤 is regular, then the 𝑤-limit of a sequence is unique. In general if lim_𝑛→∞𝑤_𝜆(𝑥_𝑛,𝑥)=0, for some 𝜆>0, then we may not have lim_𝑛→∞𝑤_𝜆(𝑥_𝑛,𝑥)=0, for all 𝜆>0. Therefore, as it is done in modular function spaces, we will say that 𝑤 satisfies Δ₂-condition if this is the case; that is, lim_𝑛→∞𝑤_𝜆(𝑥_𝑛,𝑥)=0, for some 𝜆>0 implies lim_𝑛→∞𝑤_𝜆(𝑥_𝑛,𝑥)=0, for all 𝜆>0. In [2, 3], one will find a discussion about the connection between 𝑤-convergence and metric convergence with respect to the Luxemburg distances. In particular, we have lim_𝑛→∞𝑑_𝑤𝑥_𝑛,𝑥=0ifflim_𝑛→∞𝑤_𝜆𝑥_𝑛,𝑥=0,∀𝜆>0, (15) for any {𝑥_𝑛}∈𝑋_𝑤 and 𝑥∈𝑋_𝑤. And in particular we have 𝑤-convergence and 𝑑_𝑤 convergence are equivalent if and only if the modular 𝑤 satisfies the Δ₂-condition. Moreover if the modular 𝑤 is convex, then we know that 𝑑^∗_𝑤 and 𝑑_𝑤 are equivalent which implies lim_𝑛→∞𝑑^∗_𝑤𝑥_𝑛,𝑥=0ifflim_𝑛→∞𝑤_𝜆𝑥_𝑛,𝑥=0,∀𝜆>0, (16) for any {𝑥_𝑛}∈𝑋_𝑤 and 𝑥∈𝑋_𝑤 [2, 3].
Let (𝑋,𝑤) be a modular metric space. Throughout the rest of this work, we will assume that 𝑤 satisfies the Fatou property; that is, if lim_𝑛→∞𝑤₁𝑥_𝑛,𝑥=0,lim_𝑛→∞𝑤₁𝑦_𝑛,𝑦=0, (17) then we must have 𝑤₁𝑥,𝑦≤liminf_𝑛→∞𝑤₁𝑥_𝑛,𝑦_𝑛. (18) For any 𝑥∈𝑋_𝑤 and 𝑟≥0, we define the modular ball 𝐵_𝑤(𝑥,𝑟)=𝑦∈𝑋_𝑤;𝑤₁𝑥,𝑦≤𝑟. (19) Note that if 𝑤 satisfies the Fatou property, then modular balls are 𝑤-closed. An admissible subset of 𝑋_𝑤 is defined as an intersection of modular balls. Denote by 𝒜_𝑤(𝑋_𝑤) the family of admissible subsets of 𝑋_𝑤. Note that 𝒜_𝑤(𝑋_𝑤) is stable by intersection. At this point we introduce some notation which will be used throughout the remainder of this work. For a subset 𝐴 of a modular metric space 𝑋_𝑤 set cov_𝑤(𝐴)={𝐵∶𝐵isamodularballand𝐴⊂𝐵}. (20) Recall that 𝐴 is 𝑤-bounded if 𝛿_𝑤(𝐴)=sup{𝑤₁(𝑥,𝑦);𝑥,𝑦∈𝐴}<∞.

Definition 5. Let (𝑋,𝑤) be a modular metric space. One will say that 𝒜_𝑤(𝑋_𝑤) is compact if any family (𝐴_𝛼)_𝛼∈Γ of elements of 𝒜_𝑤(𝑋_𝑤) has a nonempty intersection provided ⋂_𝛼∈𝐹𝐴_𝛼≠∅ for any finite subset 𝐹⊂Γ.

Remark 6. Note that if 𝒜_𝑤(𝑋_𝑤) is compact, then 𝑋_𝑤 is 𝑤-complete.

