A novel iterative approach for split feasibility problem

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Introduction
Fixed point theorems are really important because of their applications.Banach fixed point result is the most famous fixed point theorem.It has only two conditions, underlying mapping must be contraction and involved space must be complete.Its wide applications to different fields of mathematics as well as out side mathematics are well known.Researchers generalized this great theorem in two ways: either by weakening the involved contraction condition or via generalizing the metric structure.Banach fixed point theorem is not satisfied by the arbitrary nonexpansive mapping which is very natural class of mappings.Hence in 1965, Kirk [1] proved a very basic existence results in respect of nonexpansive mappings.After this, many generalization of nonexpansive mappings came into picture.Generalizations due to Suzuki [2], Garcia et al. [3] and Aoyama and Kohsaka [4] are worth mentioning.It is worth mentioning that nonexpansive mappings are always continuous but geenralized nonexapnsive mappings need not be continuous in general.In 2017, Pant and Shukla [5] introduced the class of generalized α-nonexpansive mappings and established some existence and convergence theorems for the newly introduced class of mappings.
Anothor direction of fixed point theory is to construct iteration process to reach fixed points of nonlinear mappings.Mann iteration [7] and Ishikawa iteration [8] are one and two step process.In 2000, Noor studied the convergence criteria of the three-step iteration method for solving general variational inequalities and related problems.Glowinski and Le Tallec [9] employed three-step iterative approaches to find solutions for the problem of elastoviscoplasticity, eigenvalue computation and the theory of liquid crystals.In [9], it was shown that the three-step iterative process yields better numerical results than the estimated iterations in two and one steps.For concrete application of fixed point iteration process one can see [10][11][12][13][14]. Owing to importance of these study, many three step iteration due to Noor [15], Agarwal et al. iteration [16], Abbas and Nazir iteration [17], Thakur et al. iterations [18,19], M* iteration [20], M iteration [21], K iteration [22] and K* iteration [23] came into picture.
Motivated by foregoing studies, Ullah et al. [24] and Temir and Korkut [25] introduced a new iteration involving generalized α-nonexpansive mappings.If S is a is a mapping on convex subset K of a Banach space E, then the process as follows: , where {φ n } and {ψ n } are sequences in (0,1).Authors showed that their process converges faster than the many known iterations.
In this paper, we prove some convergence results involving the iteration process (1.1) for generalized α-Reich-Suzuki nonexpansive mappings which is bigger class of mappings than the class of generalized α-nonexpansive mappings.Thus, our results are the genuine generalization of results of Ullah et al. [24] and Temir and Korkut [25].Further, we construct a numerical example which justify the our findings over the the existing iteration processes.In the last, we provide the solution of split feasibility problem.

Preliminaries
For making our paper self contained, we collect some basic definitions and needed results.
It is known that, in a uniformly convex Banach space, A(K,{a n }) consists of exactly one point.The following lemma due to Schu [26] is very useful in our subsequent discussion.
Lemma 1.Let E be a uniformly convex Banach space and {t n } be any sequence such that 0 1 d d p t q n for some p q , ∈  and for all n ≥ 1.Let {a n } and {b n } be any two sequences of E such that lim sup Recently, Pandey et al. [6] introduced generalized α-Reich-Suzuki nonexpansive mapping which properly contains the Reich-Suzuki nonexpansive and generalized α-nonexpansive mappings.Definition runs as follows: is said to be generalized α-Reich-Suzuki nonexpansive mapping if there exist an D [ , ) 0 1 and for each x y K , ∈ where and The following results are very important to get our results: Lemma 2. [6] Let K be a nonempty subset of a Banach space E and The above lemma shows that generalized α-Reich-Suzuki nonexpansive mapping satisfies condi- . Therefore generalized α-nonexpansive, Reich-Suzuki type nonexpansive and generalized α-Reich-Suzuki nonexpansive mapping belong to the class of mappings satisfying the condition (E).

Lemma 3. [19]
Let  be ageneralized α-Reich-Suzuki nonexpansive mapping defined on a nonempty closed subset K of a Banach space E with the Opial property.If a sequence {a n } converges weakly to c and lim ,

Convergence Results
First, we prove few lemmas which will be useful in obtaining convergence results.A mapping  : K K → is said to satisfy the Condition (A) ( [28]) if there exists a nondecreasing func- || Now, we present a strong convergence result using the Condition (A).Theorem 3. Let K be a nonempty closed convex subset of a uniformly convex Banach space E and  : K K → be a generalized α-Reich-Suzuki nonexpansive mapping with F ( ) . z If  satisfies the Condition (A) and {a n } is defined by the iteration process (1.1), then {a n } converges strongly to a point of F( ) Also, by Lemma 5 we have lim It follows from the Condition (A) that lim lim Since f is a non decreasing function satisfying f ( ) 0 0 = and f r ( ) > 0 for all r f ( , ) 0 , therefore lim By Theorem 2, the sequence {a n } converges strongly to a point of F( ).

