Some New Results on a Three-Step Iteration Process

: The purpose of this research work is to prove some weak and strong convergence results for maps satisfying ( E ) -condition through three-step Thakur ( J. Inequal. Appl. 2014 , 2014:328.) iterative process in Banach spaces. We also present a new example of maps satisfying ( E ) -condition, and prove that its three-step Thakur iterative process is more efﬁcient than the other well-known three-step iterative processes. At the end of the paper, we apply our results for ﬁnding solutions of split feasibility problems. The presented research work updates some of the results of the current literature.


Introduction
Let T be a selfmap on a subset W of a Banach space U = (U, ||.||). Subsequently, T is called contraction map on W if for each pair of elements w, w ∈ W, there is some real constant α ∈ [0, 1), such that ||Tw − Tw || ≤ α||w − w ||.
If (1) holds at α = 1, then T is called non-expansive. When a point g ∈ W exists with the property Tg = g, then g is called a fixed point of T. The fixed point set of T we often denote by the notation Fix(T). In 1922, Banach [1] proved that any self contraction map of a closed subset W of a Banach space has a unique fixed point. Later, the Banach result [1] was extended by Caccioppoli [2] in complete metric spaces. In 1965, Browder [3] and Gohde [4] proved that any self non-expansive map of a convex bounded closed subset W of a uniformly convex Banach space U always admits a fixed point. The Browder-Gohde result was proved by Kirk [5] in the context of reflexive Banach spaces. We know that the class of non-expansive maps is important as an application point of view. Thus, it is very natural to consider larger classes of non-expansive maps. One of the larger class of non-expansive maps was introduced by Suzuki [6] in 2008. A selfmap T on a subset W of a Banach space is said to be Suzuki map (or said to satisfy (C)-condition), if for each pair of elements w, w ∈ W, it follows that 1 2 ||w − Tw|| ≤ ||w − w || ⇒ ||Tw − Tw || ≤ ||w − w ||.
Suzuki also proved that, if a map T satisfies (C)-condition, then for all w, w ∈ W, ||w − Tw || ≤ 3||w − Tw|| + ||w − w || holds. Inspired by Suzuki (C)-condition, Garcia-Falset et al. [7] introduced (E)-condition, as follows: a selfmap T on a subset W of a Banach space is said be Garcia-Falset map (or said to satisfy (E)-condition), if for each pair of elements w, w ∈ W, there is some real constant µ ≥ 1, such that We see that any map T with (C)-condition satisfies (E)-condition with real constant µ = 3. Nevertheless, an example in the Section 4 shows that there exists maps in the class of Garcia-Falset maps which does not belong to the class of Suzuki maps. Hence, the class of Garica-Falset maps properly includes the class of Suzuki maps. Garcia-Falset et al. [7] also proved some existence theorems of fixed points for maps satisfying (E)-condition. Recently, Usurelu et al. [8] studied some fixed point results for this class of maps and using an example, they studied the visualization of convergence behaviors of some iterative processes. In this paper, we use the three-step iterative process, which is different from the iterative process used in [8] for approximating fixed points of maps of this class. We also present a new example of maps satisfying (E)-condition, and prove that its under the consideration three-step iterative process is more efficient than the other well-known three-step iterative processes. In the last section, we shall apply our results for finding solutions of split feasibility problems.
However, once the existence of fixed point for an operator is established, then the finding of this fixed point is not easy work. One of the simplest iterative method for finding fixed points is the Picard iterative method, which is, w k+1 = Tw k . The Banach-Caccioppoli result states that the unique fixed point of contractions can be obtained by using the Picard iterative method. Nevertheless, the Picard iterative method does not always work properly in the finding of fixed points of non-expansive maps. For finding fixed points of non-expansive maps and to obtain relatively better convergence speed, one deals with the different iterative methods, e.g., Mann [9], Ishikawa [10], Agarwal [11], Noor [12], Abbas [13], and others. Among the other things, Thakur et al. [14] introduced the following three-step iterative process for finding fixed points of non-expansive maps in Banach spaces, as follows: where a k , b k , c k ∈ (0, 1).
In [14], Thakur et al. proved some important strong and weak convergence theorems of the iterative process (2) for the class of non-expansive maps in the context of uniformly convex Banach spaces. Recently in 2020, Maniu [15] extended the results of Thakur et al. [14] to the setting of Suzuki maps. The purpose of this research is to extend the results of Maniu [15] to the more general setting of Garcia-Falset maps. We also study the rate of convergence of the iterative process (2) with the some well-known three-step iterative processes in the setting of Garcia-Falset maps, under different initial points and set of parameters. At the end of the paper, we shall apply our results to find the solution of split feasibility problems.