Definition 7. Let 𝑀 be a modular metric space. A function Φ∶𝑀→[0,∞) is said to be (i)𝑤-convex if {𝑥;Φ(𝑥)≤𝑟}∈𝒜_𝑤(𝑀)forany𝑟≥0; (21)(ii)a type if it is defined as Φ(𝑢)=limsup_𝑛→∞𝑤₁𝑥_𝑛,𝑢, (22)where (𝑥_𝑛) is a 𝑤-bounded sequence in 𝑀.
Types are very useful in the study of the geometry of Banach spaces and the existence of fixed point of mappings. We will say that 𝑀 is type-stable if types are 𝑤-convex. We have the following lemma.

Lemma 8. Let 𝑀 be a modular metric space such that 𝒜_𝑤(𝑀) is compact on 𝑀 which is type-stable. Then for any type Φ, there exists 𝑧∈𝑀 such that Φ(𝑧)=inf{Φ(𝑥);𝑥∈𝑀}. (23) The proof is easy and will be omitted.

3. Pointwise Contraction in Modular Metric Spaces

In [2, 3] the author defined Lipschitzian mappings in modular metric spaces and proved some fixed point theorems. In this paper, we propose a more general definition. Indeed, in the case of modular function spaces, it is proved in [6] that 𝑤_𝜆𝑇(𝑥),𝑇𝑦≤𝑤_𝜆𝑥,𝑦,forany𝜆>0 (24) if and only if 𝑑_𝑤𝑇(𝑥),𝑇𝑦≤𝑑_𝑤𝑥,𝑦. (25) Moreover an example is given such that 𝑤₁𝑇(𝑥),𝑇𝑦≤𝑤₁𝑥,𝑦, (26) but 𝑇 is not Lipschitzian with respect to 𝑑_𝑤 with constant 1.

Definition 9. Let (𝑋,𝑤) be a modular metric space. Let 𝑀 be a nonempty subset of 𝑋_𝑤. A map 𝑇∶𝑀→𝑀 is called (i)generalized contraction if there is an increasing and upper semicontinuous function 𝜓∶ℝ⁺→[0,∞) satisfying 𝜓(𝑡)<𝑡, for 𝑡>0, such that for any 𝑥,𝑦∈𝑀𝑤₁𝑇(𝑥),𝑇𝑦≤𝜓𝑤₁𝑥,𝑦; (27)(ii)generalized pointwise contraction if there is an increasing and upper semicontinuous function with respect to the second variable 𝜓∶𝑀×ℝ⁺→[0,∞) satisfying 𝜓(𝑥,𝑡)<𝑡, for 𝑡>0 and 𝑥∈𝑀, such that 𝑤₁𝑇(𝑥),𝑇𝑦≤𝜓𝑥,𝑤₁𝑥,𝑦, (28)for any 𝑥,𝑦∈𝑀.

Now, we are ready to prove an analogue to Boyd and Wong fixed point theorem [26] in modular metric spaces.

Theorem 10. Let (𝑋,𝑤) be a modular metric space. Let 𝑀 be a nonempty 𝑤-closed 𝑤-bounded subset of 𝑋_𝑤. Assume that the family 𝒜_𝑤(𝑀) is compact and 𝑇∶𝑀→𝑀 is a generalized contraction. Then 𝑇 has a unique fixed point 𝑧. Moreover the orbit {𝑇^𝑛(𝑥)} converges to 𝑧, for each 𝑥∈𝑀.

