Extensions of Göhde and Kannan Fixed Point Theorems in Strictly Convex Banach Spaces

Abstract

Let nonempty subsets E and F of a Banach space X be given, along with a mapping $\mathcal{S}:E\cup F\to E\cup F$ defined as noncyclic when $\mathcal{S}\left(E\right)\subseteq E$ and $\mathcal{S}\left(F\right)\subseteq F$. In this case, an optimal pair of fixed points is defined as a point $(p,q)\in E\times F$ where p and q are fixed points of $\mathcal{S}$ that estimate the distance between E and F. This article explores an extended version of Göhde’s fixed point problem to identify optimal fixed point pairs for noncyclic relatively nonexpansive maps in strictly convex Banach spaces, while introducing new classes of noncyclic Kannan contractions, noncyclic relatively Kannan nonexpansive contractions using the proximal projection mapping defined on union of proximal pairs, and proving additional existence results with supporting examples.

1. Introduction

In 1948, Brodskii and Milman presented the concept of normal structure.

Definition 1 

([1]). A convex subset E of a Banach space X is said to possess a normal structure if for any convex set $H\subseteq E$ with a positive diameter, there exists a point $x_0\in H$ such that

$$\underset{x\in H}{sup}\parallel x-x_0\parallel <\mathrm{diam}\left(H\right).$$

In other words, H is enclosed in a ball centered at a point in H with a radius smaller than the diameter of H.

Brodskii and Milman demonstrated that a family of isometric mappings from a convex compact subset of a Banach space into itself has a common fixed point by leveraging the concept of normal structure, which is well-established for nonempty, compact, and convex subsets of such spaces, as well as for nonempty, bounded, closed, and convex subsets of uniformly convex Banach spaces (refer to [2] for further details and examples).

Note that a Banach space X is $\left(i\right)$ strictly convex if for any two distinct points $x,y\in X$ with $\parallel x\parallel =\parallel y\parallel =1$, we have $\parallel {\displaystyle \frac{x+y}{2}}\parallel <1$; $\left(ii\right)$ uniformly convex if for any $\epsilon \in [0,2]$

$$inf\left\{1-{\displaystyle \frac{\parallel x+y\parallel }{2}}:\parallel x\parallel \le 1,\parallel y\parallel \le 1,\parallel x-y\parallel \ge \epsilon \right\}>0.$$

Hilbert spaces and $l^p$ spaces, $(1<p<\infty )$ are uniformly convex. Clearly, every uniformly convex Banach space is strictly convex, but the reverse may not necessarily be true; for instance, the Banach space $l^1$ with the norm

$$\parallel x\parallel ={\left({\parallel x\parallel }_1^2+{\parallel x\parallel }_2^2\right)}^{{\displaystyle \frac{1}{2}}},\forall x\in l^1,$$

(where ${\parallel .\parallel }_1$ and ${\parallel .\parallel }_2$ are the norms on $l^1$ and $l^2$, respectively) is strictly convex but not uniformly convex (see [3] for more details).

The primary significance of the normal structure lies in its contribution to fixed point theory for nonexpansive self-mappings, a concept first introduced by W.A. Kirk.

Theorem 1 

(Kirk’s fixed point theorem [4]). Let E be a nonempty, weakly compact and convex subset of a Banach space X and $\mathcal{S}:E\to E$ be a nonexpansive mapping, that is,

$$\parallel \mathcal{S}x-\mathcal{S}y\parallel \le \parallel x-y\parallel ,\forall x,y\in E.$$

If E possesses a normal structure, then $\mathcal{S}$ has a fixed point.

The initial exploration of fixed points for nonexpansive maps without a normal structure was conducted by Göhde.

Theorem 2 

(Göhde’s fixed point theorem [5]; see also [6]). Let E be a nonempty, bounded, closed and convex subset of a normed linear space X and $\mathcal{S}:E\to E$ be a nonexpansive map. If there is a compact subset M of E such that for any$x\in E$, the sequence $\left\{{\mathcal{S}}^{n}x\right\}$ has a limit point in M, then $\mathcal{S}$ has a fixed point.

In 1973, Kannan presented a new set of self-mappings.