Preliminaries
In this section, we shall deal with some basic definitions and early results. Let W be any nonempty subset of a Banach space U. Fix p ∈ U and assume that {w k } ⊆ U is bounded. Define r(p, {w k }) by We denote the asymptotic radius of {w k } with respect to W by r(W, {w k }) and define, as follows: We denote the asymptotic center of {w k } with respect to W by A(W, {w k }) and define, as follows: The asymptotic center of {w k } with respect to W is nonempty and convex whenever W is convex weakly compact, (see, e.g., [16,17] and others). One of the well known property of the set A(W, {w k }) is the singletoness property in the frame work of uniformly convex Banach spaces [18].
Recall that a Banach space U is said to have Opial's property [19], if, for any weakly convergent sequence {t k } in U with a weak limit t ∈ U, follows the following strict inequality The following result shows that the class of Suzuki maps is a sub-class of Garcia-Falset maps. Lemma 1. [7] Let W be a nonempty subset of a Banach space and let T : W → W satisfies (C)-condition. Subsequently, T satisfies (E)-condition with µ = 3.

Lemma 3. [7]
Let T be a selfmap on a subset W of a Banach space having Opial property. Let T satisfy the (E)-condition. If {w k } is weakly convergent to g and lim k→∞ ||w k − Tw k || = 0, then g ∈ Fix(T).
The following characterization is due to Schu [20].

Convergence Results in Banach Spaces
This section contains some weak and strong convergence results of the iterative process (2) for operators satisfying (E)-condition. Throughout the section, U will stand for uniformly convex Banach space.

Lemma 5.
Let W be a nonempty convex closed subset of U and T : W → W be a map satisfying (E)-condition with Fix(T) = ∅. If {w k } is generated by (2), then lim k→∞ ||w k − g|| exists for every g ∈ Fix(T).
Proof. Let g ∈ Fix(T). By Lemma 2, we have While using the above inequilities, we have Thus, {||w k − g||} is bounded and non-increasing, which implies that lim k→∞ ||w k − g|| exists for each g ∈ Fix(T). Now, we establish the following result which will be used throughout in the upcoming theorems. Theorem 1. Let W be a nonempty closed convex subset of U and let T : Proof. Let {w k } be bounded and lim k→∞ ||Tw k − w k || = 0. Let g ∈ A(W, {w k }). We shall prove that Tg = g. Since T satisfies (E)-condition, we have It follows that Tg ∈ A(W, {w k }). Since A(W, {w k }) is singleton set, we have Tg = g. Hence, Fix(T) = ∅.
Conversely, we assume that Fix(T) = ∅ and g ∈ Fix(T). We shall prove that {w k } is bounded and lim k→∞ ||w k − Tw k || = 0. By Lemma 5, lim k→∞ ||w k − g|| exists and {w k } is bounded. Put From the proof of Lemma 5, it follows that By Lemma 2, we have lim sup Again, from the proof of Lemma 5, It follows that Accordingly, we can get ||w k+1 − g|| ≤ ||z k − g||.
From (4) and (6), we get From (7), we have Applying Lemma 4, we obtain Using compactness of the domain W, we establish the following strong convergence of {w k } generated by (2) for maps satisfying (E)-condition. Proof. By compactness of W we can construct a subsequence {w k l } of {w k } such that lim k→∞ ||w k l − u|| = 0, for some u ∈ W. Because the map T satisfies (E)-condition, one can find some real constant µ ≥ 1, such that ||w k l − Tu|| ≤ µ||w k l − Tw k l || + ||w k l − u||.
In the view of Theorem 1, lim k→∞ ||w k l − Tw k l || = 0. Now, using lim k→∞ ||w k l − Tw k l || = 0 and lim k→∞ ||w k l − u|| = 0, we have from (8), lim k→∞ ||w k l − Tu|| = 0. Now, the uniqueness of limits in Banach space follows that Tu = u. Hence, u is the fixed point of T. By Lemma 5, lim n→∞ ||w k − u|| exists. Hence, u is the strong limit of {w k }. Proof. The proof is elementary and, hence, omitted.
The next theorem requires condition I of Sentor and Dotson [21]. The detail definition is given below. Definition 1. [21] Let W be a nonempty subset of U. A selfmap T of W is said to satisfy condition I if there is a nondecreasing function ξ with the properties ξ(a) = 0 if and only if a = 0, ξ(a) > 0 for every a ∈ (0, ∞) and ||w − Tw|| ≥ ξ(dist(w, Fix(T))) for each w ∈ W.
From (9) Proof. By Theorem 1, the sequence {w k } is bounded and lim k→∞ ||Tw k − w k || = 0. U is reflexive because U is uniform convex. Now, by reflexivity of U, we can construct a weakly convergent subsequence {w k s } of {w k } with some weak limit u 1 ∈ W. By Lemma 3, we conclude that u 1 ∈ Fix(T). We claim that {w k } converges weakly to u 1 . Assume that u 1 is not the weak limit of {w k }. Accordingly, we choose another weakly convergent subsequence {w k t } of {w k } with some weak limit u 2 ∈ W and assume that u 2 = u 1 . Again, by Lemma 3, u 2 ∈ Fix(T). Using Lemma 5 and Opial condition, we have This is a contradiction. Hence u 1 is the weak limit of {w k } and fixed point of T.