Proof. Let 𝑥∈𝑀, we define the 𝑤-type Φ(𝑢)=limsup_𝑛→∞𝑤₁𝑇^𝑛(𝑥),𝑢,foreach𝑢∈𝑀. (29) Since 𝒜_𝑤(𝑀) is compact, then for any 𝑥∈𝑀, we have Ω(𝑥)=_𝑛≥1cov_𝑤𝑇^𝑘(𝑥);𝑘≥𝑛≠∅. (30) Since 𝑇 is a generalized contraction, for any 𝑥,𝑦∈𝑀, we have 𝑤₁𝑇^𝑛(𝑥),𝑇^𝑛𝑦≤𝜓𝑤₁𝑇^𝑛−1(𝑥),𝑇^𝑛−1𝑦, (31) which shows that {𝑤₁(𝑇^𝑛(𝑥),𝑇^𝑛(𝑦))} is decreasing and bounded below. Therefore 𝑤₁(𝑇^𝑛(𝑥),𝑇^𝑛(𝑦)) converges to 𝐿≥0. Thus we get 𝐿≤𝜓(𝐿) since 𝜓 is upper semicontinuous. Our assumptions on 𝜓 force 𝐿=0, which means that lim_𝑛→∞𝑤₁𝑇^𝑛(𝑥),𝑇^𝑛𝑦=0forany𝑥,𝑦∈𝑀. (32) Now let 𝑧 be a fixed point of 𝑇; that is, 𝑇(𝑧)=𝑧, and then we have for any 𝑥∈𝑀lim_𝑛→∞𝑤₁𝑇^𝑛(𝑥),𝑇^𝑛(𝑧)=0. (33) Since 𝑇^𝑛(𝑧)=𝑧, then lim_𝑛→∞𝑤₁(𝑇^𝑛(𝑥),𝑧)=0; that is, if 𝑇 has a fixed point 𝑧, then any orbit {𝑇^𝑛(𝑥)}𝑤-converges to 𝑧. Note that if 𝑢 is another fixed point of 𝑇, then lim_𝑛→∞𝑤₁(𝑇^𝑛(𝑢),𝑧)=0, so 𝑤₁(𝑢,𝑧)=0; that is, 𝑢=𝑣. Therefore 𝑇 has at most one fixed point. Let us finish the proof of the theorem by showing that 𝑇 has a fixed point. Fix 𝑥∈𝑀. Let 𝑧∈Ω(𝑥). We have Φ(𝑧)=limsup_𝑛→∞𝑤₁𝑇^𝑛(𝑥),𝑧≤𝛿_𝑤(𝑀)<∞. (34) Let 𝜀>0. There exists 𝑛₀≥1, such that for every 𝑛≥𝑛₀, 𝑤₁𝑇^𝑛(𝑥),𝑧≤Φ(𝑧)+𝜀. (35) Now, for 𝑛≥𝑛₀, ℎ≥1, and since 𝜓 is increasing, we get 𝑤₁𝑇^𝑛+ℎ(𝑥),𝑇^𝑛(𝑧)≤𝜓^ℎ𝑤₁𝑇^𝑛(𝑥),𝑧≤𝜓^ℎ(Φ(𝑧)+𝜀). (36) In particular, we have 𝑇^𝑛+ℎ(𝑥)∈𝐵_𝑤(𝑇^ℎ(𝑧),𝜓^ℎ(Φ(𝑧)+𝜀)), for any ℎ≥𝑛₀. So Ω(𝑥)⊂cov_𝑤𝑇^𝑛+ℎ(𝑥);𝑛≥𝑛₀⊂𝐵_𝑤𝑇^ℎ(𝑧),𝜓^ℎ(Φ(𝑧)+𝜀). (37) Since 𝑧∈Ω(𝑇^𝑛(𝑥)), we get 𝑤₁𝑧,𝑇^ℎ(𝑧)≤𝜓^ℎ(Φ(𝑧)+𝜀). (38) Since 𝜓 is upper semicontinuous, if we let 𝜀 go to 0, we get 𝑤₁𝑧,𝑇^ℎ(𝑧)≤𝜓^ℎ(Φ(𝑧)),foranyℎ≥1. (39) Note that lim_ℎ→∞𝜓^ℎ(Φ(𝑧))=0. Indeed we have 𝜓^ℎ(Φ(𝑧))≤𝜓^ℎ−1(Φ(𝑧)); that is, {𝜓^ℎ(Φ(𝑧))} is decreasing. Since Φ is a positive function, the sequence {𝜓^ℎ(Φ(𝑧))} converges to some 𝐿≥0. Since 𝜓 is upper semicontinuous, we get 𝜓(𝐿)≤𝐿. Our assumptions on 𝜓 will imply 𝐿=0. Therefore, we have lim_𝑛→∞𝑤₁𝑧,𝑇^ℎ(𝑧)=0. (40) Similarly, we have 𝑤₁(𝑇(𝑧),𝑇^ℎ+1(𝑧))≤𝜓(𝑤₁(𝑧,𝑇^ℎ(𝑧))), thus lim_𝑛→∞𝑤₁𝑇(𝑧),𝑇^ℎ+1(𝑧)=0. (41) Since 𝑤 is regular, the uniqueness of 𝑤-limit implies 𝑇(𝑧)=𝑧, (42) which prove that 𝑧 is a fixed point by 𝑇. In particular Ω(𝑥) is reduced to one point.

