Certain Novel Best Proximity Theorems with Applications to Complex Function Theory and Integral Equations

Abstract

Let $\mathcal{E}$ and $\mathcal{F}$ be nonempty disjoint subsets of a metric space $(M,d)$. For a non-self-mapping $\phi :\mathcal{E}\to \mathcal{F}$, which is fixed-point free, a point $\varkappa \in \mathcal{E}$ is said to be a best proximity point for the mapping $\phi$ whenever the distance of the point $\varkappa$ to its image under $\phi$ is equal to the distance between the sets, $\mathcal{E}$ and $\mathcal{F}$. In this article, we establish new best proximity point theorems and obtain real extensions of Edelstein’s fixed point theorem in metric spaces, Krasnoselskii’s fixed point theorem in strictly convex Banach spaces, Dhage’s fixed point theorem in strictly convex Banach algebras, and Sadovskii’s fixed point problem in strictly convex Banach spaces. We then present applications of these best proximity point results to complex function theory, as well as the existence of a solution of a nonlinear functional integral equation and the existence of a mutually nearest solution for a system of integral equations.

1. Introduction

In 1911, Brouwer proved that if $\mathcal{E}\subseteq {\mathbb{R}}^n$ is a nonempty, bounded, closed, and convex set and $\phi :\mathcal{E}\to \mathcal{E}$ is continuous, then $\phi$ has a fixed point; that is, there exists a point $\varkappa \in \mathcal{E}$ for which $\phi \varkappa = \varkappa$.

In 1930, Schauder [1] first established an infinite–dimensional generalization of Brouwer’s fixed point theorem (FPT for brief) as follows.

Theorem 1. 

Let $\mathcal{E}$ be a nonempty, compact, and convex subset of a Banach space $\mathcal{X}$. If $\phi :\mathcal{E}\to \mathcal{E}$ is continuous, then φ has a fixed point.

We mention that if $\mathcal{E}$ is a nonempty subset of a Banach space $\mathcal{X}$, then a function $\phi :\mathcal{E}\to \mathcal{X}$ is a compact operator whenever $\phi$ maps bounded sets into relatively compact sets. In this situation, we can consider the following more useful version of Schauder’s FPT.

Theorem 2 

(Schauder’s FPT). Let $\mathcal{E}$ be a nonempty, bounded, closed, and convex subset of a Banach space $\mathcal{X}$. If $\phi :\mathcal{E}\to \mathcal{E}$ is a continuous and compact operator, then φ has a fixed point.

The Schauder’s FPT is one of the most powerful tools in dealing with nonlinear problems in analysis, and, in particular, it has played a major role in the development of fixed point theory and the theory of differential equations.

Due to the fact that Schauder’s FPT has fundamental importance, the theorem has been generalized in various directions by different methods. These generalizations can be divided into two kinds. The first one is purely topological, and the second one is of probabilistic interest in connection with stochastic analysis and stochastic finance.

Let $(M,d)$ be a metric space. A mapping $\phi :\mathcal{E}\subseteq M\to M$ is said to be a contraction if there exists a constant $\alpha \in (0,1)$ such that

$$d(\phi x,\phi y) \le \alpha d(x,y),\mathrm{for}\mathrm{all}x,y\in \mathcal{E}.$$

Moreover, the mapping $\phi$ is called a contractive mapping provided that

$$d(\phi x,\phi y) < d(x,y),\mathrm{for}\mathrm{all}x\ne y\in \mathcal{E}.$$

Clearly, the class of contractive maps contains the family of contractions.

In 1922, Banach established the following FPT in his thesis in order to guarantee the existence of a solution to an integral equation.

Theorem 3 

(Banach’s FPT). Let $(M,d)$ be a complete metric space and $\phi :M\to M$ be a contraction mapping. Then φ has a unique fixed point, and for each $x\in M$, the iterate sequence $\left\{{\phi }^{n}x\right\}$ converges to the fixed point.

Banach’s FPT is extremely helpful to solve integral and differential equations, providing a constructive method to approximate their solutions with an adjustable accuracy.

It is worth noticing that Theorem 3 does not hold for contractive maps. For instance, if we consider the space $\mathcal{C}[0,1]$ consisting of the real-valued continuous functions defined on $[0,1]$ equipped with the supremum norm and if we set

$$M = \left\{f\in \mathcal{C}[0,1];f\left(1\right) = 1\right\},$$

Then M is a closed subset of $\mathcal{C}[0,1]$, and so it is a complete metric space. Now, it is easy to see that the self-mapping $\phi :M\to M$ defined by

$$\phi \left(f\right)\left(t\right) = tf\left(t\right),\mathrm{for}\mathrm{all}t\in [0,1],$$

is a contractive mapping that is fixed-point-free.

In 1962, Edelstein ([2]) extended the Banach contraction principle to contractive self-mappings as follows.

Theorem 4 

(Edelstein’s FPT). Let $(M,d)$ be a compact metric space and $\phi :M\to M$ be a contractive mapping. Then φ has a unique fixed point, and for each $x\in M$, the iterate sequence $\left\{{\phi }^{n}x\right\}$ converges to the fixed point.

In [3], Krasnoselskii observed that in some of the problems, the integration of a perturbed differential operator gives rise to a sum of two applications, a contraction and a compact application. So, he combined Banach’s FPT and Schauder’s FPT and obtained the next widely used result.

Theorem 5 

(Krasnoselskii’s FPT). Let $\mathcal{E}$ be a nonempty, closed, and convex subset of a Banach space $\mathcal{X}$. Suppose $\phi :\mathcal{E}\to \mathcal{X}$ and $\psi :\mathcal{E}\to \mathcal{X}$ are two maps such that

(i)

φ is a contraction;

(ii)

ψ is a continuous and compact operator;

(iii)

$\phi \left(\mathcal{E}\right)+\psi \left(\mathcal{E}\right)\subseteq \mathcal{E}$.

Then the operator $\phi +\psi$ has a fixed point; that is, there exists an element $\varkappa \in \mathcal{E}$ for which $\phi \varkappa +\psi \varkappa = \varkappa$.

Krasnoselskii’s FPT is useful in establishing existence results in some mathematical problems. Since then, a huge number of papers have appeared, contributing generalizations or modifications of Krasnoselskii’s fixed point theorem and their applications.

Meanwhile, a large class of problems, for instance in integral equations and stability theory, has been adapted by Krasnoselskii’s fixed point method. Several extensions of the theorem have been made in the literature in the course of time by modifying the conditions (i), (ii), and (iii).

Inspired by Krasnoselskii’s FPT, Dhage in [4] proved the following FPT for multiplication of two maps in the framework of Banach algebras.

We recall that a Banach space $\mathcal{X}$ is a Banach algebra provided that there exists an operator $.:\mathcal{X}\times \mathcal{X}\to \mathcal{X}$ with $.(x,y) = x.y$ for all $x,y\in \mathcal{X}$ which is associative and bilinear and that

$$\parallel x.y\parallel \le \parallel x\parallel \parallel y\parallel ,\mathrm{for}\mathrm{all}x,y\in \mathcal{X}.$$

Theorem 6 

(Dhage’s FPT). Let $\mathcal{E}$ be a nonempty, bounded, closed, and convex subset of a Banach algebra $\mathcal{X}$ and let $\phi :\mathcal{E}\to \mathcal{X}$, $\psi :\mathcal{E}\to \mathcal{X}$ be two operators such that

(i)

φ is a contraction with the contractive constant $\alpha \in (0,1)$;

(ii)

ψ is a continuous and compact operator;

(iii)

$\phi \left(\mathcal{E}\right)\psi \left(\mathcal{E}\right)\subseteq \mathcal{E}$.

If $\alpha M < 1$, where $M = sup\{\psi x:x\in \mathcal{E}\}$, then $\phi .\psi$ has a fixed point, i.e., there exists a point $\varkappa \in \mathcal{E}$ such that $\phi \varkappa .\psi \varkappa = \varkappa$.

The hybrid FPTs in Banach algebras are also useful for proving the existence theorems for certain nonlinear differential and integral equations. Here in this paper, we illustrate the applicability of an extension of Dhage’s FPT (Corollary 4) by considering nonlinear functional integral equations (see Examples 3 and 4).

The current paper consists of five sections. Section 1 is the introduction, and Section 2 describes the concepts of proximal pairs, projection operators, and best proximity points for non-self-maps and relates some basic facts.

In Section 3, we present the best proximity version of Edelstein, Krasnoselskii, and Dhage FPTs (Theorems 4–6).

Section 4 is devoted to obtaining a generalization of Sadovskii’s FPT, which leads to obtaining extensions of Darbo and Schauder FPTs in the setting of strictly convex Banach spaces.

Finally, in Section 5, we use our existence results to give an application to complex function theory and find a solution to a class of nonlinear functional integral equations.

We also define a notion of mutually nearest solutions for a system of integral equations and study their existence by applying a best proximity version of Schauder’s FPT.

2. Preliminaries

We recall that a Banach space $\mathcal{X}$ is said to be strictly convex if for any two distinct points $u,v\in \mathcal{X}$ such that $\parallel u\parallel = \parallel v\parallel = 1$, it is the case that $\parallel \frac{u+v}{2}\parallel < 1$. Hilbert spaces and $l^p$ spaces $(1 < p < \infty )$ are instances of strictly convex Banach spaces.

Let $\mathcal{E}$ and $\mathcal{F}$ be nonempty subsets of a metric space $(M,d)$. We will say that a pair $(\mathcal{E},\mathcal{F})$ has a property if and only if both the sets $\mathcal{E}$ and $\mathcal{F}$ have that property. For instance, $(\mathcal{E},\mathcal{F})$ being closed means that both $\mathcal{E}$ and $\mathcal{F}$ are closed subsets of $\mathcal{X}$. We set

$$\mathfrak{D}(\mathcal{E},\mathcal{F}): = inf\left\{d\right(u,v):(u,v)\in \mathcal{E}\times \mathcal{F}\}.$$

Also, the proximal pair of the pair $(\mathcal{E},\mathcal{F})$ is denoted by $({\mathcal{E}}_0,{\mathcal{F}}_0)$, where

$$\begin{array}{cc}\hfill {\mathcal{E}}_0& = \{u\in \mathcal{E}:\mathrm{there}\mathrm{exists}{v}^{\prime }\in \mathcal{F}|d(u,v^{\prime }) = \mathfrak{D}(\mathcal{E},\mathcal{F})\},\hfill \\ \hfill {\mathcal{F}}_0& = \{v\in \mathcal{F}:\mathrm{there}\mathrm{exists}{u}^{\prime }\in \mathcal{E}|d(u^{\prime },v) = \mathfrak{D}(\mathcal{E},\mathcal{F})\}.\hfill \end{array}$$

In general, the proximal pairs may be empty. However, if $(\mathcal{E},\mathcal{F})$ is a compact pair in a metric space M or $(\mathcal{E},\mathcal{F})$ is a bounded and closed pair in a reflexive Banach space $\mathcal{X}$, then its proximal pair is nonempty.

