Research Article Fixed Points of Nonlinear and Asymptotic Contractions in the Modular Space

A fixed point theorem for nonlinear contraction in the modular space is proved. Moreover, a fixed point theorem for asymptotic contraction in this space is studied.


Introduction
The theory of modular space was initiated by Nakano [1] in connection with the theory of order spaces and was redefined and generalized by Musielak and Orlicz [2].By defining a norm, particular Banach spaces of functions can be considered.Metric fixed theory for these Banach spaces of functions has been widely studied (see [3]).Another direction is based on considering and abstractly given functional which control the growth of the functions.Even though a metric is not defined, many problems in fixed point theory for nonexpansive mappings can be reformulated in modular spaces.
In this paper, a fixed point theorem for nonlinear contraction in the modular space is proved.Moreover, Kirk's fixed point theorem for asymptotic contraction is presented in this space.In order to do this and for the sake of convenience, some definitions and notations are recalled from [1][2][3][4][5][6].
Definition 1.3.A modular ρ defines a corresponding modular space, that is, the space X ρ given by where δ ρ (B) is called the ρ-diameter of B. (5) Say that ρ has Fatou property if whenever Example 1.5.Let (X ρ ,ρ) be a modular space, then the function d ρ defined on X ρ × X ρ by is a metric and (X ρ ,d ρ ) is a metric space.
Remark 1.6.Let (X ρ ,d ρ ) be a metric space which is given in Example 1.5 and let {x n } be a Cauchy sequence in it.This means that A. Razani et al. (1.8) and this shows that (1.9) Therefore and this proves that (X ρ ,d ρ ) is a complete metric space.In addition, it implies that all nonconstant sequences for large indices that are convergent must be convergent to zero.
Theorem 1.7.Suppose that (X ρ ,ρ) is a modular space and T : X ρ → X ρ satisfies the following condition: for all x, y ∈ X ρ , where ψ : P → [0,∞) is upper semicontinuous from the right on P and for all t ∈ P − {0}, ψ(t) < t and Then 0 is the only fixed point of T.
Proof.We use the metric d ρ and note that the closure of P which is denoted by P is with respect to metric d ρ .This metric and the mapping T satisfy the conditions of [7, Theorem 1], so the proof is complete.

A fixed point of nonlinear contraction
The Banach contraction mapping principle shows the existence and uniqueness of a fixed point in a complete metric space.this has been generalized by many mathematicians such as Arandelović [8], Edelstein [9], Ćirić [10], Rakotch [11], Reich [12], Kirk [13], and so forth.In addition, Boyd and Wong [7] studied mappings which are nonlinear contractions in the metric space.It is necessary to mention that the applications of contraction, generalized contraction principle for self-mappings, and the applications of nonlinear contractions are well known.In this section, an existence fixed point theorem for nonlinear contractions in modular spaces is proved as follows.
Theorem 2.1.Let X ρ be a ρ-complete modular space, where ρ satisfies the Δ 2 -condition.Assume that ψ : R + → [0,∞) is an increasing and upper semicontinuous function satisfying Let B be a ρ-closed subset of X ρ and T : B → B a mapping such that there exist c, for all x, y ∈ B. Then T has a fixed point.
Proof.Let x ∈ X ρ .At first, we show that the sequence {ρ(c(T n x − T n−1 x))} converges to 0. For n ∈ N, we have (2.3) then which is a contradiction, so a = 0. Now, we show that {T n x} is a ρ-Cauchy sequence for x ∈ X ρ .Suppose that {lT n x} is not a ρ-Cauchy sequence.Then, there are an > 0 and sequences of integers {m k }, {n k }, with m k > n k ≥ k, and such that We can assume that Let m k be the smallest number exceeding n k for which (2.6) holds, and Obviously, Σ k = ∅ and since Σ k ⊂ N, then by Well ordering principle, the minimum element of Σ k is denoted by m k , and clearly (2.7) holds.Now, let α 0 ∈ R + be such that l/c + 1/α 0 = 1, then we have (2.9) If k tends to infinity, and by (2.10) Thus, as k → ∞, we obtain ≤ ψ( ), which is a contradiction for > 0. Therefore {lT n x} is a ρ-Cauchy sequence, and by Δ 2 -condition, {T n x} is a ρ-Cauchy sequence, and by the fact that X ρ is ρ-complete, there is a z ∈ B such that ρ(T n x − z) → 0 as n → +∞.Now, it is enough to show that z is a fixed point of T. Indeed, The proof is complete.
Corollary 2.2.Let X ρ be a ρ-complete modular space where ρ satisfies the Δ 2 -condition.Let B be a ρ-closed subset of X ρ and let T : B → B be a mapping such that there exist c,k,l ∈ R + , c > l and k ∈ (0,1), for all x, y ∈ B. Then T has a fixed point.
Corollary 2.3.Let X ρ be a ρ-complete modular space, where ρ is s-convex and satisfies the Δ 2 -condition.Also, assume that B ⊆ X ρ is a ρ-closed subset of X ρ and T : B → B is a mapping such that there exist c,k,l ∈ R + with c > max{l,kl}, for all x, y ∈ B. Then T has a fixed point.
Proof.Consider l 0 to be one constant such that c > l 0 > max {l, kl}.Then we have Thus we get where c > l 0 and k 0 = (lk/l 0 ) s < 1.So by using Corollary 2.2, the proof is complete.