Definition 2. 

Let E be a nonempty subset of a Banach space X. A mapping $\mathcal{S}:E\to E$ is said to be Kannan nonexpansive if

$$\parallel \mathcal{S}x-\mathcal{S}y\parallel \le {\displaystyle \frac{1}{2}}\left(\parallel x-\mathcal{S}x\parallel +\parallel y-\mathcal{S}y\parallel \right),\forall x,y\in E.$$

It is important to highlight that Kannan nonexpansive mappings might not be continuous, indicating that the classes of nonexpansive and Kannan nonexpansive mappings are distinct, leading to a different version of the fixed point Theorem 1.

Theorem 3 

(Kannan’s fixed point theorem [7]). Let E be a nonempty, weakly compact and convex subset of a Banach space X and $\mathcal{S}:E\to E$ be a Kannan nonexpansive mapping such that for any nonempty, closed, convex and $\mathcal{S}$-invariant subset H of E with $\mathrm{diam}\left(H\right)>0$, we have

$$sup\{\parallel x-\mathcal{S}x\parallel :x\in H\}<\mathrm{diam}\left(H\right).$$

Then, $\mathcal{S}$ has a unique fixed point.

Soardi ([8]) demonstrated a counterpart result of Kirk’s fixed point theorem for Kannan nonexpansive maps using normal structure, which Wong later enhanced by introducing the concept of close-to-normal structures to generalize normal structures (see [9]).

This paper outlines its structure as follows: In Section 2, we recall some fundamental notions and notations which will be used in our next discussions. Section 3 extends Göhde’s fixed point theorem to noncyclic relative mappings (resulting in a version of Theorem 4 that does not rely on a geometric understanding of proximal normal structure). Section 4 introduces a new category of mappings called noncyclic Kannan contractions, utilizing proximal projection mappings, and examines the existence and convergence of optimal pairs of fixed points for these mappings. Finally, in Section 5, we extend the class of noncyclic Kannan contractions to noncyclic relatively Kannan nonexpansive maps and present a generalization of Kannan’s fixed point theorem.

2. Preliminaries

In 2005, Eldred et al. ([10]) generalized Theorem 1 to include noncyclic relatively nonexpansive maps by defining the concept of proximal normal structure, and before presenting their main result, we must clarify certain notions and notations.

Let E and F be two nonempty subsets of a Banach space X, and we say that the pair $(E,F)$ has a property if both sets satisfy that property; for example, $(E,F)$ is considered a closed pair if both E and F are closed.

A mapping $\mathcal{S}:E\cup F\to E\cup F$ is said to be noncyclic provided that $\mathcal{S}\left(E\right)\subseteq E$ and $\mathcal{S}\left(F\right)\subseteq F$. In this situation, a point $(p,q)\in E\times F$ is called an optimal pair of fixed points of $\mathcal{S}$ if

$$p=\mathcal{S}p,q=\mathcal{S}q,\parallel p-q\parallel =\mathrm{D}(E,F):=inf\{\parallel x-y\parallel :(x,y)\in E\times F\}.$$

The collection of all optimal fixed point pairs for the noncyclic mapping $\mathcal{S}$ is represented as ${\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right)$. A mapping $\mathcal{S}:E\cup F\to E\cup F$ is called relatively nonexpansive if

$$\parallel \mathcal{S}x-\mathcal{S}y\parallel \le \parallel x-y\parallel ,\forall (x,y)\in E\times F.$$

It is important to note that relatively nonexpansive maps are not necessarily continuous. So, in the case that the mapping $\mathcal{S}:E\cup F\to E\cup F$ is noncyclic relatively nonexpansive, which is not continuous on $E\cup F$ necessarily, the problem is to find two distinct fixed points in E and F which approximate the distance between E and F, while many fixed point theorems are discussed on continuous maps.

Example 1. 

