Existence and convergence of best proximity points for generalized pseudo-contractive and Lipschitzian mappings via an Ishikawa-type iterative scheme

In this article, we prove the existence of the best proximity point for the class of nonself generalized pseudo-contractive and Lipschitzian mappings. Also, we approximate the best proximity point through the proposed Ishikawa’s iteration process for the case of nonself-mappings. Finally, we provide an example to illustrate our main result


Introduction
Assume that M and N are nonempty subsets of a metric space (X, d).If M ∩ N = ∅, then the mapping f from M to N does not have a solution for the fixed-point equation f (η) = η.When the fixed-point equation does not possess a solution, then it is attempted to determine an approximate solution η such that the error d(η, f η) is minimum.In this situation, the best proximity-point theorems guarantee the existence and uniqueness of such an optimization for the fixed-point equations.Naturally, the best proximity point for the nonself-mappings is defined as follows: Definition 1.1 Let M, N be two nonempty and disjoint subsets of a metric space (X, d).
A mapping : M → N is said to have a best proximity point if there exist Many researchers have proved the existence results on the best proximity points for various kinds of contractions.For such results, one may refer to [2, 4, 6-8, 12, 13, 15-18].Recently, researchers have shown an interest in approximating the best proximity points through well-known iterative processes that may be seen in [1, 3, 9-11, 14, 19, 20].
On the other hand, numerous research articles have been published on the convergence of fixed points for the class of self-and nonself-contractive-type mappings in metric spaces, Hilbert spaces, and several classes of Banach spaces.For further exploration of this topic, we refer to the monograph [5] and the references cited therein.
A fundamental result in metric fixed-point theory is the following theorem, which uses the Picard iteration method.Theorem 1.2 [5] Let (X, d) be a complete metric space and : X → X be a contraction, that is an operator satisfying d( η, ω) ≤ ad(η, ω), for any η, ω ∈ X, with a ∈ [0, 1) fixed.Then, has a unique fixed point.
One of the effective methods for approaching the fixed point of a mapping : X → X is the Ishikawa iteration scheme, starting with any η 0 ∈ X and for n ≥ 0 defined by where γ n , δ n ∈ [0, 1].In this direction, we state the following theorem on the iterative approximation of a fixed point that was proved by Ishikawa [11], for Lipschitzian pseudocontractive mapping.
Theorem 1.3 [11] Let K be a convex and compact subset of a Hilbert space H and let : K → K be Lipschitzian pseudo-contractive and let η 1 ∈ K .Then, the Ishikawa iteration {η n }, defined by where {γ n }, {δ n } are sequences of positive numbers satisfying converges strongly to a fixed point of .
The next result gives sufficient conditions to obtain a fixed point without assuming the Lipschitzian condition.
Theorem 1.4 [5] Let K be a closed, bounded, and convex subset of a real uniformly convex Banach space H. Let : K → K a strongly pseudo-contractive that has at least a fixed point η * .Let η 1 ∈ K , then the Ishikawa iteration {η n }, defined by where {γ n }, {δ n } are sequences of positive numbers satisfying converges strongly to a fixed point of .
Motivated by Theorems 1.3 and 1.4, a natural question arises: how can one construct the Ishikawa iteration for nonself-mappings that approximate the best proximity point of such mappings?In this context, we will initiate the construction of the Ishikawa iteration process for nonself-mappings and investigate the convergence results for the best proximity point.
Before presenting the iterative approximation for the best proximity point, let us establish the existence of a best proximity point.To do so, we will recall some basic notions and definitions: Let M and N be two subsets of a Hilbert space H with inner product •, • and norm • : In [13], Kirk et al. proved the following lemma that guarantees the nonemptiness of M 0 and N 0 .Lemma 1.5 Let X be a reflexive Banach space and M be a nonempty, closed, bounded, and convex subset of X, and N be a nonempty, closed, and convex subset of X.Then, M 0 and N 0 are nonempty and satisfy P N (M 0 ) ⊆ N 0 , P M (N 0 ) ⊆ M 0 .Definition 1. 6 Let H be a Hilbert space with inner product •, • and norm • .An operator : H → H is said to be Lipschitzian if there exists a constant s > 0 such that, for all η, ω in H, ηω ≤ s ηω .Definition 1.7 Let H be a Hilbert space with inner product •, • and norm • .An operator : H → H is said to be a generalized pseudo-contraction if there exists a constant r > 0 such that, for all η, ω in H, ( 1 ) 2. If r = 1, then a generalized pseudo-contraction reduces to a pseudo-contraction.
Definition 1.9 Let H be a Banach space with norm • .An operator : H → H is said to be strongly pseudo-contraction if there exists a constant t > 1 such that holds for all η, ω in H and c > 0.
In this work, we begin by providing a set of sufficient conditions for the existence of a best proximity point for nonself-Lipschitzian, generalized pseudo-contractive mappings.Subsequently, we construct the Ishikawa iteration for nonself-mappings and establish convergence results for the best proximity point of Lipschitzian pseudo-contractive nonselfmappings.To support our main result, we present an illustrative example.
Furthermore, we delve into the convergence of the best proximity point for strongly pseudo-contractive mappings without imposing the Lipschitzian condition.This discussion expands the scope of our findings and highlights the applicability of our results in a broader class of mappings.

Main results
Let us prove the existence result of the best proximity point for nonself-generalized pseudo-contractive and Lipschitzian mapping in the Hilbert space settings.Theorem 2.1 Let M, N be two closed and convex subsets of a real Hilbert space H assume M to be bounded.Let : M → N be a generalized, pseudo-contractive, and Lipschitzian mapping with corresponding constants r and s such that 0 < r < 1, s > 1.If (M 0 ) ⊆ N 0 , then has a unique best proximity point.
Remark 2.2 1.If 0 < s < 1, then is a contraction nonself-mapping and the result follows from [15]. 2. If s = 1, then is a nonexpansive nonself-mapping and the result follows from [18].Now, we define a construction of Ishikawa iteration for the case of nonself-mapping: Let M, N be two convex subsets of a Hilbert space H. Let us define : M → N and assume (M 0 ) ⊆ N 0 .Consider the projective operator P M 0 : M 0 → M 0 .Let η 1 ∈ M 0 , then the Ishikawa iteration {η n }, is defined by where γ n , δ n ∈ [0, 1].Next, we extend the convergence result of Theorem 1.3, for the case of nonselfmappings, by using the proposed Ishikawa iteration for nonself-mappings.