Before we state our next result, we will need to define the concept of Chebyshev center and radius in modular metric spaces. Let (𝑋,𝑤) be a modular metric space and let 𝐴⊂𝑋 be a nonempty 𝑤-bounded subset. For any 𝑥∈𝐴, define 𝑟_𝑥(𝐴)=sup𝑤₁𝑥,𝑦;𝑦∈𝐴. (43) The Chebyshev radius of 𝐴 is defined by 𝑅_𝑤(𝐴)=inf𝑟_𝑥(𝐴);𝑥∈𝐴. (44) Obviously we have 𝑅_𝑤(𝐴)≤𝑟_𝑥(𝐴)≤𝛿_𝑤(𝐴), for any 𝑥∈𝐴. The Chebyshev center of 𝐴 is defined as 𝐶_𝑤(𝐴)=𝑥∈𝐴;𝑟_𝑥(𝐴)=𝑅_𝑤(𝐴). (45) The following result is a generalization of Kirk's fixed point theorem [18] on pointwise contraction mappings.

Theorem 11. Let (𝑋,𝑤) be a modular metric space. Let 𝑀 be a nonempty 𝑤-closed 𝑤-bounded subset of 𝑋_𝑤. Assume that the family 𝒜_𝑤(𝑀) is compact and 𝑇∶𝑀→𝑀 is a generalized pointwise contraction. Then 𝑇 has a unique fixed point 𝑧. Moreover the orbit {𝑇^𝑛(𝑥)} converges to 𝑧, for each 𝑥∈𝑀.