Consider the Banach space$X=({\mathbb{R}}^2,\parallel .{\parallel }_{\infty })$and let$E=\left\{\right(x,0):0\le x\le 1\},F=\left\{\right(x,1):0\le x\le 1\}$. Define$\mathcal{S}:E\cup F\to E\cup F$with

$$\mathcal{S}(x,0)=\left\{\begin{array}{c}(\sqrt{x},0),\mathit{if}x\in \mathbb{Q}\cap [0,1],\hfill \\ (x^2,0),\mathit{if}x\in {\mathbb{Q}}^c\cap [0,1],\hfill \end{array}\right.\mathcal{S}(x,1)=\left\{\begin{array}{c}(0,1),\mathit{if}x\in \mathbb{Q}\cap [0,1],\hfill \\ (1,1),\mathit{if}x\in {\mathbb{Q}}^c\cap [0,1].\hfill \end{array}\right.$$

Clearly, $\mathcal{S}$ is noncontinuous with  $\mathcal{S}\left(E\right)\subseteq E$, $\mathcal{S}\left(F\right)\subseteq F$ . Additionally, for any $\left((x,0),(y,1)\right)\in E\times F$, it is easy to see that 

$${\parallel \mathcal{S}(x,0)-\mathcal{S}(y,1)\parallel }_{\infty }=1={\parallel (x,0)-(y,1)\parallel }_{\infty },$$

 which implies that $\mathcal{S}$ is noncyclic relatively nonexpansive.

For a nonempty pair  $(E,F)$, its proximal pair is denoted by $(E_0,F_0)$, where 

$$E_0=\{x\in E:\parallel x-y^{\prime }\parallel =\mathrm{D}(E,F)\mathit{for}\mathit{some}{y}^{\prime }\in F\},$$

$$F_0=\{y\in F:\parallel x^{\prime }-y\parallel =\mathrm{D}(E,F)\mathit{for}\mathit{some}{x}^{\prime }\in E\}.$$

Proximal pairs may be empty; however, when $(E,F)$ forms a weakly compact and convex pair in a Banach space X, the pair $(E_0,F_0)$ will also be nonempty, closed, and convex (as noted in [11] for further sufficient conditions regarding the nonemptiness of proximal pairs in hyperconvex metric spaces).

Definition 3. 

A pair $(E,F)$ in a normed linear space X is proximinal if for every  $(x,y)\in E\times F$ , there is an element $(x^{\prime },y^{\prime })\in E\times F$ such that 

$$\parallel x-y^{\prime }\parallel =\mathrm{D}(E,F)=\parallel x^{\prime }-y\parallel ,$$

  or equivalently, $E_0=E,F_0=F$.

Throughout this paper, we will adopt the notations outlined below:

$$\begin{array}{cc}\hfill {\delta }_x\left(E\right)& =sup\{\parallel x-y\parallel :y\in E\}\mathrm{for}\mathrm{all}x\in X,\hfill \\ \hfill \delta (E,F)& =sup\{\parallel x-y\parallel :x\in E,y\in F\},\hfill \\ \hfill \mathrm{diam}\left(E\right)& =\delta (E,E).\hfill \end{array}$$

The closed and convex hull of a set E will be denoted by $\overline{\mathrm{con}}\left(E\right)$. Also, the closed ball centered at $x\in X$ with radius $r>0$ is displayed by $\mathcal{B}(x;r)$.

The upcoming lemma will play a key role in our future discussions.

Lemma 1 

([12]). Let $(E,F)$ be a nonempty pair in a normed linear space X. Then

$$\delta (E,F)=\delta \left(\overline{\mathrm{con}}\left(E\right),\overline{\mathrm{con}}\left(F\right)\right).$$

Definition 4 

([10]). A convex pair $(E,F)$ in a Banach space X is said to have a proximal normal structure if for every nonempty, bounded, closed, convex and proximinal pair $(H_1,H_2)\subseteq (E,F)$ for which $\mathrm{D}(H_1,H_2)=\mathrm{D}(E,F)$ and $\delta (H_1,H_2)>\mathrm{D}(E,F)$, there exists $(p,q)\in H_1\times H_2$ such that

$$max\{{\delta }_p\left(H_2\right),{\delta }_q\left(H_1\right)\}<\delta (H_1,H_2).$$

In Definition 4, when $E=F$, it highlights the normal structure concept for the set E, with [10] demonstrating that every nonempty, bounded, closed, and convex pair in a uniformly convex Banach space X possesses a proximal normal structure, while every nonempty, compact, and convex pair in a Banach space X also has this property (refer to [13] for further details).