Definition 1.  

A nonempty pair $(\mathcal{E},\mathcal{F})$ in a metric space $(M,d)$ is said to be proximinal if ${\mathcal{E}}_0 = \mathcal{E}$ and ${\mathcal{F}}_0 = \mathcal{F}$.

For a nonempty subset $\mathcal{E}$ of a metric space M, a metric projection operator${\mathbb{P}}_{\mathcal{E}}:M\to 2^{\mathcal{E}}$ is defined with

$${\mathbb{P}}_{\mathcal{E}}\left(u\right): = \left\{v\in \mathcal{E}:d(u,v) = \mathfrak{D}\left(\right\{u\},\mathcal{E})\right\},\mathrm{for}\mathrm{all}u\in M.$$

It is worth mentioning that if $\mathcal{E}$ is a nonempty, bounded, closed, and convex subset of a reflexive and strictly convex Banach space $\mathcal{X}$, then the metric projection ${\mathbb{P}}_{\mathcal{E}}$ is a single-valued map from $\mathcal{X}$ onto $\mathcal{E}$.

The next proposition plays a fundamental role in the proof of our main corollaries in this paper.

Proposition 1 

([5]). Assume $\mathcal{X}$ is a strictly convex Banach space and $(\mathcal{E},\mathcal{F})$ is a nonempty, closed, and convex pair in $\mathcal{X}$ for which ${\mathcal{E}}_0$ is nonempty. Define a map $\mathbb{P}:{\mathcal{E}}_0\cup {\mathcal{F}}_0\to {\mathcal{E}}_0\cup {\mathcal{F}}_0$ with

$$\mathbb{P}\left(u\right) = \left\{\begin{array}{c}{\mathbb{P}}_{{\mathcal{E}}_0}\left(u\right);\mathrm{if}u\in {\mathcal{F}}_0,\hfill \\ {\mathbb{P}}_{{\mathcal{F}}_0}\left(u\right);\mathrm{if}u\in {\mathcal{E}}_0.\hfill \end{array}\right.$$

Then the following statements hold:

(i)

$\parallel u-\mathbb{P}u\parallel = \mathfrak{D}(\mathcal{E},\mathcal{F})$ for any $u\in {\mathcal{E}}_0\cup {\mathcal{F}}_0$ and $\mathbb{P}$ is cyclic on ${\mathcal{E}}_0\cup {\mathcal{F}}_0$, i.e., $\mathbb{P}\left({\mathcal{E}}_0\right)\subseteq {\mathcal{F}}_0$ and $\mathbb{P}\left({\mathcal{F}}_0\right)\subseteq {\mathcal{E}}_0$;

(ii)

${\mathbb{P}}_{{\mathcal{E}}_0}$ and ${\mathbb{P}}_{{\mathcal{F}}_0}$ are isometry;

(iii)

${\mathbb{P}}_{{\mathcal{E}}_0}$ and ${\mathbb{P}}_{{\mathcal{F}}_0}$ are affine;

(iv)

${\mathbb{P}}_{{\mathcal{E}}_0}^2 = I_{{\mathbb{E}}_0}$ and ${\mathbb{P}}_{{\mathcal{F}}_0}^2 = I_{{\mathcal{F}}_0}$, where $I_A$ denotes the identity mapping on a nonempty subset A of $\mathcal{X}$.

Here, we recall a geometric concept on a nonempty pair of subsets of a metric space which was first introduced in [6].

Definition 2.  

Let $(\mathcal{E},\mathcal{F})$ be a pair of nonempty subsets of a metric space $(M,d)$ such that ${\mathcal{E}}_0\ne \varnothing$. The pair $(\mathcal{E},\mathcal{F})$ is said to have the P-property if and only if

$$\begin{array}{c}\hfill d(u_1,v_1) = \mathfrak{D}(\mathcal{E},\mathcal{F}) = d(u_2,v_2)\Rightarrow d(u_1,u_2) = d(v_1,v_2),\end{array}$$

where $u_1,u_2\in {\mathcal{E}}_0$ and $v_1,v_2\in {\mathcal{F}}_0$.

It is clear that for a nonempty subset $\mathcal{E}$ of a metric space $(M,d)$, the pair $(\mathcal{E},\mathcal{E})$ has the P-property.

Remark 1.  

It was proved in [7] that every nonempty and convex pair in a strictly convex Banach space $\mathcal{X}$ has the P-property. Furthermore, it is worth noticing that the strict convexity assumption of the Banach space $\mathbb{X}$ in Proposition 1 can be replaced by the P-property of $(\mathcal{E},\mathcal{F})$ (see [7] for more details).

Definition 3.  

Let $(\mathcal{E},\mathcal{F})$ be a pair of nonempty disjoint subsets of a metric space $(M,d)$ and $\phi :\mathcal{E}\to \mathcal{F}$ be a non-self-mapping. A point $\varkappa \in \mathcal{E}$ is said to be a best proximity point of φ provided that

$$d(\varkappa ,\phi \varkappa ) = \mathfrak{D}(\mathcal{E},\mathcal{F}).$$

The relevance of best proximity points is that they provide optimal solutions for the problem of best approximation between two disjoint sets.

We mention that in 2011, Sadiq Basha [8] introduced a class of non-self-maps, called proximal contractions of the first and second kinds, to investigate the existence, uniqueness, and convergence of a best proximity point (see also [9] for more details).

3. Generalizations of Edelstein, Krasnoselskii, and Dhage FPTs

In this section, we prove three theorems, which are connected with Edelstein, Krasnoselskii and Dhage FPTs. We begin by introducing the novel family of non-self proximal contractions.

3.1. Best Proximity Version of Edelstein’s FPT

Definition 4. 

Let $(\mathcal{E},\mathcal{F})$ be a pair of nonempty disjoint subsets of a metric space $(M,d)$. A map $\phi :\mathcal{E}\to \mathcal{F}$ is called Edelstein proximal contractive if

$$\left\{\begin{array}{c}d(u_1,\phi x_1) = \mathfrak{D}(\mathcal{E},\mathcal{F}),\hfill \\ d(u_2,\phi x_2) = \mathfrak{D}(\mathcal{E},\mathcal{F}),\hfill \end{array}\right.\Rightarrow \left\{\begin{array}{c}d(u_1,u_2) < d(x_1,x_2),\mathrm{if}{x}_1\ne x_2,\hfill \\ d(u_1,u_2) = 0,\mathrm{if}{x}_1 = x_2,\hfill \end{array}\right.$$

where $x_1,x_2,u_1,u_2\in \mathcal{E}$.

Our first main result is the following best proximity point theorem.

Theorem 7. 

Let $(\mathcal{E},\mathcal{F})$ be a pair of nonempty subsets of a metric space $(M,d)$ such that ${\mathcal{E}}_0$ is nonempty and compact. Assume that $\phi :\mathcal{E}\to \mathcal{F}$ and $f:\mathcal{E}\to \mathcal{E}$ satisfy the following conditions:

(1)

φ is an Edelstein proximal contractive mapping;

(2)

$\phi \left({\mathcal{E}}_0\right)\subseteq {\mathcal{F}}_0$;

(3)

${\mathcal{E}}_0\subseteq f\left({\mathcal{E}}_0\right)$;

(4)

f is an isometry.

Then there exists a unique element $\varkappa \in {\mathcal{E}}_0$ such that

$$d(f\varkappa ,\phi \varkappa ) = \mathfrak{D}(\mathcal{E},\mathcal{F}).$$

Proof. 

For an element $x\in {\mathcal{E}}_0$, since ${\mathcal{E}}_0\subseteq f\left({\mathcal{E}}_0\right)$ and $\phi \left({\mathcal{E}}_0\right)\subseteq {\mathcal{F}}_0$, there is a point $u\in {\mathcal{E}}_0$ such that $d(fu,\phi x) = \mathfrak{D}(\mathcal{E},\mathcal{F})$. Let $u^{\prime }\in {\mathcal{E}}_0$ be such that $d(f{u}^{\prime },\phi x) = \mathfrak{D}(\mathcal{E},\mathcal{F})$. Because of the reality that $\phi$ is an Edelstein proximal contractive mapping and f is an isometry, $u = u^{\prime }$.

Then for any $x\in {\mathcal{E}}_0$, there exists only one point $\chi x\in {\mathcal{E}}_0$ for which $d(f\chi x,\phi x) = \mathfrak{D}(\mathcal{E},\mathcal{F})$. So, we can consider a self-map $\chi :{\mathcal{E}}_0\to {\mathcal{E}}_0$ such that

$$d(f\chi x,\phi x) = \mathfrak{D}(\mathcal{E},\mathcal{F}),\mathrm{for}\mathrm{all}x\in {\mathcal{E}}_0.$$

Note that for any $x_1,x_2\in {\mathcal{E}}_0$ with $x_1\ne x_2$, we have $d(f\chi x_1,\phi x_1) = \mathfrak{D}(\mathcal{E},\mathcal{F}) = d(f\chi x_2,\phi x_2)$ and thus

$$d(\chi x_1,\chi x_2) = d(f\chi x_1,f\chi x_2) < d(x_1,x_2),$$

which ensures that $\chi$ is a contractive self-map. Since ${\mathcal{E}}_0$ is compact, we can use Edelstein’s FPT to obtain a unique fixed point for the mapping $\chi$, say $\varkappa \in {\mathcal{E}}_0$. In this situation, we conclude that

$$d(f\varkappa ,\phi \varkappa ) = d(f\chi \varkappa ,\phi \varkappa ) = \mathfrak{D}(\mathcal{E},\mathcal{F}),$$

and the result follows. □

The next best proximity point results are obtained from Theorem 7 immediately.

Corollary 1. 

Let $(\mathcal{E},\mathcal{F})$ be a pair of nonempty subsets of a metric space $(M,d)$ such that ${\mathcal{E}}_0$ is nonempty and compact. Assume that $\phi :\mathcal{E}\to \mathcal{F}$ is an Edelstein proximal contractive mapping such that $\phi \left({\mathcal{E}}_0\right)\subseteq {\mathcal{F}}_0$. Then φ has a unique best proximity point.

Corollary 2. 

Let $(\mathcal{E},\mathcal{F})$ be a pair of nonempty subsets of a metric space $(M,d)$ such that ${\mathcal{E}}_0$ is nonempty and compact and $(\mathcal{E},\mathcal{F})$ has the P-property. Assume that $\phi :\mathcal{E}\to \mathcal{F}$ is a contractive mapping such that $\phi \left({\mathcal{E}}_0\right)\subseteq {\mathcal{F}}_0$. Then φ has a unique best proximity point.

