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A new iterative approximation scheme for Reich–Suzuki-type nonexpansive operators with an application

In this article, we propose a faster iterative scheme, called the AH iterative scheme, for approximating ﬁxed points of contractive-like mappings and Reich–Suzuki-type nonexpansive mappings. We show that the AH iterative scheme converges faster than a number of existing iterative schemes for contractive-like mappings. The w 2 -stability result of the new iterative scheme is established and a supportive example is provided to illustrate the notion of w 2 -stability. Then, we prove weak and several strong convergence results of our new iterative scheme for ﬁxed points of Reich–Suzuki-type nonexpansive mappings. Further, we carry out a numerical experiment to illustrate the eﬃciency of our new iterative scheme. As an application, we use our main result to prove the existence of a solution of a mixed-type nonlinear integral equation. An illustrative example to validate the result in our application is also given. Our results extend and generalize several related results in the existing literature.


Introduction
Throughout the paper, let N be the set of all natural numbers, R a set of all real numbers, V a nonempty subset of a Banach space M. A mapping U : V → V is called a contraction if there exists ζ ∈ [0, 1) such that Up -Uq ≤ ζ pq , for all p, q ∈ V. If ζ = 1, then U is called a nonexpansive mapping. A point p ∈ V is said to be a fixed point of U if it satisfies Up = p . We denote by (U) the set of all fixed points of U.
The major ideas of fixed-point theory can be divided into two categories. One is to find the necessary and sufficient conditions under which an operator admits fixed points. The other is to determine such fixed points by using some schematic algorithms. Formally, the first category is usually referred to as the existence part and the second one is known as the computation or approximation part. Another important concept of fixed-point theory that is less well known is the study of the behaviors of fixed points such as stability, data dependency, etc.
For some decades now, the fixed-point theory has been revealed as a very powerful and useful tool in the study of nonlinear phenomenon. In particular, fixed-point techniques have been applied in diverse areas of biology, chemistry, economics, engineering, game theory, physics, etc., (see [4, 5, 13, 23-26, 30, 31] and the references therein).
In [6], Berinde provided the class of weak contractions that properly includes the class of Zamfirescu operators [47]. This class of mappings is also known by many authors as almost contraction mappings. Definition 1.1 A mapping U : V → V is called a weak contraction if there exist ζ ∈ (0, 1) and L ≥ 0 such that Up -Uq ≤ ζ pq + L p -Up , for all p, q ∈ V. (1.1) In [17], Imoru and Olantiwo gave a definition of a class of mapping considered to be a generalization of the classes of mappings studied by Berinde [6], Osilike and Udomene [28] and some other existing classes of contraction mappings as follows. In recent years, many extensions and generalizations of nonexpansive mappings have been studied by several authors due to their importance in terms of applications.
In 2008, Suzuki [35] introduced an interesting generalization of nonexpansive mappings and obtained some existence and convergence results. Such mappings are commonly known as mappings satisfying condition (C).

Definition 1.4 A mapping
In 2019, Pant and Pandey [29] considered the class of Reich-Suzuki-type nonexpansive mappings as follows. Definition 1.5 A mapping U : V → V is said to be Reich-Suzuki-type nonexpansive if there exists a real number ∈ [0, 1) such that for each p, q ∈ V, (1.4) Clearly, every mapping satisfying condition (C) is a Reich-Suzuki-type nonexpansive mapping with = 0. However, the converse is not true, as shown in [29].
Very recently, Ahmad et al. [3] introduced a novel iterative scheme known as the JK iterative scheme as follows: where {r v } and {k v } are sequences in (0, 1). The authors established some weak and strong convergence results for mappings satisfying condition (C). They further showed numerically that the JK iterative scheme converges faster than the S [2] and Thakur [37] iterative schemes.
Motivated and inspired by the research in this direction, we propose a new fourstep iterative scheme called the AH iterative scheme, to approximate the fixed points of contractive-like mappings and Reich-Suzuki-type nonexpansive mappings as follows: where {r v } and {k v } are sequences in (0, 1). The purpose of this article is to prove that the AH iterative scheme (1.6) converges faster than the JK iterative scheme (1.5) for contractive-like mappings. Numerically, we further show that the AH iterative scheme (1.6) converges faster than a number of existing iterative schemes. Also, we prove that our proposed iterative scheme defined by (1.6) is w 2 -stable and the stability result is supported with an example. Again, we establish weak and strong convergence results of the AH iterative scheme (1.6) for Reich-Suzuki-type nonexpansive mappings. Further, we use a new example of Reich-Suzuki-type nonexpansive mappings to show that the AH iterative scheme (1.6) outperforms some existing prominent iterative schemes. Finally, we use our main results to establish the existence of the solution of a nonlinear integral equation in Banach spaces.

