Approximating fixed point of({\lambda},{\rho})-firmly nonexpansive mappings in modular function spaces

In this paper, we first introduce an iterative process in modular function spaces and then extend the idea of a {\lambda}-firmly nonexpansive mapping from Banach spaces to modular function spaces. We call such mappings as ({\lambda},{\rho})-firmly nonexpansive mappings. We incorporate the two ideas to approximate fixed points of ({\lambda},{\rho})-firmly nonexpansive mappings using the above mentioned iterative process in modular function spaces. We give an example to validate our results.


Introduction
Fixed point theory has several applications in different disciplines and therefore it has been a flourishing area of research. The metric fixed pint theory in the framework of Banach spaces usually involves a close link of geometric and topological conditions. Fixed point theory in modular function spaces and metric fixed point theory are near relatives because former provides modular equivalents of norm and metric concepts. Modular spaces are extensions of the classical Lebesgue and Orlicz spaces, and in many instances conditions cast in this framework are more natural and more easily verified than their metric analogs. For more discussion, see for example, Khamsi and Kozlowski [3].
Nowadays, a vigorous research activity is developed in the area of numerical reckoning fixed points for suitable classes of nonlinear operators: see, for example, [9,10] , and applications to image recovery and variational inequalities: see [11,12,13,14]. Existence of fixed points in modular function spaces has been studied by many researchers, for example, Khamsi and Kozlowski [3] and the references therein. Dhompongsa et al. [2] have proved the existence of fixed point of ρ-contractions under certain conditions. Buthina and Kozlowski [1], for the first time, proved results on approximating fixed points in modular function spaces through Mann and Ishikawa iterative processes. Some work for multivalued mappings in modular function spaces using Mann iterative process was done by Khan and Abbas [5]. Khan [4] introduced an iterative process for approximation of fixed points of certain mappings in Banach spaces. This process is independent of both Mann and Ishikawa iterative processes in the sense that neither reduces to the other under the given conditions. Moreover, it is faster than all of Picard, Mann and Ishikawa iterative processes in case of contractions [4]. We extend this process to the framework of modular function spaces. On the other hand, λ-firmly nonexpansive mappings in Banach spaces have attracted many researchers. For a discussion on such mappings, see for example Ruiz et al. [6] and the references cited therein. As far as we know, no work has been done until now on this kind of mappings in modular function spaces. We thus introduce the idea of the so-called (λ, ρ)-firmly nonexpansive mappings, in short (λ, ρ)-FNEM. We approximate the fixed points of such mappings using the above mentioned iterative process in modular function spaces. This will create new results in modular function spaces.

Preliminaries
Here is a brief note on modular function spaces to make the discussion selfcontained. This has mainly been extracted from Khamsi and Kozlowski [3].
A set A ∈ Σ is said to be ρ-null if ρ(g1 A ) = 0 for every g ∈ E. A property p(ω) is said to hold ρ-almost everywhere (ρ-a.e.) if the set {ω ∈ Ω : p(ω) does not hold} 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 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).
It is easy to see that ρ : M →[0, ∞] posseses the following properties: ρ is called a convex modular if, in addition, the following property is satisfied: Definition 2. Let ρ be a regular function pseudomodular. We say that ρ is a regular convex function modular if ρ(f ) = 0 implies f = 0 ρ-a.e.
The class of all nonzero regular convex function modulars defined on Ω is denoted by ℜ. The convex function modular ρ defines the modular function space L ρ as Generally, the modular ρ is not sub-additive and therefore does not behave as a norm or a distance. However, the modular space L ρ can be equipped with an F -norm defined by In case ρ is convex modular, defines a norm on the modular space L ρ , and is called the Luxemburg norm.
If ρ is convex and satisfies the ∆ 2 -condition, then L ρ = E ρ . Moreover, ρ satisfies the ∆ 2 -condition if and only if F -norm convergence and modular convergence are equivalent.
Note that, ρ-convergence does not imply ρ-Cauchy since ρ does not satisfy the triangle inequality. In fact, one can show that this will happen if and only if ρ satisfies the ∆ 2 -condition.
• ρ-closed if the ρ-limit of a ρ-convergent sequence of D always belongs to D.
• ρ-a.e. closed if the ρ-a.e. limit of a ρ-a.e. convergent sequence of D always belongs to D. • ρ-compact if every sequence in D has a ρ-convergent subsequence in D.
• ρ-a.e. compact if every sequence in D has a ρ-a.e. convergent subsequence in D.
A sequence {t n } ⊂ (0, 1) is called bounded away from 0 if there exists a > 0 such that t n ≥ a for every n ∈ N. Similarly, {t n } ⊂ (0, 1) is called bounded away from 1 if there exists b < 1 such that t n ≤ b for every n ∈ N. The following lemma can be seen as an analogue of a famous lemma due to Schu [7] in Banach spaces.
The set of all fixed points of T will be denoted by F ρ (T ). The ρ-distance from an f ∈ L ρ to a set D ⊂ L ρ is given as follows: The following definition is a modular space version of the condition (I) of Senter and Dotson [8]. Let D ⊂ L ρ . A mapping T : D → D is said to satisfy condition (I) if there exists a nondecreasing function ℓ : [0, ∞) → [0, ∞) with ℓ(0) = 0, ℓ(r) > 0 for all r ∈ (0, ∞) such that The folowing general theorem ([3, Theorem 5.7]) confirms the existence fixed points of ρ-nonexpansive mappings. Theorem 1. Assume ρ ∈ ℜ satisfy (U U C1). Let D be a ρ-closed, ρ-bounded convex and nonempty subset of L ρ . Then, any T : D → D pointwise asymptotically nonexpansive mapping has a fixed point. Moreover, the set of all fixed points F (T ) is ρ-closed and convex.