Proof. 

Note that if $d(u_1,\phi x_1) = \mathfrak{D}(\mathcal{E},\mathcal{F}) = d(u_2,\phi x_2)$ were $x_1,x_2,u_1,u_2\in {\mathcal{E}}_0$ with $x_1\ne x_2$, then by the fact that $\phi$ is a contractive map and $(\mathcal{E},\mathcal{F})$ has the P-property,

$$d(u_1,u_2) = d(\phi x_1,\phi x_2) < d(x_1,x_2).$$

In the case that $x_1 = x_2$, we use again the P-property on the pair $(\mathcal{E},\mathcal{F})$ to obtain $u_1 = u_2$. Therefore, $\phi$ is an Edelstein proximal contractive mapping, and the result follows from Theorem 7. □

Example 1. 

Consider $\mathcal{X} = \mathcal{C}[0,1]$, which consists of all real-valued continuous functions defined on $[0,1]$, endowed with the norm

$$\parallel f\parallel = {\parallel f\parallel }_2+{\parallel f\parallel }_{\infty },\mathrm{for}\mathrm{all}f\in \mathcal{X}.$$

Then $(\mathcal{X},\parallel .\parallel )$ is a Banach space, and because of the appearance of ${\parallel .\parallel }_2$ we can see that $(\mathcal{X},\parallel .\parallel )$ is strictly convex. Also, in view of the fact that the norms $\parallel .\parallel$ and ${\parallel .\parallel }_{\infty }$ are equivalent, $(\mathcal{X},\parallel .\parallel )$ is not reflexive. Set

$$\mathcal{E} = \left\{f\in \mathcal{X}:f\left(1\right) = 1\right\},\mathcal{E} = \left\{g\in \mathcal{X}:g\left(1\right) = 0\right\}.$$

Then $(\mathcal{E},\mathcal{F})$ is a closed and convex pair in a strictly convex Banach space $\mathcal{X}$, and so $(\mathcal{E},\mathcal{F})$ has the P-property. By ${\mathfrak{D}}_{\infty }(\mathcal{E},\mathcal{F})$ and ${\mathfrak{D}}_2(\mathcal{E},\mathcal{F})$, we denote the distance between the sets $\mathcal{E}$ and $\mathcal{F}$ w.r.t. the norms ${\parallel .\parallel }_{\infty }$ and ${\parallel .\parallel }_2$. Clearly, ${\mathfrak{D}}_{\infty }(\mathcal{E},\mathcal{F}) = 1$. Moreover, if we consider

$$f_n\left(t\right) = t^n,g_n\left(t\right)\equiv 0,\mathrm{for}\mathrm{all}t\in [0,1],$$

then $(f_n,g_n)\in \mathcal{E}\times \mathcal{F}$ and that

$${\mathfrak{D}}_2(\mathcal{E},\mathcal{F}) \le {\parallel f_n-g_n\parallel }_2 = {\left({\int }_0^1{t}^{2n}dt\right)}^{\frac{1}{2}} = \frac{1}{\sqrt{2n+1}}{\to }^{n\to +\infty }0,$$

which implies that ${\mathfrak{D}}_2(\mathcal{E},\mathcal{F}) = 0$. Therefore, $\mathfrak{D}(\mathcal{E},\mathcal{F}) = 1$. Define $\phi :\mathcal{E}\to \mathcal{F}$ with

$$\phi f\left(t\right) = (1-t)f\left(t\right),\mathrm{for}\mathrm{all}t\in [0,1].$$

Then φ is a contractive non-self-mapping but does not have a best proximity point because of the fact that ${\mathcal{E}}_0 = \varnothing$.

3.2. Best Proximity Version of Krasnoselskii’s FPT

The next result gives us a sufficient condition that assures the existence of a best proximity point for the sum of two non-self-mappings.

Theorem 8. 

Let $(\mathcal{E},\mathcal{F})$ be a pair of nonempty, closed and convex subsets of a strictly convex Banach space $\mathcal{X}$ such that ${\mathcal{E}}_0$ is nonempty and compact. Suppose $\phi :\mathcal{E}\to \mathcal{X}$ and $\psi :\mathcal{F}\to \mathcal{X}$ are two maps satisfying the following assumptions:

(i)

φ is a contractive mapping;

(ii)

ψ is a continuous and compact operator;

(iii)

$\phi \left({\mathcal{E}}_0\right)+\psi \left({\mathcal{F}}_0\right)\subseteq {\mathcal{E}}_0$.

Then there is a point $\vartheta \in {\mathcal{F}}_0$ such that ${\parallel \vartheta -(I-\phi )}^{-1}\psi \vartheta \parallel = \mathfrak{D}(\mathcal{E},\mathcal{F})$, where I is an identity mapping.

Proof. 

As a result of [7], the pair $({\mathcal{E}}_0,{\mathcal{F}}_0)$ is closed, and it is easy to see that it is also convex. For an arbitrary and fixed element $v\in {\mathcal{F}}_0$, define ${\phi }_{v}u = \phi u+\psi v$ for any $u\in {\mathcal{E}}_0$. By assumption (iii), ${\phi }_v$ is a self-map on ${\mathcal{E}}_0$.

Moreover, if $u_1$ and $u_2$ are disjoint points in ${\mathcal{E}}_0$, then by the fact that $\phi$ is a contractive mapping, we obtain

$$\parallel {\phi }_v{u}_1-{\phi }_v{u}_2\parallel = \parallel \phi u_1-\phi u_2\parallel < \parallel u_1-u_2\parallel ,$$

That is, ${\phi }_v$ is a contractive self-mapping on the compact set ${\mathcal{E}}_0$. Edelstein’s FPT guarantees the existence of a unique fixed point for the mapping ${\phi }_v$, which is in correspondence to v, say ${\varkappa }_v\in {\mathcal{F}}_0$. Hence,

$${\varkappa }_v = {\phi }_v\left({\varkappa }_v\right) = \phi {\varkappa }_v+\psi v,\mathrm{for}\mathrm{all}v\in {\mathcal{F}}_0,$$

and so, $\psi \left({\mathcal{F}}_0\right)\subseteq (I-\phi )\left({\mathcal{E}}_0\right)$. Furthermore, for distinct elements $u_1,u_2\in {\mathcal{E}}_0$ we have

$$\parallel (I-\phi )u_1-(I-\phi )u_2\parallel \ge \parallel u_1-u_2\parallel -\parallel \phi u_1-\phi u_2\parallel >0,$$

and

$$\begin{array}{c}\hfill \parallel (I-\phi )u_1-(I-\phi )u_2\parallel \le \parallel u_1-u_2\parallel +\parallel \phi u_1-\phi u_2\parallel < 2\parallel u_1-u_2\parallel .\end{array}$$

This implies that $(I-\phi )$ is a homeomorphism on ${\mathcal{E}}_0$ and thereby ${(I-\phi )}^{-1}\left(\psi \left({\mathcal{F}}_0\right)\right)\subseteq {\mathcal{E}}_0$. Now, by considering the proximal projection operator defined as $\mathbb{P}$ on ${\mathcal{E}}_0\cup {\mathcal{F}}_0$, which is cyclic, we obtain

$$\mathbb{P}{(I-\phi )}^{-1}\psi \left({\mathcal{F}}_0\right)\subseteq \mathbb{P}\left({\mathcal{E}}_0\right)\subseteq {\mathcal{F}}_0.$$

Since $\psi$ is a compact and continuous operator, the mapping $\mathbb{P}{(I-\phi )}^{-1}\psi :{\mathcal{F}}_0\to {\mathcal{F}}_0$ is compact and continuous too. Apply Schauder’s FPT to find a fixed point for this map; that is,

$$\mathrm{there}\mathrm{exists}\vartheta \in {\mathcal{F}}_0\mathrm{such}\mathrm{that}\mathbb{P}{(I-\phi )}^{-1}\psi \left(\vartheta \right) = \vartheta ,$$

and so, by the property (i) of the proximal projection operator, we conclude that

$${\parallel \vartheta -(I-\phi )}^{-1}{\psi \left(\vartheta \right)\parallel = \parallel \mathbb{P}(I-\phi )}^{-1}\psi \left(\vartheta \right)-{(I-\phi )}^{-1}\psi \left(\vartheta \right)\parallel = \mathfrak{D}(\mathcal{E},\mathcal{F}),$$

and this completes the proof. □

Remark 2. 

As we mentioned in Remark 1, we can replace the condition of strict convexity of the Banach space $\mathcal{X}$ in Theorem 8 with the P-property of the pair $(\mathcal{E},\mathcal{F})$.

The next corollary is an extension of Krasnoselskii’s FPT.

Corollary 3. 

Let $\mathcal{E}$ be a nonempty, compact, and convex subset of a Banach space $\mathcal{X}$. Suppose $\phi ,\psi :\mathcal{E}\to \mathcal{X}$ are two maps that satisfy the following conditions:

(1)

φ is a contractive mapping;

(2)

ψ is a continuous;

(3)

$\phi \left(\mathcal{E}\right)+\psi \left(\mathcal{E}\right)\subseteq \mathcal{E}$.

Then the mapping ${(I-\phi )}^{-1}\psi$ has a fixed point; that is,

$$\mathit{there}\mathit{exists}\varkappa \in \mathcal{E}\mathit{such}\mathit{that}\phi \varkappa +\psi \varkappa = \varkappa .$$

Proof. 

We consider $\mathcal{E} = \mathcal{F}$ in Theorem 8 and note that $(\mathcal{E},\mathcal{E})$ has the P-property. Then the result follows. □

The next example is presented to illustrate Theorem 8.

Example 2. 