Preliminaries
Let M * be the dual of a Banach space M and ·, · denotes the generalized duality pairing between M and M * . Then, the multivalued mapping J : M → 2 M * is the normalized duality mapping defined for each p ∈ M by J(p) = q ∈ M * : p, q = p 2 = q 2 . (2.1) Then, a Banach space M is said to be smooth if the limit exists for each p, q ∈ D. In this case, the norm of M is called Gâteaux differentiable. It is well known that J is single valued if M is smooth [12]. Suppose for each p ∈ D, the limit of (2.2) exists and is attained uniformly for q ∈ D, the norm of M is called Fréchet differentiable in this case. It is also well known that for all p, q ∈ M, where J is the Fréchet derivative of the functional 1 2 · 2 at p ∈ M and b is an increasing function defined on [0, ∞) such that lim v↓0 A Banach space M is said to be uniformly convex if for each ∈ (0, 2], there exists δ > 0 such that for p, q ∈ M satisfying p ≤ 1, q ≤ 1 and pq > , we have p+q Let V be a nonempty closed convex subset of a Banach space M, and {p v } is a bounded sequence in M. For p ∈ M, we put The asymptotic radius of {p v } relative to V is defined by The asymptotic center of {p v } relative to V is given as: In a uniformly convex Banach space, it is well known that A(V, {p v }) consists of exactly one point.
Let V be a nonempty closed convex subset of a Banach space M. A mapping U : V → V is said to be demiclosed with respect to p ∈ M, if for each sequence {p v } that is weakly convergent to p ∈ V and {Up v } converges strongly to q implies that Up = q. Definition 2.1 ([7]) Let {δ v } and {γ v } be two sequences of real numbers that converge to δ and γ , respectively, and assume that there exists

Lemma 2.6 ([33]) Suppose M is a uniformly convex Banach space and {ι
If U is a Reich-Suzuki-type nonexpansive mapping with (U) = ∅, then the following hold: (i) If U is a Reich-Suzuki-type nonexpansive mapping, then for every choice of p ∈ V and p ∈ (U), it follows that Up -Up ≤ pp . (ii) If U satisfies condition (C), then U is a Reich-Suzuki-type nonexpansive mapping.
If U is a Reich-Suzuki-type nonexpansive mapping, then for all p, q ∈ V, the following inequality holds: We now offer a numerical example that satisfies the inequality of the above lemma but does not satisfy the condition (C). Example 2.9 Let (R, · ) be a Banach space with the usual norm and (1) The mapping U does not satisfy the condition (C) and hence is not a nonexpansive mapping. If we take p = 1 3 and q = 1, then On the other hand, (2) Now, we show that U satisfies condition (2.4). For this, the following conditions are considered: Case IV: If p ∈ [-1, 0) and q = 1 3 , we have Hence, U satisfies the condition (2.4) with 3+ 1-≥ 1.