Numerical Example
In this section, we construct following example of a generalized α-Reich Suzuki nonexpansive mapping.[ , ] .
For x = 1 11 and y = 1, we have Thus  does not satisfy condition (C).Now we show that  satisfies the condition (E).We consider different cases as follows:    Thus  satisfy the condition (E) with P t 1 and has a fixed point 0. Now, using above example, we will show that Iteration (1.1) converges faster than S-iteration, Abbas iteration, Thakur New iteration, M-iteration and K*-iteration.Let D n 0 75 ., E n 0 45 ., J n 0 15 .for all n ∈  and x 1 0 3 = ., then we get the following Table 1 of iteration values and graph.

Application
If and nonempty set.Censor and Elfving [29] solved the class of inverse problems by using SFP.In 2002, Byrne [30] introduced CQ-algorithm for solving the SFP.In this, the iterative step a k is calculated as follows: where 0 2 Any a C ∈ is the solution of SFP if and only if it solves the following fixed point equation: For details, one can refer [32,33].
As  is a nonexpansive map and owing to Lemma 6, it generalized α-nonexpansive mapping for α = 0. Hence by using Theorem 1, we get the following main result:

Lemma 4 .
Let  be a generalized α-Reich-Suzuki nonexpansive mapping defined on a nonempty closed convex subset K of a Banach space E with F ( ) . z Let {a n } be the iterative sequence defined by the iteration process (1.1).Then, lim exists for all p F ∈ ( ). Proof.Let p F ∈ ( ).In view of second part of Lemma 2,  is quasi-nonexpansive hence we have

Theorem 4 .Theorem 5 .
If {a n } is the sequence generated by the iterative algorithm (1.1) with  then, {a n } converges weakly to the solution of SFP(5.1).By using Theorem 2, we have the following convergence theorem: If {a n } is the sequence generated by the iterative algorithm (1.1) with  n } converges strongly to the solution of SFP(5.1)  if and only if lim inf ( , )In this paper we studied the convergence behaviour of an iteration scheme introduced by Ullah et al. and Temir and Korkut[25] in respect of generalized α-Reich-Suzuki nonexpansive mapping.Theorems 1, 2 and 3 of our paper are main convergence theorems which generalized the Theorems 3.3 and 3.4 of Ullah et al.[24] and Theorem 2.2 and 2.3 of Temir and Korkut[25].
be a nonempty closed convex subset of a Banach E, and let {a n } be a bounded sequence in E.
If {a n } is the iterative sequence defined by the iteration process (1.1), {a n } converges weakly to a fixed point of  .If {a n } is defined by the iteration process (1.1), then {a n } converges strongly to a point of F( ) if and only if lim inf ( , ( )) .For the converse part, assume that lim inf ( , ( )) .
Lemma 5. Let  be a generalized α-Reich-Suzuki nonexpansive mapping defined on a nonempty closed convex subset K of a uniformly convex Banach space E. Let {a n } be the iterative sequence defined by the iteration process(1.1).Then, F ( )  z if and only if {a n } is bounded and lim .This implies that p A K a n ∈ ( ,{ }).Since E is uniformly convex, A K a n ( ,{ }) is singleton, therefore we get p p = .Theorem 1.Let  be a generalized α-Reich-Suzuki nonexpansive mapping defined on a nonempty closed convex subset K of a Banach space E which satisfies the Opial's condition with F ( ) . z o f || || exists.In order to show the weak convergence of the iteration process (1.1) to a fixed point of  , we will prove that {a n } has a unique weak sub sequential limit in F( ) For this, let { } a n j and { } a n k be two subsequences of {a n } which converges weakly to u and v respectively.By Lemma 4, we have lim of  0 Proof.If the sequence {a n } converges to a point p F ∈ ( )  , then it is obvious that lim inf ( , ( )) . of  0 So, {u j } is a Cauchy sequence in F ( ).  By Lemma 2, F ( )  is closed, so {u j } converges to some u F ∈ ( ). of || || exists by Lemma 4, therefore {a n } converges strongly to u F ∈ ( )  .
C and Q are nonempty, closedand convex subsets of two real Hilbert spaces H 1 and H 2 respectively