Consider the Banach space $\mathcal{X} = ({\mathbb{R}}^2,\parallel .{\parallel }_1)$ and let

$$\mathcal{E} = \left\{(x,0):0 \le x \le 1\right\},\mathcal{F} = \left\{(y,1)\in {\mathbb{R}}^2:-1 \le y \le \frac{1}{2}\right\}.$$

Obviously, $(\mathcal{E},\mathcal{F})$ is a bounded, closed, and convex pair with $\mathfrak{D}(\mathcal{E},\mathcal{F}) = 1$. Also, if ${\mathbf{x}}_1 = (x_1,0),{\mathbf{x}}_2 = (x_2,0)\in \mathcal{E}$ and ${\mathbf{y}}_1 = (y_1,1),{\mathbf{y}}_2 = (y_2,1)\in \mathcal{F}$ are such that

$$\parallel {\mathbf{x}}_1-{\mathbf{y}}_1{\parallel }_1 = \mathfrak{D}(\mathcal{E},\mathcal{F}) = {\parallel {\mathbf{x}}_2-{\mathbf{y}}_2\parallel }_1,$$

then we must have $x_1 = y_1$ and $x_2 = y_2$ and so,

$$\parallel {\mathbf{x}}_1-{\mathbf{x}}_2{\parallel }_1 = |x_1-x_2| = |y_1-y_2| = \parallel {\mathbf{y}}_1-{\mathbf{y}}_2{\parallel }_1,$$

that is, $(\mathcal{E},\mathcal{F})$ has the P-property. Furthermore,

$${\mathcal{E}}_0 = \left\{(x,0):0 \le x \le \frac{1}{2}\right\},{\mathcal{F}}_0 = \left\{(y,1)\in {\mathbb{R}}^2:0 \le y \le \frac{1}{2}\right\}.$$

Define $\phi :\mathcal{E}\to \mathcal{X},\psi :\mathcal{F}\to \mathcal{X}$ with

$$\phi (x,0) = \left(\frac{x^2}{2},0\right),\psi (y,1) = ({sin}^{2}y,0),forall(x,y)\in [0,1]\times [-1,\frac{1}{2}].$$

Then for any $x_1\ne x_2\in [0,1]$ we have

$$\begin{array}{cc}\hfill \parallel \phi (x_1,0)-\phi (x_2,0){\parallel }_1& = \frac{1}{2}|x_1^2-x_2^2| = \frac{1}{2}|x_1-x_2\left|\right|x_1+x_2|\hfill \\ \hfill & < |x_1-x_2| = \parallel (x_1,0)-(x_2,0){\parallel }_1,\hfill \end{array}$$

which implies that φ is a contractive mapping. Clearly, ψ is continuous and compact. Moreover, for $\left((x,0),(y,1)\right)\in {\mathcal{E}}_0\times {\mathcal{F}}_0$, we have $x,y\in [0,\frac{1}{2}]$, and so,

$$\phi (x,0)+\psi (y,1) = (\underset{\le \frac{1}{2}}{\underbrace{\frac{x^2}{2}+{sin}^{2}y}},0)\in {\mathcal{E}}_0,$$

that is, $\phi \left({\mathcal{E}}_0\right)+\psi \left({\mathcal{F}}_0\right)\subseteq {\mathcal{E}}_0$. Therefore, all of the conditions of Theorem 8 hold, and there exists a point $\vartheta \in {\mathcal{F}}_0$ such that ${\parallel \vartheta -(I-\phi )}^{-1}{\psi \left(\vartheta \right)\parallel }_1 = \mathfrak{D}(\mathcal{E},\mathcal{F})$.

To find this point, by a simple calculation, we can find that ${(I-\phi )}^{-1}(x,0) = (1-\sqrt{1-2x},0)$ for all $x\in [0,\frac{1}{2}]$, and so,

$${(I-\phi )}^{-1}\psi (y,1) = {(I-\phi )}^{-1}({sin}^{2}y,0) = \left(1-\sqrt{1-2{sin}^{2}y},0\right),forally\in [0,\frac{1}{2}].$$

Now, if we consider $\vartheta = (0,1)$, then $\psi \left(\vartheta \right) = (0,0)$, and so ${(I-\phi )}^{-1}\psi \left(\vartheta \right) = (0,0)$, which implies that

$${\parallel \vartheta -(I-\phi )}^{-1}{\psi \left(\vartheta \right)\parallel }_1 = {\parallel (0,1)-(0,0)\parallel }_1 = 1 = \mathfrak{D}(\mathcal{E},\mathcal{F}).$$

3.3. Best Proximity Version of Dehage’s FPT

The third main theorem of this section is devoted to the best proximity version of Dhage’s FPT as below.

Theorem 9. 

Let $\mathcal{X}$ be a strictly convex Banach algebra and $(\mathcal{E},\mathcal{F})$ be a nonempty, bounded, closed, and convex pair in $\mathcal{X}$ such that ${\mathcal{E}}_0$ is nonempty and compact. Assume that $\phi :\mathcal{E}\to \mathcal{X}$ and $\psi :\mathcal{F}\to \mathcal{X}$ are two maps satisfying the following conditions:

(i)

φ is a contractive mapping;

(ii)

ψ is a continuous and compact operator with $L: = \parallel \psi \left(\mathcal{F}\right)\parallel < 1$;

(iii)

$\phi \left({\mathcal{E}}_0\right)\psi \left({\mathcal{F}}_0\right)\subseteq {\mathcal{E}}_0$.

Then

$$\mathit{there}\mathit{exists}\vartheta \in {\mathcal{F}}_0\mathit{such}\mathit{that}\parallel \vartheta -\left(\phi {\mathbb{P}}_{{\mathcal{E}}_0}\vartheta \right).\left(\psi \vartheta \right)\parallel = \mathfrak{D}(\mathcal{E},\mathcal{F}).$$

Proof. 

Let $v\in {\mathcal{F}}_0$ be an arbitrary point, and for any $u\in {\mathcal{E}}_0$, define ${\widehat{\phi }}_v\left(u\right) = \left(\phi u\right).\left(\psi v\right)$. It follows from condition (iii) that $\widehat{\phi }$ is a mapping from ${\mathcal{E}}_0$ into itself.

Also, for any disjoint elements $u_1,u_2\in {\mathcal{E}}_0$ using the fact that $\mathcal{X}$ is a Banach algebra and $\phi$ is a contractive mapping, we have

$$\parallel {\widehat{\phi }}_v\left(u_1\right)-{\widehat{\phi }}_v\left(u_2\right)\parallel = \parallel \left(\phi u_1\right).\left(\psi v\right)-\left(\phi u_2\right).\left(\psi v\right)\parallel \le \underset{\le 1}{\underbrace{\parallel \psi v\parallel }}\parallel \phi u_1-\phi u_2\parallel < \parallel u_1-u_2\parallel .$$

Thus, for any $v\in {\mathcal{F}}_0$, the mapping ${\widehat{\phi }}_v:{\mathcal{E}}_0\to {\mathcal{E}}_0$ is contractive on a compact domain. By using Edelstein’s FPT, ${\widehat{\phi }}_v$ has a unique fixed point, which will be denoted by ${\widehat{\varkappa }}_v\in {\mathcal{E}}_0$. Thereby,

$${\widehat{\varkappa }}_v = {\widehat{\phi }}_v\left({\widehat{\varkappa }}_v\right) = \left(\phi {\widehat{\varkappa }}_v\right).\left(\psi v\right),\mathrm{for}\mathrm{all}v\in {\mathcal{F}}_0.$$

So, we can define a mapping $g:{\mathcal{F}}_0\to {\mathcal{E}}_0$ for which

$$gv = {\widehat{\varkappa }}_v = \left(\phi {\widehat{\varkappa }}_v\right).\left(\psi v\right),\mathrm{for}\mathrm{all}v\in {\mathcal{F}}_0.$$

The map g has the following properties:

♠ g is continuous. To show this, let $\left\{v_n\right\}$ be a sequence in ${\mathcal{F}}_0$ such that $v_n\to v\in {\mathcal{F}}_0$.Then

$$\begin{array}{cc}\hfill \parallel g{v}_n-gv\parallel & = \parallel \left(\phi {\widehat{\varkappa }}_{v_n}\right).\left(\psi v_n\right)-\left(\phi {\widehat{\varkappa }}_v\right).\left(\psi v\right)\parallel \hfill \\ \hfill & \le \parallel \left(\phi {\widehat{\varkappa }}_{v_n}\right).\left(\psi v_n\right)-\left(\phi {\widehat{\varkappa }}_v\right).\left(\psi v_n\right)\parallel +\parallel \left(\phi {\widehat{\varkappa }}_v\right).\left(\psi v_n\right)-\left(\phi {\widehat{\varkappa }}_v\right).\left(\psi v\right)\parallel \hfill \\ \hfill & \le \parallel \phi {\widehat{\varkappa }}_{v_n}-\phi {\widehat{\varkappa }}_v\parallel \parallel \psi v_n\parallel +\parallel \phi {\widehat{\varkappa }}_v\parallel \parallel \psi v_n-\psi v\parallel \hfill \\ \hfill & \le L\parallel {\widehat{\varkappa }}_{v_n}-{\widehat{\varkappa }}_v\parallel +\parallel \phi {\widehat{\varkappa }}_v\parallel \parallel \psi v_n-\psi v\parallel .\hfill \end{array}$$

Hence,

$$\parallel g{v}_n-gv\parallel \le \frac{1}{1-L}\parallel \phi {\widehat{\varkappa }}_v\parallel \parallel \psi v_n-\psi v\parallel .$$

If in the above inequality $n\to \infty$, by the continuity of $\psi$, we obtain $g{v}_n\to gv$ and so, g is continuous.

♠ g is a compact operator. In this regard, let $x_0\in {\mathcal{E}}_0$ be a fixed element. Then for any $u\in {\mathcal{E}}_0$, we have

$$\parallel \phi u\parallel \le \parallel \phi u-\phi x_0\parallel +\parallel \phi x_0\parallel \le \parallel u-x_0\parallel +\parallel \phi x_0\parallel \le \underset{: = M}{\underbrace{\mathrm{diam}\left({\mathcal{E}}_0\right)+\parallel \phi x_0\parallel }},$$

which concludes that $\phi$ is bounded on ${\mathcal{E}}_0$ with the bound $M>0$. Now suppose that $\left\{v_n\right\}$ is a sequence in ${\mathcal{F}}_0$. Since $\psi$ is compact, we may assume that ${lim\; sup}_{m,n\to +\infty }\parallel \psi v_n-\psi v_m\parallel = 0$. Then for any $m,n\in \mathbb{N}$ we have

$$\begin{array}{cc}\hfill \parallel g{v}_m-g{v}_n\parallel & = \parallel \left(\phi {\widehat{\varkappa }}_{v_m}\right).\left(\psi v_m\right)-\left(\phi {\widehat{\varkappa }}_{v_n}\right).\left(\psi v_n\right)\parallel \hfill \\ \hfill & \le \parallel \left(\phi {\widehat{\varkappa }}_{v_m}\right).\left(\psi v_m\right)-\parallel \left(\phi {\widehat{\varkappa }}_{v_n}\right).\left(\psi v_m\right)\parallel +\parallel \left(\phi {\widehat{\varkappa }}_{v_n}\right).\left(\psi v_m\right)-\left(\phi {\widehat{\varkappa }}_{v_n}\right).\left(\psi v_n\right)\parallel \hfill \\ \hfill & \le \parallel \psi v_m\parallel \parallel \phi {\widehat{\varkappa }}_{v_m}-\phi {\widehat{\varkappa }}_{v_n}\parallel +\parallel \phi {\widehat{\varkappa }}_{v_n}\parallel \parallel \psi v_m-\psi v_n\parallel \hfill \\ \hfill & \le L\parallel {\widehat{\varkappa }}_{v_m}-{\widehat{\varkappa }}_{v_n}\parallel +M\parallel \psi v_m-\psi v_n\parallel ,\hfill \end{array}$$

which deduces that

$$\underset{m,n\to \infty }{lim\; sup}\parallel g{v}_m-g{v}_n\parallel \le \underset{m,n\to \infty }{lim\; sup}\frac{M}{1-L}\parallel \psi v_m-\psi v_n\parallel = 0.$$

This implies that $g\left({\mathcal{F}}_0\right)$ is totaly bounded, and so it is relatively compact. Hence, g is a compact operator.