Rate of convergence
In this section, we show both analytically and numerically that the AH iterative scheme (1.6) converges faster than the JK iterative scheme (1.5) for contractive-like mappings.
Now, we give a nontrivial example to compare the rate of convergence of the AH iterative scheme (1.6) with some leading iterative schemes in the literature.
Using MATLAB R2015a, we obtain Tables 1-3 and Fig. 1. From Tables 1-3, we can easily see that all the iterative schemes with control parameters r v = 0.8, k v = 0.6, l v = 0.5, v ∈ N and starting point (2, 2.5, 3) converge to p = (0, 0, 0). Obviously, our iterative scheme (1.6) requires the least number of iterations as compared to other iterative schemes. Also, from the graphical point of view in Fig. 1, it is evident that the AH iterative scheme (1.6) converges faster than other iterative schemes.
In [8], Berinde observed that the concept of stability in Definition 4.1 is not precise because of the sequence {h v } that is arbitrarily taken. To overcome this limitation, Berinde [8] observed that it would be more natural if {h v } were an approximate sequence of {p v }. Therefore, any iteration process that is stable will also be weakly stable but the converse is generally not true.
then we shall say that (4.3) is weakly U-stable or weakly stable with respect to U.
In 2010, Timis [38] studied a new concept of weak stability that is obtained from Definition 4.3 by replacing the approximate sequence with the notion of the equivalent sequence that is more general.
then we shall say that (4.4) is weakly w 2 -stable with respect to U.
It is shown in [38] with an example that any equivalent sequence is an approximative sequence but the reverse is not true.
In this section, we prove that the AH iterative scheme (1.6) is w 2 -stable with respect to U for contractive-like mappings.
In order to support the analytical proof of Theorem 4.6, we provide the following illustrative example.
It is not difficult to see that which shows that lim v→∞ p vh v = 0. If follows that {p v } and {h v } are equivalent sequences.
Clearly, lim v→∞ v = 0. Therefore, the sequence {p v } generated by the AH iterative scheme (1.6) is w 2 -stable with respect U.

Convergence results
In this section, we prove weak and strong convergence theorems of the AH iterative scheme (1.6) for Reich-Suzuki-type nonexpansive mappings.  In particular, inf{ p ρ 0p : p ∈ (U)} < ε 2 . Therefore, there exists p ∈ (U) such that This implies that the sequence {p v } is Cauchy in V. Since V is closed, there must be an element q ∈ V such that lim v→∞ p v = q. Now, lim v→∞ d(p v , (U)) = 0 gives that d(q, (U) = 0, that is q ∈ (U).
A strong convergence on a compact domain is established in the following way: As i → ∞, one can see that p v i → Uq. It follows that Uq = q, i.e., q ∈ (U). According to Lemma 5.1, lim v→∞ p vq exists, that is, q forms a strong limit for {p v }.
A strong convergence theorem using a condition (I) of the operators is the following: Up vp v = 0. Since all the requirements of Theorem 5.5 are shown, one concludes that the sequence {p v } is strongly convergent in the fixed-point set of U.

Numerical result
In this section, we give an example of a Reich-Suzuki-type nonexpansive mapping that does not satisfy the condition (C). Further, we compare the convergence of the AH iterative scheme with some leading iterative schemes in the literature.
Example 6.1 Let (R, · ) be a Banach space with the usual norm and V = [5,7]. Let U : V → V be a mapping defined by if p = 7.
(6.1) (i) The mapping U does not satisfy the condition (C). For this, let p = 6 and q = 7, we have On the other hand, (ii) Now, to demonstrate that U is a Reich-Suzuki-type nonexpasive mapping, the following cases are considered.
Hence, U is a Reich-Suzuki-type nonexpansive mapping and has fixed point 5.
From Table 4 and Fig. 2, it can be clearly seen that the AH iterative converges faster to the fixed point of U than other iterative schemes.   Table 4 7 Application In this section, we consider the application of our main results to the following nonlinear mixed Volterra-Fredholm-type integral equation: contractive-like mappings. We have provided an example to illustrate the notion of w 2stability of the AH iterative scheme with respect to U. Also, we have proved weak and several strong convergence theorems for Reich-Suzuki-type nonexpansive mappings in uniformly convex Banach spaces. A new example of Reich-Suzuki-type nonexpansive mappings has been provided to compare the convergence behavior of the AH iterative method (1.6) with some well-known iterative schemes. As an application, we used our main results to establish the existence of solution of a mixed-type nonlinear integral equation. Finally, we illustrated the result in our application with an interesting example.