The subsequent theorem extends Kirk’s fixed point theorem for noncyclic relatively nonexpansive maps, a key finding from [10].

Theorem 4 

(Theorem 2.2 of [10]). Let $(E,F)$ be a nonempty, weakly compact convex pair in a strictly convex Banach space X, and suppose  $(E,F)$ has a proximal normal structure. Assume that  $\mathcal{S}:E\cup F\to E\cup F$ is a noncyclic relatively nonexpansive mapping. Then ${\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right)\ne \varnothing$.

Another important tool that will assist in proving our main results is metric projection. For a nonempty subset E of X a metric projection operator, ${\mathcal{P}}_E:X\to 2^E$ is defined as

$${\mathcal{P}}_E\left(x\right):=\{y\in E:\parallel x-y\parallel =\mathrm{D}(\left\{x\right\},E)\},$$

where $2^E$ denotes the set of all subsets of E. It is well-known that if E is a nonempty, closed and convex subset of a reflexive and strictly convex Banach space X, then the metric projection ${\mathcal{P}}_E$ is single-valued from X to E.

The following proposition outlines key characteristics of a metric projection operator applied to a union of proximal sets.

Proposition 1 

([14]). Let $(E,F)$ be a nonempty, closed and convex pair in a strictly convex Banach space X such that $E_0\ne \varnothing$. Define $\mathcal{P}:E_0\cup F_0\to E_0\cup F_0$ as

$$\mathcal{P}\left(x\right)=\left\{\begin{array}{c}{\mathcal{P}}_{E_0}\left(x\right)\mathit{if}x\in F_0,\hfill \\ {\mathcal{P}}_{F_0}\left(x\right)\mathit{if}x\in E_0.\hfill \end{array}\right.$$

(1)

Then the following statements hold:

(i) 

$\parallel x-\mathcal{P}x\parallel =\mathrm{D}(E,F)$ for any $x\in E_0\cup F_0$ and $\mathcal{P}$ is cyclic, that is, $\mathcal{P}\left(E_0\right)\subseteq F_0,\mathcal{P}\left(F_0\right)\subseteq E_0$;

(ii) 

$\mathcal{P}{|}_{E_0},\mathcal{P}{|}_{F_0}$ are isometric;

(iii) 

$\mathcal{P}{|}_{E_0},\mathcal{P}{|}_{F_0}$ are affine;

(iv) 

${\mathcal{P}}^2{|}_{E_0}=i_{E_0},{\mathcal{P}}^2{|}_{F_0}=i_{F_0}$, where $i_M$ denotes the identity mapping on a nonempty subset M of X.

The mapping $\mathcal{P}:E_0\cup F_0\to E_0\cup F_0$ defined in (1) is called a proximal projection mapping.

3. Göhde’s Theorem for Noncyclic Relatively Nonexpansive Maps

We start our key findings in this paper with the upcoming lemma.

Lemma 2. 

Let $(E,F)$ be a nonempty, bounded, closed and convex pair in a strictly convex Banach space X such that  $E_0$ is nonempty and let $\mathcal{S}:E\cup F\to E\cup F$ be a noncyclic relatively nonexpansive mapping. Then the pair $(E_0,F_0)$ is  $\mathcal{S}$-invariant, that is, $\mathcal{S}$ is noncyclic on $E_0\cup F_0$. Also  $\mathcal{S}$ and  $\mathcal{P}$ commute, where $\mathcal{P}$ is a proximal projection mapping.

Proof. 