Consider the composition operator ${\mathbb{P}}_{{\mathcal{E}}_0}g:{\mathcal{F}}_0\to {\mathcal{F}}_0$, which is compact and continuous on a bounded, closed, and convex set ${\mathcal{F}}_0$. Schauder’s FPT leads to the existence of a fixed point for ${\mathbb{P}}_{{\mathcal{E}}_0}g$, say $\vartheta \in {\mathcal{F}}_0$, that is, ${\mathbb{P}}_{{\mathcal{E}}_0}g\vartheta = \vartheta$ and so, $g\vartheta = {\mathbb{P}}_{{\mathcal{E}}_0}\vartheta$ because of the fact that ${\mathbb{P}}^2 = I$. Thereby,

$$\begin{array}{cc}\hfill \mathfrak{D}(\mathcal{E},\mathcal{F})& = \parallel \vartheta -{\mathbb{P}}_{{\mathcal{E}}_0}\vartheta \parallel = \parallel \vartheta -g\vartheta \parallel \hfill \\ \hfill & = \parallel \vartheta -\phi \left(g\vartheta \right).\psi \left(\vartheta \right)\parallel \hfill \\ \hfill & = \parallel \vartheta -\phi \left({\mathbb{P}}_{{\mathcal{E}}_0}\vartheta \right).\psi \left(\vartheta \right)\parallel ,\hfill \end{array}$$

and we are finished. □

Remark 3. 

We can replace the strict convexity of a Banach space $\mathcal{X}$ with the P-property of the pair $(\mathcal{E},\mathcal{F})$ in Theorem 9.

The following novel fixed point result is a direct consequence of Theorem 9.

Corollary 4. 

Let $\mathcal{E}$ be a nonempty, compact and convex subset of a Banach algebra $\mathcal{X}$, and suppose $\phi ,\psi :\mathcal{E}\to \mathcal{X}$ are maps that satisfy the following assumptions:

(1)

φ is a contractive mapping;

(2)

ψ is a continuous operator with $L: = \parallel \psi \left(\mathcal{E}\right)\parallel < 1$;

(3)

$\phi \left(\mathcal{E}\right)\psi \left(\mathcal{E}\right)\subseteq \mathcal{E}$.

Then the multiplication mapping $\phi .\psi$ has a fixed point, i.e.,

$$\mathit{there}\mathit{exists}\varkappa \in \mathcal{E}\mathit{such}\mathit{that}\left(\phi \varkappa \right).\left(\psi \varkappa \right) = \varkappa .$$

4. Sadovskii Proximal Condensing Operators

In this section, we apply a concept of measure of noncompactness to introduce a new family of condensing operators that satisfy the Sadovskii contractive condition.

To this end, we recall the notion of measure of noncompactness, which was used to extend Schauder’s FPT. Throughout this section, $\mathcal{B}\left(\mathcal{X}\right)\left(\mathcal{K}\left(\mathcal{X}\right)\right)$ stands for the set of all nonempty and bounded (compact) subsets of a Banach space $\mathcal{X}$.

Definition 5. 

A function $\mu :\mathcal{B}\left(\mathcal{X}\right)\to [0,+\infty )$ is said to be a measure of noncompactness (MNC) if it satisfies the following axioms:

(1)

The family $ker\left(\mu \right): = \left\{\mathcal{E}\in \mathcal{B}\left(\mathcal{X}\right):\mu \left(\mathcal{E}\right) = 0\right\}$ is nonempty and $ker\left(\mu \right)\subseteq \mathcal{K}\left(\mathcal{X}\right)$;

(2)

$\mu \left(\mathcal{E}\right) = \mu \left(\overline{\mathcal{E}}\right)$, $\mathcal{E}\in \mathcal{B}\left(\mathcal{X}\right)$;

(3)

If $\mathcal{E}\subseteq \mathcal{F}$, then $\mu \left(\mathcal{E}\right)\subseteq \mu \left(\mathcal{F}\right)$, where $\mathcal{E},\mathcal{F}\in \mathcal{B}\left(\mathcal{X}\right)$;

(4)

$\mu \left(\mathrm{con}\left(\mathcal{E}\right)\right) = \mu \left(\mathcal{E}\right)$, where $\mathrm{con}\left(\mathcal{E}\right)$ denotes the convex hull of the set $\mathcal{E}\subseteq \mathcal{X}$;

(5)

If $\underset{n\to \infty }{lim}\mu \left({\mathcal{E}}_n\right) = 0$ for a nonincreasing sequence $\left\{{\mathcal{E}}_n\right\}$ of nonempty, bounded and closed subsets of $\mathcal{X}$, then

$${\mathcal{E}}_{\infty }: = \bigcap _{n \ge 1}{\mathcal{E}}_n\ne \varnothing .$$

Note that ${\mathcal{E}}_{\infty }\in ker\left(\mu \right)$.

A trivial example of MNCs is the function

$$\mu \left(\mathcal{E}\right) = \mathrm{diam}\left(E\right),\mathrm{for}\mathrm{all}\mathcal{E}\in \mathcal{B}\left(\mathcal{X}\right).$$

We refer to [10] for more interesting examples and applications of MNCs.

Consider a non-self-map $\phi :\mathcal{E}\to \mathcal{F}$ where $(\mathcal{E},\mathcal{F})$ is a nonempty, bounded, closed, and convex pair in a Banach space $\mathcal{X}$ with ${\mathcal{E}}_0\ne \varnothing$ and $\phi \left({\mathcal{E}}_0\right)\subseteq {\mathcal{F}}_0$.

By ${\mathcal{N}}_{(\mathcal{E},\mathcal{F})}\left(\phi \right)$ we denote the set of all pairs $(\mathcal{C},\mathcal{D})\subseteq (\mathcal{E},\mathcal{F})$ such that $(\mathcal{C},\mathcal{D})$ is a nonempty, closed, convex and proximinal pair with $\phi \left(\mathcal{C}\right)\subseteq \mathcal{D}$ and $\mathfrak{D}(\mathcal{C},\mathcal{D}) = \mathfrak{D}(\mathcal{E},\mathcal{F})$. It is worth noticing that $({\mathcal{E}}_0,{\mathcal{F}}_0)\in {\mathcal{N}}_{(\mathcal{E},\mathcal{F})}\left(\phi \right)\ne \varnothing$.

We are now in a position to introduce a novel family of non-self-mappings by using the concept of MNC as follows.

Definition 6.  

Let $(\mathcal{E},\mathcal{F})$ be a nonempty, bounded, closed, and convex pair in a Banach space $\mathcal{X}$ such that ${\mathcal{E}}_0\ne \varnothing$, and let μ be an MNC on $\mathcal{X}$. We say that $\phi :\mathcal{E}\to \mathcal{F}$ is a Sadovskii proximal condensing operator if $\phi \left({\mathcal{E}}_0\right)\subseteq {\mathcal{F}}_0$ and

$$\mu \left(\phi \left(\mathcal{C}\right)\right) < \mu \left(\mathcal{D}\right),forall(\mathcal{C},\mathcal{D})\in {\mathcal{N}}_{(\mathcal{E},\mathcal{F})}\left(\phi \right)with\mu \left(\mathcal{D}\right)>0.$$

Note that if $\mathcal{E} = \mathcal{F}$ in the above definition, then we get the concept of $\mu$-condensing operator, which was considered in [11].

Here is the main result of this section.

Theorem 10. 

Suppose $(\mathcal{E},\mathcal{F})$ is a nonempty, bounded, closed, and convex pair in a strictly convex Banach space $\mathcal{X}$ such that ${\mathcal{E}}_0\ne \varnothing$, and let μ be an MNC on $\mathcal{X}$. If $\phi :\mathcal{E}\to \mathcal{F}$ is a continuous Sadovskii proximal condensing operator, then φ has a best proximity point.

Proof. 

Let $(u_0,v_0)\in \mathcal{E}\times \mathcal{F}$ be such that $\parallel u_0-v_0\parallel = \mathfrak{D}(\mathcal{E},\mathcal{F})$. Then $v_0 = \mathbb{P}{u}_0$. Set

$$\Pi : = \left\{({\mathcal{C}}_i,{\mathcal{D}}_i)\in {\mathcal{N}}_{(\mathcal{E},\mathcal{F})}\left(\phi \right)\mathrm{such}\mathrm{that}(u_0,v_0)\in {\mathcal{C}}_i\times {\mathcal{D}}_i\right\}.$$

Note that $({\mathcal{E}}_0,{\mathcal{F}}_0)\in \Pi \ne \varnothing$. Let

$$({\mathcal{K}}_1,{\mathcal{K}}_2): = \bigcap _i({\mathcal{C}}_i,{\mathcal{D}}_i);({\mathcal{C}}_i,{\mathcal{D}}_i)\in \Pi .$$

Then $(u_0,v_0)\in {\mathcal{K}}_1\times {\mathcal{K}}_2$, and so, $\mathfrak{D}({\mathcal{K}}_1,{\mathcal{K}}_2) = \mathfrak{D}(\mathcal{E},\mathcal{F})$. Clearly, $({\mathcal{K}}_1,{\mathcal{K}}_2)$ is closed, convex, and $\phi$-invariant.

To show the proximinality of the pair $({\mathcal{K}}_1,{\mathcal{K}}_2)$, assume that $x\in {\mathcal{K}}_1$. Then $x\in \bigcap {\mathcal{C}}_i$ for each index i where $({\mathcal{C}}_i,{\mathcal{D}}_i)\in \Pi$. Since the pair $({\mathcal{C}}_i,{\mathcal{D}}_i)$ is proximinal for all i, there is a point $y_i\in {\mathcal{D}}_i$ for which $\parallel x-y_i\parallel = \mathfrak{D}(\mathcal{E},\mathcal{F})$.

The strict convexity assumption of the Banach space $\mathcal{X}$ implies that $y_i = y$ for all i, and so $y\in \bigcap {\mathcal{D}}_i$. Thus, the pair $({\mathcal{K}}_1,{\mathcal{K}}_2)$ is also proximinal, which ensures that $({\mathcal{K}}_1,{\mathcal{K}}_2)\in \Pi$ and that $({\mathcal{K}}_1,{\mathcal{K}}_2)$ is a minimal element of $\Pi$ w.r.t. the reverse inclusion relation.

