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Existence and convergence of best proximity points for generalized pseudocontractive and Lipschitzian mappings via an Ishikawatype iterative scheme
Fixed Point Theory and Algorithms for Sciences and Engineering volume 2023, Article number: 19 (2023)
Abstract
In this article, we prove the existence of the best proximity point for the class of nonself generalized pseudocontractive and Lipschitzian mappings. Also, we approximate the best proximity point through the proposed Ishikawa’s iteration process for the case of nonselfmappings. Finally, we provide an example to illustrate our main result.
1 Introduction
Assume that M and N are nonempty subsets of a metric space \((X,d)\). If \(M \cap N = \emptyset \), then the mapping f from M to N does not have a solution for the fixedpoint equation \(f(\eta ) = \eta \). When the fixedpoint equation does not possess a solution, then it is attempted to determine an approximate solution η such that the error \(d(\eta , f\eta )\) is minimum. In this situation, the best proximitypoint theorems guarantee the existence and uniqueness of such an optimization for the fixedpoint equations. Naturally, the best proximity point for the nonselfmappings is defined as follows:
Definition 1.1
Let M, N be two nonempty and disjoint subsets of a metric space \((X,d)\). A mapping \(\Gamma : M \rightarrow N\) is said to have a best proximity point if there exist \(\eta ^{*} \in M\) such that \(d(\eta ^{*},\Gamma \eta ^{*}) = d(M,N)\).
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 wellknown 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 nonselfcontractivetype 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 fixedpoint theory is the following theorem, which uses the Picard iteration method.
Theorem 1.2
[5] Let \((X, d)\) be a complete metric space and \(\Gamma : X \rightarrow X\) be a contraction, that is an operator satisfying
with \(a \in [0,1)\) fixed. Then, Γ has a unique fixed point.
One of the effective methods for approaching the fixed point of a mapping \(\Gamma : X \rightarrow X\) is the Ishikawa iteration scheme, starting with any \(\eta _{0} \in X\) and for \(n \geq 0\) defined by
where \(\gamma _{n}, \delta _{n} \in [0, 1]\). 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 \(\Gamma : K \rightarrow K\) be Lipschitzian pseudocontractive and let \(\eta _{1} \in K\). Then, the Ishikawa iteration \(\{\eta _{n}\}\), defined by
where \(\{\gamma _{n}\}\), \(\{\delta _{n}\}\) are sequences of positive numbers satisfying
converges strongly to a fixed point of Γ.
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 \(\Gamma : K \rightarrow K\) a strongly pseudocontractive that has at least a fixed point \(\eta ^{*}\). Let \(\eta _{1} \in K\), then the Ishikawa iteration \(\{\eta _{n}\}\), defined by
where \(\{\gamma _{n}\}\), \(\{\delta _{n}\}\) are sequences of positive numbers satisfying
converges strongly to a fixed point of Γ.
Motivated by Theorems 1.3 and 1.4, a natural question arises: how can one construct the Ishikawa iteration for nonselfmappings that approximate the best proximity point of such mappings? In this context, we will initiate the construction of the Ishikawa iteration process for nonselfmappings 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 \(\langle \cdot , \cdot \rangle \) and norm \(\Vert \cdot \Vert \):
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 \(P_{N}(M_{0}) \subseteq N_{0}\), \(P_{M}(N_{0}) \subseteq M_{0}\).
Definition 1.6
Let H be a Hilbert space with inner product \(\langle \cdot , \cdot \rangle \) and norm \(\Vert \cdot \Vert \). An operator \(\Gamma : H \rightarrow H \) is said to be Lipschitzian if there exists a constant \(s > 0\) such that, for all η, ω in H,
Definition 1.7
Let H be a Hilbert space with inner product \(\langle \cdot , \cdot \rangle \) and norm \(\Vert \cdot \Vert \). An operator \(\Gamma : H \rightarrow H \) is said to be a generalized pseudocontraction if there exists a constant \(r > 0\) such that, for all η, ω in H,
Remark 1.8

1.
The condition (1), is equivalent to \(\langle \Gamma \eta  \Gamma \omega , \eta  \omega \rangle \leq r \Vert \eta  \omega \Vert ^{2}\).