Let $u\in E_0$. Then $\mathcal{P}u\in F_0$, and we have $\parallel u-\mathcal{P}u\parallel =\mathrm{D}(E,F)$. Since $\mathcal{S}$ is noncyclic relatively nonexpansive,

$$\mathrm{D}(E,F)\le \parallel \mathcal{S}u-\mathcal{S}\left(\mathcal{P}u\right)\parallel \le \parallel u-\mathcal{P}u\parallel =\mathrm{D}(E,F),$$

that is, $\mathcal{S}u\in E_0$. Thus $\mathcal{S}\left(E_0\right)\subseteq E_0$. Equivalently, $\mathcal{S}\left(F_0\right)\subseteq F_0$ and so, $(E_0,F_0)$ is $\mathcal{S}$-invariant. Also, by the property of the proximal projection mapping, for the arbitrary element $u\in E_0$, we have $\parallel \mathcal{S}u-\mathcal{P}\left(\mathcal{S}u\right)\parallel =\mathrm{D}(E,F)$. It is now derived from the strict convexity of X that $\mathcal{P}\left(\mathcal{S}u\right)=\mathcal{S}\left(\mathcal{P}u\right)$ which ensures that $\mathcal{S}$ and $\mathcal{P}$ commute on $E_0$. The same discussions show that $\mathcal{S}$ and $\mathcal{P}$ commute on $F_0$. □

We now present an extension of Göhde’s fixed point problem.

Theorem 5 

(compare with Theorem 4). Let $(E,F)$ be a nonempty, bounded, closed and convex pair in a strictly convex Banach space X such that $E_0$ is nonempty and let $\mathcal{S}:E\cup F\to E\cup F$ be a noncyclic relatively nonexpansive mapping. Suppose $M\subseteq E_0$ is a compact set such that for any $x\in E_0$, the sequence ${\left\{{\mathcal{S}}^{n}x\right\}}_{n\in \mathbb{N}}$ has a limit point in M. If $\mathcal{S}$ is continuous on $E_0$, then ${\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right)\ne \varnothing$.

Proof. 

Consider an element $(p,q)\in E_0\times F_0$ for which $\parallel p-q\parallel =\mathrm{D}(E,F)$. Now for any $\alpha \in (0,1)$, define a mapping ${\mathcal{S}}_{\alpha }:E\cup F\to E\cup F$ as

$${\mathcal{S}}_{\alpha }\left(u\right)=\left\{\begin{array}{c}\alpha \mathcal{S}u+(1-\alpha )p;\mathrm{if}u\in E,\hfill \\ \alpha \mathcal{S}u+(1-\alpha )q;\mathrm{if}u\in F.\hfill \end{array}\right.$$

Since $(E,F)$ is convex, the mapping ${\mathcal{S}}_{\alpha }$ is noncyclic on $E\cup F$. Conversely, for any $(u,v)\in E\times F$, we have

$$\begin{array}{cc}\hfill \parallel {\mathcal{S}}_{\alpha }\left(u\right)-{\mathcal{S}}_{\alpha }\left(v\right)\parallel & \le \alpha \parallel \mathcal{S}\left(u\right)-\mathcal{S}\left(v\right)\parallel +(1-\alpha )\parallel p-q\parallel \hfill \\ \hfill & \le \alpha \parallel u-v\parallel +(1-\alpha )\mathrm{D}(E,F).\hfill \end{array}$$

Let $n\in \mathbb{N}$ and $u\in E_0$. Then by considering the proximal projection mapping $\mathcal{P}$ and using the above inequality, it is clear that

$$\parallel {\mathcal{S}}_{\alpha }^n\left(u\right)-{\mathcal{S}}_{\alpha }^{n+1}\left(\mathcal{P}u\right)\parallel \le {\alpha }^n\parallel u-\mathcal{S}\left(\mathcal{P}u\right)\parallel +(1-{\alpha }^n)\mathrm{D}(E,F).$$

Letting $n\to \infty$ in the above inequality, this yields $\parallel {\mathcal{S}}_{\alpha }^n\left(u\right)-{\mathcal{S}}_{\alpha }^{n+1}\left(\mathcal{P}u\right)\parallel \to \mathrm{D}(E,F)$ for all $u\in E_0$. Choose $n_0\in \mathbb{N}$ for which

$$\parallel {\mathcal{S}}_{\alpha }^{n_0}\left(u\right)-{\mathcal{S}}_{\alpha }^{n_0+1}\left(\mathcal{P}u\right)\parallel <(1-\alpha )+\mathrm{D}(E,F).$$

(2)