Note that if $\mu \left({\mathcal{K}}_2\right) = 0$, then $\phi \left({\mathcal{K}}_1\right)\in ker\left(\mu \right)$. In this situation, the mapping $\phi :{\mathcal{K}}_1\to {\mathcal{K}}_2$ is a continuous and compact operator. By Proposition 1, since $\mathbb{P}$ is cyclic on ${\mathcal{K}}_1\cup {\mathcal{K}}_2$,

$$\mathbb{P}\phi \left({\mathcal{K}}_1\right)\subseteq \mathbb{P}\left({\mathcal{K}}_2\right)\subseteq {\mathcal{K}}_1,$$

which concludes that $\mathbb{P}\phi$ is a compact operator that maps the convex set ${\mathcal{K}}_1$ into itself, continuously. It now follows from Schauder’s FPT that

$$\mathrm{there}\mathrm{exists}\varkappa \in {\mathcal{K}}_1\mathrm{such}\mathrm{that}\mathbb{P}\phi \varkappa = \varkappa .$$

Then

$$\parallel \varkappa -\phi \varkappa \parallel = \parallel \mathbb{P}\phi \varkappa -\phi \varkappa \parallel = \mathfrak{D}(\mathcal{E},\mathcal{F}),$$

and the result follows in this case.

So, assume that $\mu \left({\mathcal{K}}_2\right)>0$. Define

$$\left\{\begin{array}{c}{\mathcal{L}}_2 = \overline{\mathrm{con}}\left(\phi \left({\mathcal{K}}_1\right)\cup \left\{v_0\right\}\right),\hfill \\ {\mathcal{L}}_1 = \mathbb{P}\left({\mathcal{L}}_2\right).\hfill \end{array}\right.$$

Note that $(u_0,v_0)\in {\mathcal{L}}_1\times {\mathcal{L}}_2$, which deduces that $\mathfrak{D}({\mathcal{L}}_1,{\mathcal{L}}_2) = \mathfrak{D}(\mathcal{E},\mathcal{F})$. By definition, the pair $({\mathcal{L}}_1,{\mathcal{L}}_2)$ is proximinal. Also, ${\mathcal{L}}_2$ is closed and convex.

We show that ${\mathcal{L}}_1$ is closed. Let ${\left\{x_n\right\}}_{n \ge 1}$ be a sequence in ${\mathcal{L}}_1$ such that $x_n\to x\in {\mathcal{K}}_1$. Then for each $n\in \mathbb{N}$, there is a point $y_n\in {\mathcal{L}}_2$ such that $x_n = \mathbb{P}{y}_n$. Since ${\mathbb{P}}_{{\mathcal{E}}_0}$ is continuous, $\mathbb{P}{x}_n\to \mathbb{P}x$. Also, by the fact that ${\mathbb{P}}^2 = I$, we obtain $y_n = \mathbb{P}{x}_n\to \mathbb{P}x\in {\mathcal{L}}_2$. So, $x = {\mathbb{P}}^{2}x\in \mathbb{P}\left({\mathcal{L}}_2\right) = {\mathcal{L}}_1$, that is, ${\mathcal{L}}_1$ is closed.

To see the convexity of ${\mathcal{L}}_1$, it is sufficient to note that by Proposition 1, ${\mathbb{P}}_{{\mathcal{F}}_0}$ is affine, and the result follows. Furthermore, ${\mathcal{L}}_2 = \overline{\mathrm{con}}\left(\phi \left({\mathcal{K}}_1\right)\cup \left\{v_0\right\}\right)\subseteq {\mathcal{K}}_2$ and so,

$${\mathcal{L}}_1 = \mathbb{P}\left({\mathcal{L}}_2\right)\subseteq \mathbb{P}\left({\mathcal{K}}_2\right)\subseteq {\mathcal{K}}_1.$$

Thus

$$\phi \left({\mathcal{L}}_1\right)\subseteq \phi \left({\mathcal{K}}_1\right)\subseteq \overline{\mathrm{con}}\left(\phi \left({\mathcal{K}}_1\right)\cup \left\{v_0\right\}\right) = {\mathcal{L}}_2,$$

which ensures that $({\mathcal{L}}_1,{\mathcal{L}}_2)$ is $\phi$-invariant. Hence, $({\mathcal{L}}_1,{\mathcal{L}}_2)\in \Pi$ and by the minimality of $({\mathcal{K}}_1,{\mathcal{K}}_2)$, we must have

$${\mathcal{L}}_1 = {\mathcal{K}}_1,{\mathcal{L}}_2 = {\mathcal{K}}_2.$$

Now, by this reality that $\phi$ is a Sadovskii proximal condensing operator, we have

$$\begin{array}{cc}\hfill \mu \left(\phi \left({\mathcal{K}}_1\right)\right)& < \mu \left({\mathcal{K}}_2\right) = \mu \left({\mathcal{L}}_2\right)\hfill \\ \hfill & = \mu \left(\overline{\mathrm{con}}\left(\phi \left({\mathcal{K}}_1\right)\cup \left\{v_0\right\}\right)\right)\hfill \\ \hfill & = \mu \left(\phi \left({\mathcal{K}}_1\right)\right),\hfill \end{array}$$

which is a contradiction, and this completes the proof. □

Remark 4.  

It is remarkable to note that we can replace the strict convexity assumption of the Banach space $\mathcal{X}$ in Theorem 10 with the P-property of the pair $(\mathcal{E},\mathcal{F})$.

The next results are obtained from Theorem 10.

Corollary 5

(Sadovskii’s FPT; [12]). Suppose $\mathcal{E}$ is a nonempty, bounded, closed, and convex subset of a Banach space $\mathcal{X}$, and let μ be an MNC on $\mathcal{X}$. If $\phi :\mathcal{E}\to \mathcal{E}$ is a continuous map such that

$$\mu \left(\phi \left(\mathcal{C}\right)\right) < \mu \left(\mathcal{C}\right),$$

for all nonempty set $\mathcal{C}\subseteq \mathcal{E}$ with $\mu \left(\mathcal{C}\right)>0$, then φ has a fixed point.

Proof. 

By considering $\mathcal{E} = \mathcal{F}$ in Theorem 10, the result follows. Note that we do not need the strict convexity of the Banach space $\mathcal{X}$, because $(\mathcal{E},\mathcal{F})$ has the P-property. □

Corollary 6  

([13]). Suppose $(\mathcal{E},\mathcal{F})$ is a nonvoid, bounded, closed, and convex pair in a strictly convex Banach space $\mathcal{X}$ such that ${\mathcal{E}}_0\ne \varnothing$ and let μ be an MNC on $\mathcal{X}$. If $\phi :\mathcal{E}\to \mathcal{F}$ is a continuous map such that $\phi \left({\mathcal{E}}_0\right)\subseteq {\mathcal{F}}_0$ and

$$\mu \left(\phi \left(\mathcal{C}\right)\right) \le r\mu \left(\mathcal{D}\right),forall(\mathcal{C},\mathcal{D})\in {\mathcal{N}}_{(\mathcal{E},\mathcal{F})}\left(\phi \right),$$

for some $r\in (0,1)$, then φ has a best proximity point.

Corollary 7  

(Darbo’s FPT; [14]). Suppose $\mathcal{E}$ is a nonempty, bounded, closed, and convex subset of a Banach space $\mathcal{X}$, and let μ be an MNC on $\mathcal{X}$. If $\phi :\mathcal{E}\to \mathcal{E}$ is a continuous map such that there exists $r\in (0,1)$ for which

$$\mu \left(\phi \left(\mathcal{C}\right)\right) \le r\mu \left(\mathcal{C}\right),$$

for all nonempty set $\mathcal{C}\subseteq \mathcal{E}$. Then φ has a fixed point.

The next result is the best proximity version of Schauder’s FPT.

Corollary 8.  

Let $(\mathcal{E},\mathcal{F})$ be a nonempty, bounded, closed, and convex pair in a strictly convex Banach space $\mathcal{X}$ such that ${\mathcal{E}}_0\ne \varnothing$. Suppose $\phi :\mathcal{E}\to \mathcal{F}$ is a compact and continuous operator for which $\phi \left({\mathcal{E}}_0\right)\subseteq {\mathcal{F}}_0$. Then φ has a best proximity point.

5. Applications

In the latest section of this paper, we present some applications related to the previous results.

5.1. Application to Complex Function Theory

Theorem 11. 

Consider a nonempty, bounded, closed, and convex pair $(\mathcal{E},\mathcal{F})$ of subsets of a domain $\mathcal{D}$ in the complex plane. Suppose φ is analytic in $\mathcal{D}$, which maps $\mathcal{E}$ into $\mathcal{F}$, and $\psi :\mathcal{F}\to \mathcal{E}$ is a mapping for which $\left|\phi \right(z)-\psi (\omega \left)\right| \le |z-\omega |$ for all $(z,\omega )\in \mathcal{E}\times \mathcal{F}$. If $|{\phi }^{\prime }\left(z\right)| < 1$ for all $z\in \mathcal{E}$, then there exists a unique point $\omega \in \mathcal{E}$ such that

$$|\omega -\phi \omega | = \mathfrak{D}(\mathcal{E},\mathcal{F}).$$

Proof. 

Since $(\mathcal{E},\mathcal{F})$ is a compact pair, $({\mathcal{E}}_0,{\mathcal{F}}_0)$ is nonempty. Moreover, since the complex plane with Euclidean norm is strictly convex and the pair $(\mathcal{E},\mathcal{F})$ is convex, by a result of [7], $(\mathcal{E},\mathcal{F})$ has the P-property.

Now let $z\in {\mathcal{E}}_0$ be an arbitrary element. Then there is a point $\omega \in {\mathcal{F}}_0$ such that $|z-\omega | = \mathfrak{D}(\mathcal{E},\mathcal{F})$. From the hypothesis of the theorem, we must have

$$\left|\phi \right(z)-\psi (\omega \left)\right| \le |z-\omega | = \mathfrak{D}(\mathcal{E},\mathcal{F}),$$

which implies that $\phi z\in {\mathcal{F}}_0$ and so, $\phi \left({\mathcal{E}}_0\right)\subseteq {\mathcal{F}}_0$. Furthermore, for any $z_1\ne z_2\in {\mathcal{E}}_0$ we have

$$|\phi \left(z_2\right)-\phi \left(z_1\right)| = |{\int }_{z_1}^{z_2}{\phi }^{\prime }\left(\zeta \right)d\zeta | < |z_2-z_1|,$$

that is, $\phi$ is a contractive non-self-mapping. Hence, the result follows by invoking Corollary 2. □

Corollary 9. 