2.
If \(r = 1\), then a generalized pseudocontraction reduces to a pseudocontraction.
Definition 1.9
Let H be a Banach space with norm \(\Vert \cdot \Vert \). An operator \(\Gamma : H \rightarrow H \) is said to be strongly pseudocontraction 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 nonselfLipschitzian, generalized pseudocontractive mappings. Subsequently, we construct the Ishikawa iteration for nonselfmappings and establish convergence results for the best proximity point of Lipschitzian pseudocontractive nonselfmappings. To support our main result, we present an illustrative example.
Furthermore, we delve into the convergence of the best proximity point for strongly pseudocontractive 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.
2 Main results
Let us prove the existence result of the best proximity point for nonselfgeneralized pseudocontractive 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 \(\Gamma : M \rightarrow N\) be a generalized, pseudocontractive, and Lipschitzian mapping with corresponding constants r and s such that \(0 < r < 1\), \(s>1\). If \(\Gamma (M_{0}) \subseteq N_{0}\), then Γ has a unique best proximity point.
Proof
Let \(\lambda \in (0,1)\) satisfying, \(0 < \lambda < \frac{2(1r)}{(12r + s^{2})}\). We consider a projection operator on \(M_{0}\), that is, \(P_{M_{0}} : \Gamma (M_{0}) \rightarrow M_{0}\). Also, we define an averaged operator \(F : M_{0} \rightarrow M_{0}\), associated with \(P_{M_{0}} \Gamma \),
Since Γ is generalized, pseudocontractive, and Lipschitzian, we have
Let us assume \(u = \Gamma \eta  P_{M_{0}} \Gamma \eta \) and \(v = \Gamma \omega  P_{M_{0}} \Gamma \omega \). Now, we claim that \(u =v\). Suppose \(u \neq v\), then by the strict convexity of H, we have
which is a contradiction. Therefore, \(u = v\). This implies that, \(\Gamma \eta  \Gamma \omega = P_{M_{0}} \Gamma \eta  P_{M_{0}} \Gamma \omega \). Therefore, from (3), we obtain
Then, \(\Vert F\eta  F\omega \Vert \leq ((1\lambda )^{2} + 2 \lambda (1 \lambda ) r + \lambda ^{2} s^{2} )^{1/2} \Vert \eta  \omega \Vert \).
Now, from \(0 < \lambda < \frac{2(1r)}{(12r + s^{2})}\), we obtain
This implies that F is contraction. By Theorem 1.2, F has a unique fixed point \(p^{*} \in M_{0}\). Then, \(P_{M_{0}} \Gamma p^{*} = p^{*}\). This implies that \(d(p^{*}, \Gamma p^{*}) = d(M,N)\). □
Remark 2.2

1.
If \(0< s<1\), then Γ is a contraction nonselfmapping and the result follows from [15].

2.
If \(s=1\), then Γ is a nonexpansive nonselfmapping and the result follows from [18].
Now, we define a construction of Ishikawa iteration for the case of nonselfmapping:
Let M, N be two convex subsets of a Hilbert space H. Let us define \(\Gamma : M \rightarrow N\) and assume \(\Gamma (M_{0}) \subseteq N_{0}\). Consider the projective operator \(P_{M_{0}} \Gamma : M_{0} \rightarrow M_{0}\). Let \(\eta _{1} \in M_{0}\), then the Ishikawa iteration \(\{\eta _{n}\}\), is defined by
where \(\gamma _{n}, \delta _{n} \in [0,1]\).
Next, we extend the convergence result of Theorem 1.3, for the case of nonselfmappings, by using the proposed Ishikawa iteration for nonselfmappings.
Theorem 2.3
Let M, N be two closed and convex subsets of a Hilbert space H and assume M to be compact. Let \(\Gamma : M \rightarrow N\) be a pseudocontractive and Lipschitzian mapping with \(\Gamma (M_{0}) \subseteq N_{0}\). Let \(\eta _{1} \in M_{0}\), then the Ishikawa iteration \(\{\eta _{n}\}\), defined in (4), with \(\{\gamma _{n}\}\), \(\{\delta _{n}\}\) are sequences of positive numbers satisfying
converges strongly to a best proximity point of Γ.