Proof. Since 𝒜_𝑤(𝑀) is compact, there exists a minimal nonempty 𝐾∈𝒜_𝑤(𝑀) such that 𝑇(𝐾)⊂𝐾. It is easy to check that cov_𝑤(𝑇(𝐾))=𝐾. Let us prove that 𝛿_𝑤(𝐾)=0; that is, 𝐾 is reduced to one point. Indeed since 𝑀 is 𝑤-bounded, then 𝛿_𝑤(𝐾)<∞; that is, 𝐾 is also 𝑤-bounded. Let 𝑧∈𝐾; then we have 𝐾⊂𝐵_𝑤(𝑧,𝑟_𝑧(𝐾)). Since 𝑇 is a generalized pointwise contraction, there exists an increasing upper semicontinuous mapping with respect to the second variable 𝜓∶𝑀×ℝ⁺→[0,∞) such that 𝑤₁𝑇(𝑥),𝑇𝑦≤𝜓𝑥,𝑤₁𝑥,𝑦forany𝑥,𝑦∈𝑀. (46) In particular, we have 𝑇(𝐾)⊂𝐵_𝑤(𝑇(𝑥),𝜓(𝑥,𝑟_𝑥(𝐾)), for any 𝑥∈𝐾, which implies cov_𝑤(𝑇(𝐾))⊂𝐵_𝑤𝑇(𝑥),𝜓𝑥,𝑟_𝑥(𝐾). (47) So, 𝐾⊂𝐵_𝑤(𝑇(𝑥),𝜓(𝑥,𝑟_𝑥(𝐾)), which implies 𝑟_𝑇(𝑥)(𝐾)≤𝜓𝑥,𝑟_𝑥(𝐾)forany𝑥∈𝐾. (48) This will force 𝛿_𝑤(𝐾)=0. Indeed fix 𝑧∈𝐾 and define 𝐾_𝑧=𝑥∈𝐾;𝑟_𝑥(𝐾)≤𝑟_𝑧(𝐾). (49) Clearly 𝐾_𝑧 is not empty since 𝑧∈𝐾_𝑧. Moreover we have 𝐾_𝑧=_𝑥∈𝐾𝐵_𝑤𝑥,𝑟_𝑧(𝐾)∩𝐾∈𝒜_𝑤(𝑀). (50) Since 𝑟_𝑇(𝑥)(𝐾)≤𝜓(𝑥,𝑟_𝑥(𝐾))≤𝑟_𝑥(𝐾) for any 𝑥∈𝐾, we get 𝑇(𝐾_𝑧)⊂𝐾_𝑧. The minimality behavior of 𝐾 implies 𝐾_𝑧=𝐾. In particular we have 𝑟_𝑥(𝐾)=𝑟_𝑧(𝐾) for any 𝑥∈𝐾. Hence 𝛿_𝑤(𝐾)=sup_𝑥∈𝑘𝑟_𝑥(𝐾)=𝑟_𝑧(𝐾), for any 𝑧∈𝐾. Since 𝑟_𝑥(𝐾)≤𝜓(𝑥,𝑟_𝑥(𝐾)), for any 𝑥∈𝐾, we get 𝛿_𝑤(𝐾)≤𝜓𝑧,𝛿_𝑤(𝐾). (51) Assume 𝛿_𝑤(𝐾)≠0, then 𝛿_𝑤(𝐾)≤𝜓(𝑧,𝜓(𝑥,𝛿_𝑤(𝐾))<𝛿_𝑤(𝐾), which gives a contradiction. Thus 𝛿_𝑤(𝐾)=0; that is, 𝐾 is reduced to one point which is a fixed point by 𝑇 since 𝐾 is 𝑇-invariant. Hence 𝑇 has a fixed point. Next we prove that 𝑇 has a unique fixed point. Let 𝑧 and 𝑢 be two fixed points of 𝑇. We have 𝑤₁(𝑧,𝑢)=𝑤₁(𝑇(𝑧),𝑇(𝑢))≤𝜓𝑧,𝑤₁(𝑧,𝑢). (52) Our assumptions on 𝜓 will then imply 𝑤₁(𝑧,𝑢)=0, that is, 𝑧=𝑢. Next we finish the proof of our theorem by showing that, for any 𝑥∈𝑀, the orbit {𝑇^𝑛(𝑥)}𝑤-converges to a fixed point of 𝑇. Indeed for any 𝑥∈𝑀, we have 𝑤₁𝑧,𝑇^𝑛(𝑥)≤𝜓𝑧,𝑤₁𝑧,𝑇^𝑛−1(𝑥)<𝑤₁𝑧,𝑇^𝑛−1(𝑥); (53) that is, {𝑤₁(𝑧,𝑇^𝑛(𝑥))} is decreasing. Let 𝐿=lim_𝑛→∞𝑤₁(𝑧,𝑇^𝑛(𝑥)). Suppose that 𝐿≠0. Since 𝜓 is upper semicontinuous with respect to the second variable, we get 𝐿≤𝜓(𝑧,𝐿)<𝐿, (54) which is a contradiction. Thus lim_𝑛→∞𝑤₁(𝑧,𝑇^𝑛(𝑥))=0; that is, the orbit {𝑇^𝑛(𝑥)}𝑤-converges to the fixed point 𝑧.