Let $\mathcal{E}$ be a nonempty, bounded, closed, and convex subset of a domain $\mathcal{D}$ in the complex plane. Assume that φ is analytic in $\mathcal{D}$, which maps $\mathcal{E}$ into itself. If $|{\phi }^{\prime }\left(z\right)| < 1$ for all $z\in \mathcal{E}$, then φ has a unique fixed point.

Proof. 

It is sufficient to consider $\mathcal{E} = \mathcal{F}$ and $\psi = \phi$ in Theorem 11. □

5.2. Application to Nonlinear Functional Integral Equations

In the continuation of this section, we focus on hybrid FPTs obtained in Corollary 3 and Corollary 4 to guarantee the existence of solutions of two classes of nonlinear functional integral equations.

The first class of integral equations, which is considered in Example 3, contains the sum of two operators, and the second class of integral equations, which appears in Example 4, consists of the multiplication of two operators.

In this way, for the first family of integral equations, we need to use an extension of Krasnoselskii’s FPT, which was concluded in Corollary 3, and for the second category of nonlinear functional integral equations, we apply a generalization of Dehage’s FPT, which was deduced in Corollary 4.

Example 3. 

Given a closed interval $J = [0,1]\subseteq \mathbb{R}$ and $a\in (0,\frac{1}{2}]$, the following nonlinear functional integral equation (NFIE)

$$u\left(t\right) = sin\left(\frac{a}{1+\left|u\right(t\left)\right|}\right)+a{\int }_0^t\kappa \left(s,u\left(s\right)\right)ds,forallt\in J,$$

(1)

where $\kappa :J\times \mathbb{R}\to \mathbb{R}$ is a continuous and bounded function, has a solution.

Proof. 

Consider a Banach space $\mathcal{X} = C\left(J\right)$ endowed with the supremum norm and define

$$\mathcal{E} = \left\{u\in \mathcal{X}:|u\left(x_1\right)-u\left(x_2\right)| \le (L+1)|x_1-x_2|,u\left(0\right) = sina\right\},$$

where $\left|\kappa \right(t,s\left)\right| \le L$ for some $L>0$. By using Arzela–Ascoli’s theorem, we find that $\mathcal{E}$ is a compact subset of $\mathcal{X}$, and it is easy to see that $\mathcal{E}$ is convex. Let $\phi :\mathcal{E}\to \mathcal{X}$ and $\psi :\mathcal{E}\to \mathcal{X}$ be defined by

$$\left\{\begin{array}{c}\phi u\left(t\right) = sin\left(\frac{a}{1+\left|u\right(t\left)\right|}\right),\mathrm{for}\mathrm{all}t\in J,\hfill \\ \psi u\left(t\right) = a{\int }_0^t\kappa \left(s,u\left(s\right)\right)ds,\mathrm{for}\mathrm{all}t\in J.\hfill \end{array}\right.$$

Note that $\psi$ is continuous. Also, for all $u,v\in \mathcal{E}$ we have

$$\begin{array}{cc}\hfill \left|\phi u\right(t)-\phi v(t\left)\right|& = |sin\left(\frac{a}{1+\left|u\right(t\left)\right|}\right)-sin\left(\frac{a}{1+\left|v\right(t\left)\right|}\right)|\hfill \\ \hfill & \le |\frac{a}{1+\left|u\right(t\left)\right|}-\frac{a}{1+\left|v\right(t\left)\right|}|\hfill \\ \hfill & = \frac{a\left|\right|u\left(t\right)|-|v\left(t\right)\left|\right|}{(1+|u\left(t\right)\left|\right)(1+|v\left(t\right)\left|\right)}\hfill \\ \hfill & \le a\left|u\right(t)-v(t\left)\right|,\mathrm{for}\mathrm{all}t\in J,\hfill \end{array}$$

which deduces that

$${\parallel \phi u-\phi v\parallel }_{\infty } < {\parallel u-v\parallel }_{\infty },$$

that is, $\phi$ is a contractive map. Moreover, for any $u\in \mathcal{E}$ we have $\left(\phi +\psi \right)u\left(0\right) = sina$ and

$$\begin{array}{cc}\hfill |\left(\phi u+\psi v\right)\left(x_1\right)& - \left(\phi u+\psi v\right)\left(x_2\right)| = |\left(sin\left(\frac{a}{1+\left|u\right(x_1\left)\right|}\right)+a{\int }_0^{x_1}\kappa \left(s,v\left(s\right)\right)ds\right)\hfill \\ \hfill & - \left(sin\left(\frac{a}{1+\left|u\right(x_2\left)\right|}\right)+a{\int }_0^{x_2}\kappa \left(s,v\left(s\right)\right)ds\right)|\hfill \\ \hfill & \le |sin\left(\frac{a}{1+\left|u\right(x_1\left)\right|}\right)-sin\left(\frac{a}{1+\left|u\right(x_2\left)\right|}\right)|+a{\int }_{x_1}^{x_2}\left|\kappa \left(s,v\left(s\right)\right)\right|ds\hfill \\ \hfill & \le a\left|\right|u\left(x_1\right)|-|u\left(x_2\right)\left|\right|+aL|x_1-x_2|\hfill \\ \hfill & \le a(L+1)|x_1-x_2|+aL|x_1-x_2|\hfill \\ \hfill & \le (L+1)|x_1-x_2|,\hfill \end{array}$$

which implies that $\phi \left(\mathcal{E}\right)+\psi \left(\mathcal{E}\right)\subseteq \mathcal{E}$. Now by applying Corollary 3, the NFIE (1) has a solution. □

Example 4. 

For some $a\in (0,\frac{1}{2}]$ consider the following NFIE

$$u\left(t\right) = sin\left(\frac{a}{1+\left|u\right(t\left)\right|}\right){\int }_0^t\kappa \left(s,u\left(s\right)\right)ds,forallt\in J: = [0,1],$$

(2)

where $\kappa :J\times \mathbb{R}\to \mathbb{R}$ is a continuous function. If $\left|\kappa \right(t,s\left)\right| \le 1$ for all $(t,s)\in J\times \mathbb{R}$, then Equation (2) has a solution.

Proof. 

Consider a Banach algebra $\mathcal{X} = C\left(J\right)$ and define

$$\mathcal{E} = \left\{u\in \mathcal{X}:|u\left(x_1\right)-u\left(x_2\right)| \le |x_1-x_2|,u\left(0\right) = 0\right\}.$$

Arzela–Ascoli’s theorem implies that $\mathcal{E}$ is a compact subset of $\mathcal{X}$. Let $\phi :\mathcal{E}\to \mathcal{X}$ and $\psi :\mathcal{E}\to \mathcal{X}$ be defined as

$$\left\{\begin{array}{c}\phi u\left(t\right) = sin\left(\frac{a}{1+\left|u\right(t\left)\right|}\right),\mathrm{for}\mathrm{all}t\in J,\hfill \\ \psi u\left(t\right) = {\int }_0^t\kappa \left(s,u\left(s\right)\right)ds,\mathrm{for}\mathrm{all}t\in J.\hfill \end{array}\right.$$

By a similar proof of Example 3, $\phi :\mathcal{E}\to \mathcal{X}$ is contractive. Moreover, $\psi :\mathcal{E}\to \mathcal{X}$ is continuous and

$${\parallel \psi u\parallel }_{\infty } = \underset{t\in J}{sup}\left|\psi u\left(t\right)\right| = \underset{t\in J}{sup}\left|{\int }_0^t\kappa \left(s,u\left(s\right)\right)ds\right| \le 1,\mathrm{for}\mathrm{all}u\in \mathcal{E},$$

and so, ${\parallel \psi \left(\mathcal{E}\right)\parallel }_{\infty } \le 1$. We also note that $\left(\phi u\right)\left(\psi u\right)\left(0\right) = 0$ for all $u\in \mathcal{X}$. Furthermore, for any $x_1,x_2\in J$ and $u,v\in \mathcal{E}$ we have

$$\begin{array}{cc}\hfill & |\left(\phi u\psi v\right)\left(x_1\right)-\left(\phi u\psi v\right)\left(x_2\right)|\hfill \\ & = |sin\left(\frac{a}{1+\left|u\right(x_1\left)\right|}\right){\int }_0^{x_1}\kappa \left(s,v\left(s\right)\right)ds-sin\left(\frac{a}{1+\left|u\right(x_2\left)\right|}\right){\int }_0^{x_2}\kappa \left(s,v\left(s\right)\right)ds|\hfill \\ \hfill & \le sin\left(\frac{a}{1+\left|u\right(x_1\left)\right|}\right){\int }_{x_1}^{x_2}|\kappa \left(s,v\left(s\right)\right)|ds+|sin\left(\frac{a}{1+\left|u\right(x_1\left)\right|}\right)-sin\left(\frac{a}{1+\left|u\right(x_2\left)\right|}\right)|{\int }_0^{x_2}\left|\kappa \left(s,v\left(s\right)\right)\right|ds\hfill \\ \hfill & \le a|x_2-x_1|+a|u\left(x_2\right)-u\left(x_1\right)|\hfill \\ \hfill & \le 2a|x_2-x_1|\hfill \\ \hfill & \le |x_2-x_1|.\hfill \end{array}$$

Thereby, $\phi \left(\mathcal{E}\right)\psi \left(\mathcal{E}\right)\subseteq \mathcal{E}$. Thus, all of the assumptions of Corollary 4 are satisified, and so the NFIE (2) has a solution. □

5.3. Application to a System of Integral Equations

As another application of our main conclusions, we investigate the existence of a mutually nearest solution for a system of integral equations that does not have a common solution.

To this end, let $J = [0,1]$, $R = J\times J$, and $\alpha ,\beta \in J$ be such that $\alpha >\beta$. Assume that $f_1,f_2$ are continuous non-negative real functions on R.

Also, let $a\left(t\right)\in C\left(J\right)$ be such that $a\left(t\right) \ge 0$ for all $t\in J$ and $a\left(0\right) = 0$. Let us consider the following system:

$$\left\{\begin{array}{cc}u\left(t\right)\hfill & = a\left(t\right)+\alpha +{\int }_0^t{f}_1\left(s,u\left(s\right)\right)ds,\hfill \\ v\left(t\right)\hfill & = \beta -{\int }_0^t{f}_2\left(s,v\left(s\right)\right)ds.\hfill \end{array}\right.$$

(3)

We note that the above system does not have a common solution for both equations. To define an appropriate solution for this system, we need the following requirements.