Put $w:={\mathcal{S}}_{\alpha }^{n_0}\left(u\right)$. Then $w\in E_0$, and by using Lemma 2 together with the inequality (2), we obtain $\parallel w-{\mathcal{S}}_{\alpha }\left(\mathcal{P}w\right)\parallel <(1-\alpha )+\mathrm{D}(E,F)$. In fact,

$$\begin{array}{cc}\hfill \parallel w-\mathcal{S}\left(\mathcal{P}w\right)\parallel & \le \parallel w-{\mathcal{S}}_{\alpha }\left(\mathcal{P}w\right)\parallel +\parallel {\mathcal{S}}_{\alpha }\left(\mathcal{P}w\right)-\mathcal{S}\left(\mathcal{P}w\right)\parallel \hfill \\ \hfill & <(1-\alpha )+\mathrm{D}+(1-\alpha )\parallel q-\mathcal{S}\left(\mathcal{P}w\right)\parallel \hfill \\ \hfill & \le (1-\alpha )+\mathrm{D}+(1-\alpha )\mathrm{diam}\left(F_0\right)\hfill \\ \hfill & =(1-\alpha )\left[1+\mathrm{diam}\left(F_0\right)\right]+\mathrm{D}.\hfill \end{array}$$

Set $L:=1+\mathrm{diam}\left(F_0\right)$. Then for any $n\in \mathbb{N}$, we have

$$\parallel {\mathcal{S}}^{n}w-{\mathcal{S}}^{n+1}\left(\mathcal{P}w\right)\parallel \le \parallel w-\mathcal{S}\left(\mathcal{P}w\right)\parallel <(1-\alpha )L+\mathrm{D}(E,F).$$

By using the assumption the sequence $\left\{{\mathcal{S}}^{n}w\right\}\subseteq E_0$ has a limit point in the compact set M, say ${\mathbf{x}}_{\alpha }$. Thus, for any $\epsilon >0$, there exists $k\in \mathbb{N}$ for which $\parallel {\mathcal{S}}^{k}w-{\mathbf{x}}_{\alpha }\parallel <\epsilon$. Additionally, from the continuity of $\mathcal{S}$ on the set $E_0$, there exists a real number $\delta >0$ such that for any $u\in E_0$ if $\parallel u-{\mathbf{x}}_{\alpha }\parallel <\delta$, then $\parallel \mathcal{S}u-\mathcal{S}{\mathbf{x}}_{\alpha }\parallel <\epsilon$. Since ${\mathcal{P}|}_{E_0}$ is an isometry,

$$\parallel \mathcal{S}\left(\mathcal{P}u\right)-\mathcal{S}\left(\mathcal{P}{\mathbf{x}}_{\alpha }\right)\parallel =\parallel \mathcal{P}\left(\mathcal{S}u\right)-\mathcal{P}\left(\mathcal{S}{\mathbf{x}}_{\alpha }\right)\parallel =\parallel \mathcal{S}u-\mathcal{S}{\mathbf{x}}_{\alpha }\parallel <\epsilon .$$

Let $\delta <\epsilon$. Then there is $l\in \mathbb{N}$ such that $\parallel {\mathcal{S}}^{l}w-{\mathbf{x}}_{\alpha }\parallel <\delta$, which concludes that $\parallel {\mathcal{S}}^{l+1}\left(\mathcal{P}w\right)-\mathcal{S}\left(\mathcal{P}{\mathbf{x}}_{\alpha }\right)\parallel <\epsilon$. Moreover,

$$\begin{array}{cc}\hfill \parallel \mathcal{S}\left(\mathcal{P}{\mathbf{x}}_{\alpha }\right)-{\mathbf{x}}_{\alpha }\parallel & \le \parallel \mathcal{S}\left(\mathcal{P}{\mathbf{x}}_{\alpha }\right)-{\mathcal{S}}^{l+1}\left(\mathcal{P}w\right)\parallel +\parallel {\mathcal{S}}^{l+1}\left(\mathcal{P}w\right)-{\mathcal{S}}^l\left(w\right)\parallel +\parallel {\mathcal{S}}^l\left(w\right)-{\mathbf{x}}_{\alpha }\parallel \hfill \\ \hfill & <2\epsilon +(1-\alpha )L+\mathrm{D}.\hfill \end{array}$$