4. Asymptotically Pointwise Contraction in Modular Metric Spaces

Definition 12. Let (𝑋,𝑤) be a modular metric space. Let 𝑀 be a nonempty subset of 𝑋_𝑤. A mapping 𝑇∶𝑀→𝑀 is called an asymptotic pointwise Lipschitzian mapping if there exists a sequence of mappings 𝛼_𝑛∶𝑀→[0,∞) such that 𝑤₁𝑇^𝑛(𝑥),𝑇^𝑛𝑦≤𝛼_𝑛(𝑥)𝑤₁𝑥,𝑦forany𝑥,𝑦∈𝑀. (55)(i)If {𝛼_𝑛} converges pointwise to 𝛼∶𝑀→[0,1), then 𝑇 is called an asymptotic pointwise contraction.(ii)If limsup_𝑛→∞𝛼_𝑛(𝑥)≤𝑘, with 0<𝑘<1, then 𝑇 is called strongly asymptotic pointwise contraction.
Note that if 𝑇 is asymptotic pointwise contraction, then 𝑇 has at most one fixed point. Indeed let 𝑧 and 𝑢 be two fixed points of 𝑇. Then 𝑤₁(𝑧,𝑢)=𝑤₁𝑇^𝑛(𝑧),𝑇^𝑛(𝑢)≤𝛼_𝑛(𝑧)𝑤₁(𝑧,𝑢). (56) Since limsup_𝑛→∞𝛼_𝑛(𝑧)<1, then we must have 𝑤₁(𝑧,𝑢)=0; that is, 𝑧=𝑢. A similar conclusion holds if 𝑇 is a strongly asymptotic pointwise contraction.

Theorem 13. Let (𝑋,𝑤) be a modular metric space. Let 𝑀 be a nonempty 𝑤-closed 𝑤-bounded subset of 𝑋_𝑤. Assume that the family 𝒜_𝑤(𝑀) is compact and 𝑇∶𝑀→𝑀 is a strongly asymptotic pointwise contraction. Then 𝑇 has a unique fixed point 𝑧. Moreover the orbit {𝑇^𝑛(𝑥)} converges to 𝑧, for each 𝑥∈𝑀.

Proof. Let 𝑥∈𝑀 and define the type Φ(𝑎)=limsup_𝑛→∞𝑤₁𝑇^𝑛(𝑥),𝑎, (57) for any 𝑎∈𝑀. Since 𝒜_𝑤(𝑀) is compact, then Ω(𝑥)=_𝑛≥1cov_𝑤𝑇^𝑘(𝑥);𝑘≥𝑛≠∅. (58) For any 𝑛,𝑚,ℎ≥1, we have 𝑤₁𝑇^{𝑚+𝑛+ℎ}(𝑥),𝑇^𝑚+ℎ(𝑥)≤𝛼_ℎ𝑇^𝑚(𝑥)𝑤₁𝑇^𝑛(𝑥),𝑇^𝑚(𝑥). (59) If we let 𝑛 go to infinity, we get Φ𝑇^𝑚+ℎ(𝑥)≤𝛼_ℎ𝑇^𝑚(𝑥)Φ𝑇^𝑚(𝑥). (60) Next we let ℎ go to infinity to get limsup_𝑛→∞Φ𝑇^𝑛(𝑥)≤𝑘Φ𝑇^𝑚(𝑥), (61) for some 𝑘∈(0,1), which easily implies that limsup_𝑛→∞Φ(𝑇^𝑛(𝑥))=0. Fix 𝑧∈Ω(𝑥) and notice that Φ(𝑧)≤limsup_𝑛→∞Φ𝑇^𝑛(𝑥). (62) Indeed let 𝑎∈𝑀; then for any 𝜀>0, there exists 𝑛₀≥1 such that for any 𝑛≥𝑛₀, we have 𝑤₁𝑇^𝑛(𝑥),𝑎≤Φ(𝑎)+𝜀. (63) In particular we have 𝑇^𝑛(𝑥)∈𝐵_𝑤(𝑎,Φ(𝑎)+𝜀), for any 𝑛≥𝑛₀. So Ω(𝑥)⊂cov_𝑤𝑇^𝑛(𝑥);𝑛≥𝑛₀⊂𝐵_𝑤(𝑎,Φ(𝑎)+𝜀), (64) which implies 𝑧∈𝐵_𝑤(𝑎,Φ(𝑎)+𝜀). This is true for any 𝜀>0. Hence for any 𝑎∈𝑀 we have 𝑤₁(𝑧,𝑎)≤Φ(𝑎). Therefore Φ(𝑧)=limsup_𝑛→∞𝑤₁𝑇^𝑛(𝑥),𝑧≤limsup_𝑛→∞Φ𝑇^𝑛(𝑥). (65) Therefore we have Φ(𝑧)=0; that is, {𝑇^𝑛(𝑥)}  𝑤-converges to 𝑧. This will force 𝑧 to be a fixed point of 𝑇. Indeed we have 𝑤₂(𝑧,𝑇(𝑧))≤𝑤₁𝑧,𝑇^𝑛(𝑥)+𝑤₁𝑇(𝑧),𝑇^𝑛(𝑥), (66) which implies 𝑤₂(𝑧,𝑇(𝑧))≤𝑤₁𝑧,𝑇^𝑛(𝑥)+𝛼₁(𝑧)𝑤₁𝑧,𝑇^𝑛−1(𝑥), (67) for any 𝑛≥1. Since {𝑇^𝑛(𝑥)}𝑤-converges to 𝑧, we get 𝑤₂(𝑧,𝑇(𝑧))=0; that is, 𝑇(𝑧)=𝑧, since 𝑤 is regular. Since 𝑇 has at most one fixed point, we conclude that any orbit of 𝑇𝑤-converges to the only fixed point 𝑧 of 𝑇.