Fixed points approximation of (λ, ρ)-FNEM
We first extend the idea of a λ-firmly nonexpansive mapping from Banach spaces to modular function spaces and call it (λ, ρ)-firmly nonexpansive mapping. We define the idea as follows.
Next we introduce the following iterative process in the setting of modular function spaces. For a mapping T : D → D, we define a sequence {f n } by the following iterative process: where {α n } ⊂ (0, 1) is bounded away from both 0 and 1. For details on a similar iterative process but in Banach spaces, see [4]. In this paper, using the above two ideas together, we prove our main result for approximating fixed points in modular function spaces. We give a simple numerical example to support and validate our results.
We are now in a position to give our main results as follows. Proof. Let w ∈ F ρ (T ). To prove that lim n→∞ ρ(f n − w) exists for all w ∈ F ρ (T ), consider This implies ρ (T g n − T w) ≤ ρ (g n − w) and hence Also, because T is a (λ, ρ)-FNEM, Thus lim n→∞ ρ(f n − w) exists for each w ∈ F ρ (T ). Suppose that where m ≥ 0. Note that the above calculations also give the following inequality: Next, we prove that lim n→∞ ρ(f n − T f n ) = 0. Now using 3.4, 3.2 and 3.3, we have But then ρ(f n+1 − w) ≤ ρ(g n − w) implies that as required.
Using the above result, we now prove our convergence result for approximating fixed points of (λ, ρ)-firmly nonexpansive mappings in modular function spaces using our iterative process (3.1) as follows. Proof. Since D is ρ-compact, there exists a subsequence {f n k } of {f n } such that lim k→∞ (f n k − z) = 0 for some z ∈ D. Since T is a (λ, ρ)-FNEM, using convexity of ρ, we have Applying Theorem 2, lim n→∞ ρ(f n k − T f n k ) = 0. That is, ρ( z−T z 3 ) = 0. Hence z is a fixed point of T. That is, {f n } ρ-converges to a fixed point of T.
To prove that {f n } is a ρ-Cauchy sequence in D,let ε > 0. By (3.6) , there exists a constant n 0 such that for all n ≥ n 0, Hence there exists a y ∈ F ρ (T ) such that By ∆ 2 -condition, ρ (f n+m − f n ) < ε for m, n ≥ n 0 . Hence {f n } is a ρ-Cauchy sequence in a ρ-closed subset D of L ρ , and so it converges in D. Let lim n→∞ f n = w.
We now give the following example to show the Theorem 4 is indeed valid.
n f n T f n g n f n+1 = } ρ-converges to 1, the fixed point of T, to the accuracy of 10 −5 on 22nd iteration. On furthercomputations, the accuracy increases to 10 −10 on 42nd iteration. Remark 1. In the above example, {f n } ρ-converges faster to 1 if we take α n near the fixed point. For example, if we take α n = 0.75, then the convergence to the accuracy of 10 −5 is obtained on 19th iteration. But if we take α n = 0.25, far from 1, the required accuracy is achieved on 26th iteration.

Concluding Remarks
We have proved some strong convergence results using (λ, ρ)-firmly nonexpansive mappings on a faster iterative algorithm in modular function spaces. In our opinion it would be interesting to consider the following using above ideas: (1) studying the stabiltiy and data dependency problems (2) finding applications to general variational inequalities or equilibrium problems as well as to split feasibility problems. We may suggest the redaer to combine the ideas studied in [9,10,11,12,13,14]