Numerical Example and Rate of Convergence
Example 1. Define a selfmap T on W = [0, 1] as follows: First, we are going to show that T belongs to the class of Garcia-Falset maps, that is, we shall show that |w − Tw | ≤ µ|w − Tw| + |w − w | for each pair of elements w, w ∈ W and some real constant µ ≥ 1. Fix µ = 5 and consider the following cases.
(i): For w, w ∈ W 1 , we have (iii): For w ∈ W 2 and w ∈ W 1 , we have From the above cases, one can conclude that T belongs to the class of Garcia-Falset maps. Next, we show that T does not belong to the class of Suzuki maps. We select w = 1 800 and w = 1 500 . Susbequently,  Table 1 shows that three-step Thakur iteration process [14] converges faster to the fixed point g = 0 as compared three-step Abbas [13] and three-step Noor [12] iterative processes. Table 1. Strong convergence of three-step iterative processes while using T given in Example 1. In Example 1, we set different values for parameters a k , b k and c k and set stopping criterion w k − w * < 10 −15 , where the element w * = 0 is a unique fixed point of T. The influence of initial guess and parameter for the three-step Thakur [14], three-step Abbas [13], and three-step Noor [12] iterative processes can be seen in the Tables 2-4.

Application
In this section, we are interested in finding of the solution of a split feasibility problem (in short SFP) by using the three-step iterative method (2). To do this, we assume that U 1 and U 2 are any two real Hilbert spaces, ∅ = C ⊆ U 1 and ∅ = Q ⊆ U 2 be convex and closed. Assume that L : U 1 → U 2 be a linear and bounded. Subsequently, the SFP mathematically described as finding an element w ∈ C, such that w ∈ C, Lw ∈ Q.
Next we assume that the solution set Ω associated with the SFP (10) is nonempty and let We see that the set Ω is nonempty convex as well as closed. Censor and Elfving [22] solved the class of inverse problems with the help of SFP. In the year 2002, Byrne [23] proposed the remarkable CQ-algorithm for solving the SFP. If γ ∈ (0, 2 ||L|| 2 ), P C and P Q represent the projections onto C and Q respectively and L * : U * 2 → U * 1 is the adjoint of L, then the sequence {w k } of CQ-algorithm is generated iteratively, as follows: The following facts are in [24]. Lemma 6. If T = P C [I − γL * (I − P Q )L], for γ ∈ (0, 2 ||L|| 2 ), then T is non-expansive.
By assumption, the set Ω that is associated with a SFP is nonempty, one can see that the element w * ∈ C is the solution of SFP if and only if it solves the following fixed point equation: Hence, the solution set Ω coincides with the fixed point set of the operator T, that is, Fix(T) = Ω = C ∩ L −1 Q = ∅. For details, one can refer [25,26]. Now, we present our main results.
Theorem 6. Let T = P C [I − γL * (I − P Q )L] and {w k } be a sequence defined by the iterative process (2), then {w k } converges weakly to the some solution of a SFP (10).
Proof. By Lemma 6, the operator T is non-expansive. In the view of Lemma 1, T is Garcia-Falset operator. The conclusions follows from Theorem 5.
Theorem 7. Let T = P C [I − γL * (I − P Q )L] and {w k } be a sequence defined by the iterative process (2), then {w k } converges strongly to the solution of SFP (10), provided that lim inf k→∞ d(w k , Ω) = 0.
Proof. Proof follows from Theorem 3.

Conclusions
The three-step Thakur [14] iterative process converges faster than three-step Abbas [13] and three-step Noor [12] iterative process, respectively, for the example under consideration as shown in the Tables 1-3. The class of Garcia-Falset maps is wider than the class of Suzuki maps, as shown in the Example 1. Hence, our results update the results of Maniu [15] from the setting of Suzuki maps to the general setting of Garcia-Falset maps. We have also applied our results for finding solutions of split feasibility problems. Because our iterative process converges faster than Abbas and Noor iterations and the class of Garcia-Falset maps is more general than the class of non-expansive and Suzuki maps, so our results improve and extend the corresponding results in [6,7,[12][13][14][15].