Let $C\left(J\right)$ be renormalized according to

$$\parallel h\parallel = {\parallel h\parallel }_2+{\parallel h\parallel }_{\infty },\mathrm{for}\mathrm{all}h\in C\left(J\right).$$

It is well-known that $\left(C\left(J\right),\parallel .\parallel \right)$ is a non-reflexive strictly convex Banach space such that ${\parallel h\parallel }_{\infty } \le \parallel h\parallel \le {2\parallel h\parallel }_{\infty }$ for all $h\in C\left(J\right)$. Set

$$\begin{array}{cc}\hfill \mathcal{E}& = \left\{p\in C\left(J\right):\left|p\right(t)-\beta | \le 1,p\left(0\right) = \alpha ,p\left(t\right) \ge \alpha ,\mathrm{for}\mathrm{all}t\in J\right\},\hfill \\ \hfill \mathcal{F}& = \left\{q\in C\left(J\right):\left|q\right(t)-\alpha | \le 1,q\left(0\right) = \beta ,q\left(t\right) \le \beta ,\mathrm{for}\mathrm{all}t\in J\right\}.\hfill \end{array}$$

Obviously, the pair $(\mathcal{E},\mathcal{F})$ is closed and convex in $C\left(J\right)$ and $(\alpha ,\beta )\in \mathcal{E}\times \mathcal{F}$. Suppose that $M>0$ is a common bound for the functions $f_1,f_2$ on R. Choose

$$\rho \le min\left\{1,\frac{1-|\beta -\alpha |}{M},\frac{|\beta -\alpha |}{4M}\right\},$$

and consider the closed interval $J^{\prime } = [0,\rho ]$. Now define a non-self-mapping $\phi :\mathcal{E}\to C\left(J^{\prime }\right)$ with

$$\phi \left(p\left(t\right)\right) = \beta -{\int }_0^t{f}_2(s,p\left(s\right))ds,\mathrm{for}\mathrm{all}t\in J^{\prime }.$$

(4)

Then $\phi \left(p\left(0\right)\right) = \beta$, and clearly, $\phi \left(p\left(t\right)\right) \le \beta$. Also, for any $t\in J^{\prime }$ we have

$$\begin{array}{cc}\hfill |\phi \left(p\left(t\right)\right)-\alpha |& = |\left(\beta +{\int }_0^t{f}_2(s,p\left(s\right))ds\right)-\alpha |\hfill \\ \hfill & \le |\beta -\alpha |+{\int }_0^t{f}_2(s,p\left(s\right))ds\hfill \\ \hfill & \le |\beta -\alpha |+\rho M \le 1.\hfill \end{array}$$

Thus, $\phi p\in \mathcal{F}$, and so, $\phi$ maps the set $\mathcal{E}$ into $\mathcal{F}$.

We are now in a position to introduce the following notion for the system (3).

Definition 7. 

A function $\varkappa \in C\left(J\right)$ is a mutually nearest solution for the system of integral Equation (3) provided that ϰ is a best proximity point for the non-self-mapping φ defined in Equation (4).

We now state the following existence result.

Theorem 12. 

Under the aforementioned notations and definitions, if

$$f_1(s,q\left(s\right))+f_2(s,p\left(s\right)) \le \frac{1}{2}|q\left(s\right)-p\left(s\right)|-|\beta -\alpha |,$$

(5)

provided that $\frac{1}{2}|q\left(s\right)-p\left(s\right)|>|\beta -\alpha |$, then the system (3) has a mutually nearest solution.

Proof. 

At first, let us estimate the distance between two sets $\mathcal{E}$ and $\mathcal{F}$. For any $(p,q)\in \mathcal{E}\times \mathcal{F}$, we have $p\left(t\right)-q\left(t\right) \ge \alpha -\beta >0$ and so

$${\parallel p-q\parallel }_2 = {\left({\int }_0^1{|p\left(t\right)-q\left(t\right)|}^{2}dt\right)}^{\frac{1}{2}} \ge {\left({\int }_0^1{|\beta -\alpha |}^{2}dt\right)}^{\frac{1}{2}} = |\beta -\alpha |.$$

It now follows from the definition of the norm $\parallel .\parallel$ on $C\left(J\right)$ that

$$\parallel p-q\parallel = {\parallel p-q\parallel }_2{+\parallel p-q\parallel }_{\infty } \ge 2|\beta -\alpha |.$$

On the other hand, since $(\alpha ,\beta )\in \mathcal{E}\times \mathcal{F}$, we must have $\mathfrak{D}(\mathcal{E},\mathcal{F}) = 2|\beta -\alpha |$. Thus, $(\alpha ,\beta )\in {\mathcal{E}}_0\times {\mathcal{F}}_0$. Also, since $\left(C\right(J),\parallel .\parallel )$ is strictly convex, the proximal pair $({\mathcal{E}}_0,{\mathcal{F}}_0)$ is closed ([7]), and it is easy to see that it is also convex.

We assert that $\phi \left({\mathcal{E}}_0\right)\subseteq {\mathcal{F}}_0$. In this regard, define a map $\psi :\mathcal{F}\to C\left(J^{\prime }\right)$ as follows:

$$\psi \left(q\left(t\right)\right) = \alpha +{\int }_0^t{f}_1(s,q\left(s\right))ds,\mathrm{for}\mathrm{all}t\in J^{\prime }.$$

Then $\psi \left(q\left(t\right)\right) \ge \alpha$ and $\psi \left(q\left(0\right)\right) = \alpha$. Moreover,

$$\begin{array}{cc}\hfill |\psi \left(q\left(t\right)\right)-\beta |& = |\left(\alpha +{\int }_0^t{f}_1(s,q\left(s\right))ds\right)-\beta |\hfill \\ \hfill & \le |\beta -\alpha |+{\int }_0^t{f}_1(s,q\left(s\right))ds\hfill \\ \hfill & \le |\beta -\alpha |+\rho M \le 1,\hfill \end{array}$$

which deduces that $\psi$ maps the set $\mathcal{F}$ to the set $\mathcal{E}$. Set

$$\begin{array}{cc}\hfill {\gamma }_1& = \left\{s\in J^{\prime };\frac{1}{2}|q\left(s\right)-p\left(s\right)|>|\beta -\alpha |\right\},{\gamma }_2 = J^{\prime }-{\gamma }_1.\hfill \end{array}$$

Hence for an element $(p,q)\in \mathcal{E}\times \mathcal{F}$ by choosing the constant $\rho$ we obtain

$$\begin{array}{cc}\hfill |\phi \left(p\left(t\right)\right)-\psi \left(q\left(t\right)\right)|& = |\left(\beta -{\int }_0^t{f}_2(s,p\left(s\right))ds\right)-\left(\alpha +{\int }_0^t{f}_1(s,q\left(s\right))ds\right)|\hfill \\ \hfill & \le |\beta -\alpha |+{\int }_0^t\left(f_2(s,p\left(s\right))+f_1(s,q\left(s\right))\right)ds\hfill \\ \hfill & \le |\beta -\alpha |+{\int }_{{\gamma }_1}\left(f_2(s,p\left(s\right))+f_1(s,q\left(s\right))\right)ds+{\int }_{{\gamma }_2}\left(f_2(s,p\left(s\right))+f_1(s,q\left(s\right))\right)ds\hfill \\ \hfill & \le |\beta -\alpha |+\frac{1}{2}|p\left(s\right)-q\left(s\right)|-|\beta -\alpha |+2M\rho \hfill \\ \hfill & \le \frac{1}{2}{\parallel p-q\parallel }_{\infty }+\frac{1}{2}|\beta -\alpha |\hfill \\ \hfill & \le \frac{1}{2}{\parallel p-q\parallel }_{\infty }+\frac{1}{2}{\parallel p-q\parallel }_2 = \frac{1}{2}\parallel p-q\parallel .\hfill \end{array}$$

Thereby, $\parallel \phi \left(p\right)-\psi \left(q\right){\parallel }_{\infty } \le \frac{1}{2}\parallel p-q\parallel$ which yields that

$$\parallel \phi \left(p\right)-\psi \left(q\right)\parallel \le 2\parallel \phi \left(p\right)-\psi \left(q\right){\parallel }_{\infty } \le \parallel p-q\parallel .$$

(6)

Now to show $\phi \left({\mathcal{E}}_0\right)\subseteq {\mathcal{F}}_0$, suppose that $p\in {\mathcal{E}}_0$. Then there is an element $q\in {\mathcal{F}}_0$ for which $\parallel p-q\parallel = \mathfrak{D}(\mathcal{E},\mathcal{F})$. It follows from the inequality (6) that

$$\parallel \phi \left(p\right)-\psi \left(q\right)\parallel \le \parallel p-q\parallel = \mathfrak{D}(\mathcal{E},\mathcal{F}),$$

which guarantees that $\phi \left(p\right)\in {\mathcal{F}}_0$. Furthermore, $\phi$ is continuous, and for any $p\in \mathcal{E}$, we have

$$|\phi \left(p\left(t\right)\right)| = |\beta -{\int }_0^t{f}_2(s,p\left(s\right))ds| \le \beta +{\int }_0^t{f}_2(s,p\left(s\right))ds \le \beta +M\rho ,$$

and so, the class of $\{\phi p{\}}_{p\in \mathcal{E}}$ is bounded. Furthermore, for any $p\in \mathcal{E}$ and $t_1 < t_2\in J^{\prime }$ we have

$$\begin{array}{cc}\hfill |\phi \left(p\left(t_1\right)\right)-\phi \left(p\left(t_2\right)\right)|& = |\left(\beta -{\int }_0^{t_1}{f}_2(s,p\left(s\right))ds\right)-\left(\beta -{\int }_0^{t_2}{f}_2(s,p\left(s\right))ds\right)|\hfill \\ \hfill & = {\int }_{t_1}^{t_2}{f}_2(s,p\left(s\right))ds \le M|t_2-t_1|,\hfill \end{array}$$

which concludes that $\{\phi p{\}}_{p\in \mathcal{E}}$ is equicontinuous. Hence, by applying Arzela–Ascoli’s theorem, the set $\phi \left(\mathcal{E}\right)$ is relatively compact.

Now, Corollary 8 ensures the existence of a best proximity point for the mapping $\phi$, which is a mutually nearest solution for the system (3). □

6. Conclusions

This work focused on the best proximity types of well-known fixed-point problems due to Krasnoselskii, Dhage, and Sadovskii in the framework of strictly convex Banach spaces (see Theorems 8–10). The proofs of these theorems are based on using the proximal projection operator defined in Proposition 1. As applications, we surveyed the existence of a solution for two classes of nonlinear functional integral equations, one of which contains the sum of two nonlinear operators, and another one consists of the multiplication of two nonlinear operators. We also considered a system of integral equations that does not have a common solution, and for this system, we introduced a notion of a mutually nearest solution, which is indeed a best proximity point for a corresponding operator to the system, and established the existence of such solutions by using Schauder’s best proximity theorem.