A fixed point of asymptotic contraction
The concept of "asymptotic contraction" is suggested by one of the earliest versions of Banach's principle attributed to Caccioppoli [15] and it has a long history in the nonlinear functional analysis [16].Many mathematicians (such as Chen [17], Gerhardy [18], Jachymski and J óźwik [19], Kirk [20], Suzuki [21], Xu [22], etc.) studied this concept and proved the existence of fixed points.In this section, Kirk's fixed point theorem for asymptotic contraction is proved in modular spaces.In order to do this, we need a theorem from [14] as follows.
Theorem 3.1.Let X ρ be a ρ-complete modular space.Let {F n } n be a decreasing sequence of nonempty ρ-closed subsets of X ρ with δ ρ (F n ) → 0 as n → +∞.Then n F n is reduced to one point.
Now, we state Kirk's fixed point theorem for asymptotic contraction in modular spaces (see [8]).
Theorem 3.3.Let X ρ be a ρ-complete modular space.Also, assume that ρ satisfies the Δ 2condition and the Fatou property.Let f : X ρ → X ρ be a ρ-continuous mapping and there exists a sequence {ϕ i } i of continuous functions such that ϕ i : [0,+∞) → [0,+∞) for i ∈ N and there exists c > 1 such that for all x, y ∈ X ρ .Let ϕ i → ϕ uniformly on the range of ρ, where ϕ : [0,+∞) → [0,+∞) and ϕ(r) < r for all r >0 and ϕ(0)=0.If there exists an x ∈ X ρ such that the sequence { f n (x)} n∈N is ρ-bounded, then f has a unique fixed point.
A. Razani et al. 7 Proof.Note that {ϕ i } i is continuous for all i and since {ϕ i } i converge uniformly to ϕ, then ϕ is continuous.Now for each x, y ∈ X ρ , x = y, Now, we prove that limρ( f n (x) − f n (y)) = 0 for all x, y ∈ X ρ .Otherwise, there exist x, y ∈ X ρ and ε > 0 such that Then there exists k such that Then by taking limsup from both sides of it, continuity of ϕ, and (3.4), we have ϕ(ε) ≥ ε.This is in contradiction with ϕ(ε) < ε.Therefore, (3.4) and (3.5) state that ( This is clearly a contradiction.Thus we get for all x, y ∈ X ρ .Since ρ satisfies the Δ 2 -condition, then for all x, y ∈ X ρ .This means that the sequence { f n (x)} n for all x ∈ X ρ and all n ∈ N is ρ-bounded.Now, we assume that a ∈ X ρ is arbitrary and a n = f n (a) for n ∈ N, and let We can choose α ∈ R + such that 1/α + 1/c = 1.Consider the sets defined by where L = max{c,2α}.The ρ-boundedness of {a n } implies that Y is ρ-bounded.By using (3.8), and considering the Δ 2 -condition of ρ, we get F n = ∅ for all n, and for all m and k = 1,2,...,n.By the Fatou property of ρ, and (3.10), we have Therefore x 0 ∈ F n and this means that F n is ρ-closed.
It is clear that F n+1 ⊆ F n , for all n.Now, it is enough to show that δ ρ (F n ) → 0, as n → ∞.Suppose that {x n }, {y n } are two arbitrary sequences with x n , y n ∈ F n .Consider the subsequences {x nj }, {y nj } such that lim nj →∞ ρ x nj − y nj = limsupρ x n − y n . (3.12) Then (3.17) Consequently, {F n } satisfies all conditions of Theorem 3.1, and then n F n = {z}.Since z ∈ F n for all n, then ρ(L(z − f (z))) < 1/n, for all n.Then letting n → ∞, we have ρ(L(z − f (z))) = 0. Thus L(z − f (z)) = 0. this means that f (z) = z, and the proof is complete.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
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