Proof
First, we prove that the mapping \(P_{M_{0}} \Gamma : M_{0} \rightarrow M_{0}\) is pseudocontractive. It is enough to show that \(\langle P_{M_{0}}\Gamma \eta  P_{M_{0}}\Gamma \omega , \eta  \omega \rangle \leq \Vert \eta  \omega \Vert ^{2}\), for all \(\eta , \omega \in M_{0}\). Now, we assume \(x = \Gamma \eta  P_{M_{0}} \Gamma \eta \) and \(y = \Gamma \omega  P_{M_{0}} \Gamma \omega \). Now, we claim that \(x =y\). Suppose \(x \neq y\), then by the strict convexity of H, we have
which is a contradiction. Therefore, \(x = y\). This implies that, \(\Gamma \eta  \Gamma \omega = P_{M_{0}} \Gamma \eta  P_{M_{0}} \Gamma \omega \). Since Γ is pseudocontractive, we obtain
Now, using that Γ is a Lipschitzian mapping, there exist \(s>0\), we obtain
which implies that the mapping \(P_{M_{0}} \Gamma : M_{0} \rightarrow M_{0}\) is a Lipschitzian operator. Moreover, \(M_{0}\) satisfies all the requirements of Theorem 1.3. This implies that the sequence \(\{\eta _{n} \}\) converges to a fixed point \(p^{*}\) of \(P_{M_{0}}\Gamma \). Then, \(P_{M_{0}} \Gamma p^{*} = p^{*}\). This implies that \(d(p^{*}, \Gamma p^{*}) = d(M,N)\), that is, \(p^{*}\) is a best proximity point of Γ. This completes the proof. □
The following example illustrates Theorem 2.2.
Example 2.4
Let \(H = \mathbb{R}^{2}\) be a Hilbert space with the Euclidean inner product and norm. Assume \(M = \{(0,\eta ) : 1/2 \leq \eta \leq 2\}\), \(N = \{(1,\eta ) : 1/2 \leq \eta \leq 2 \}\). Clearly, \(M_{0} = M\), \(N_{0} = N\). Now, we define \(\Gamma : M \rightarrow N\) by \(\Gamma (0,\eta ) = (1, 1/\eta )\). Then, one can easily verify that Γ is pseudocontractive and Lipschitzian. Assume \(\eta _{0} = 0.5 \), \(\gamma _{n} = \delta _{n} = \frac{1}{\sqrt{n}}\) for all \(n \geq 0\). Then,
As \(n \rightarrow \infty \), the Ishikawa iteration \((0, \eta _{n+1}) \rightarrow (0,1)\), in particular, at \((0, \eta _{118}) =(0, 1)\), reaches the best proximity point of Γ. This result is achieved by simple Matlab coding.
Finally, we approximate the best proximity point for strongly pseudocontractive nonselfmappings without Lipschitzian. This is an extended version of Theorem 1.4, for the case of nonselfmappings.
Theorem 2.5
Let M, N be two closed, bounded, and convex subsets of a real uniformly convex Banach space H. Let \(\Gamma : M \rightarrow N\) be a strongly pseudocontractive that has at least a best proximity point \(\eta ^{*}\) and assume that \(\Gamma (M_{0}) \subseteq N_{0}\). Let \(\eta _{1} \in M_{0}\), then the Ishikawa iteration \(\{\eta _{n}\}\), defined in (4), with \(\{\gamma _{n}\}\), \(\{\delta _{n}\}\) being sequences of positive numbers satisfying
converges strongly to a best proximity point of Γ.
Proof
One can easily verify that the mapping \(P_{M_{0}} \Gamma : M_{0} \rightarrow M_{0}\) is strongly pseudocontractive and the result follows by Theorem 1.4. □
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Pragadeeswarar, V., Gopi, R. Existence and convergence of best proximity points for generalized pseudocontractive and Lipschitzian mappings via an Ishikawatype iterative scheme. Fixed Point Theory Algorithms Sci Eng 2023, 19 (2023). https://doi.org/10.1186/s13663023007578
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DOI: https://doi.org/10.1186/s13663023007578