Next we relax the strong behavior of 𝑇 but assume that types are convex to obtain the following result.

Theorem 14. Let (𝑋,𝑤) be a modular metric space. Let 𝑀 be a nonempty 𝑤-closed 𝑤-bounded subset of 𝑋_𝑤. Assume that the family 𝒜_𝑤(𝑀) is compact. Assume that 𝒜_𝑤(𝑀) is compact and type-stable. Let 𝑇∶𝑀→𝑀 be an asymptotic pointwise contraction. Then 𝑇 has a unique fixed point 𝑧. Moreover the orbit {𝑇^𝑛(𝑥)} converges to 𝑧, for each 𝑥∈𝑀.

Proof. Similarly one can easily show that 𝑇 has at most one fixed point. As we did in the proof of the previous result, let 𝑥∈𝑀 and define the type Φ(𝑎)=limsup_𝑛→∞𝑤₁𝑇^𝑛(𝑥),𝑎,foreach𝑎∈𝑀. (68) Since 𝒜_𝑤(𝑀) is compact and type-stable, then there exists 𝑧∈𝑀 such that Φ(𝑧)=inf{Φ(𝑎);𝑎∈𝑀}. (69) Let us show that Φ(𝑧)=0. Indeed we have 𝑤₁𝑇^𝑛+𝑚(𝑥),𝑇^𝑚(𝑧)≤𝛼_𝑚(𝑧)𝑤₁𝑇^𝑛(𝑥),𝑧, (70) for any 𝑛,𝑚≥1. If we let 𝑛 go to infinity, we get Φ𝑇^𝑚(𝑧)≤𝛼_𝑚(𝑧)Φ(𝑧), (71) which implies Φ(𝑧)=inf{Φ(𝑎);𝑎∈𝑀}≤Φ𝑇^𝑚(𝑧)≤𝛼_𝑚(𝑧)Φ(𝑧). (72) If we let 𝑚 go to infinity, we get Φ(𝑧)≤𝛼(𝑧)Φ(𝑧). Since 𝛼(𝑧)<1, we get Φ(𝑧)=0, which implies that {𝑇^𝑛(𝑥)}𝑤-converges to 𝑧. This will force 𝑧 to be a fixed point of 𝑇. Since we already noticed that 𝑇 has at most one fixed point, then 𝑇 has a fixed point 𝑧 and any orbit 𝑤-converges to 𝑧.

Acknowledgment

The author would like to thank Professor M. A. Khamsi with whom the author had many fruitful discussions regarding this work.