By the compactness of the set M, there exists a sequence ${\left\{{\alpha }_i\right\}}_i$ such that ${\alpha }_i\to 1$ and ${\mathbf{x}}_{{\alpha }_i}\to x^{★}\in M$. Again using the continuity of $\mathcal{S}$ on $E_0$, $\mathcal{S}\left({\mathbf{x}}_{{\alpha }_i}\right)\to \mathcal{S}\left(x^{★}\right)$. Also, by the continuity of $\mathcal{P}$ on $E_0$, we must have

$$\mathcal{S}\left(\mathcal{P}{\mathbf{x}}_{{\alpha }_i}\right)=\mathcal{P}\left(\mathcal{S}{\mathbf{x}}_{{\alpha }_i}\right)\to \mathcal{P}\left(\mathcal{S}{x}^{★}\right)=\mathcal{S}\left(\mathcal{P}{x}^{★}\right).$$

Therefore,

$$\parallel \mathcal{S}\left(\mathcal{P}{x}^{★}\right)-x^{★}\parallel =\mathrm{D}(E,F)=\parallel \mathcal{P}{x}^{★}-x^{★}\parallel .$$

Strict convexity of X ensures that $\mathcal{S}\left(\mathcal{P}{x}^{★}\right)=\mathcal{P}{x}^{★}$ and so, $\mathcal{P}\left(\mathcal{S}{x}^{★}\right)=\mathcal{P}{x}^{★}$ or $\mathcal{S}{x}^{★}=x^{★}$, because $\mathcal{P}$ is an isometry on $E_0$. Hence, $(x^{★},\mathcal{P}{x}^{★})\in E_0\times F_0$ is an optimal pair of fixed points of $\mathcal{S}$, and this completes the proof. □

An example below illustrates Theorem 5.

Example 2. 

Consider the Banach space  $({\ell }^1,\parallel .\parallel )$ for which 

$$\parallel x\parallel ={\parallel x\parallel }_1+{\parallel x\parallel }_2,\forall x\in {\ell }^1,$$

where ${\parallel .\parallel }_1,{\parallel .\parallel }_2$ are the norms on ${\ell }^1$ and  ${\ell }^2$, respectively. Because of the appearance of ${\parallel .\parallel }_2$, $({\ell }^1,\parallel .\parallel )$ is strictly convex. Let

$$\begin{array}{cc}\hfill E& =\left\{\mathbf{x}=(1,\underset{x}{\underbrace{x_1,x_2,x_3,\dots }})\in {\ell }^1:\parallel x\parallel \le 1\right\},\hfill \\ \hfill F& =\left\{\mathbf{y}=(2,\underset{y}{\underbrace{y_1,y_2,y_3,\dots }})\in {\ell }^1:\parallel y\parallel \le 1\right\}.\hfill \end{array}$$

Then $(E,F)$ is a bounded, closed and convex pair in ${\ell }^1$. Furthermore, for any $(\mathbf{x},\mathbf{y})\in E\times F$, we have 

$$\parallel \mathbf{x}-\mathbf{y}\parallel =\left(1+{\displaystyle \sum _{j=1}^{\infty }}|x_j-y_j|\right)+{\left(1+{\displaystyle \sum _{j=1}^{\infty }}{|x_j-y_j|}^2\right)}^{{\displaystyle \frac{1}{2}}},$$

 which implies that $\mathrm{D}(E,F)=2$. It is remarkable to note that $E_0=E$ and  $F_0=F$. Define $\mathcal{S}:E\cup F\to E\cup F$ as 

$$\mathcal{S}\left(\mathbf{x}\right)=(1,x_1^2,x_2^2,{\displaystyle \frac{x_3}{2}},{\displaystyle \frac{x_4}{2}}\dots ),\mathcal{S}\left(\mathbf{y}\right)=(2,y_1^2,y_2^2,{\displaystyle \frac{y_3}{2}},{\displaystyle \frac{y_4}{2}}\dots ).$$

Clearly, $\mathcal{S}$ is a noncyclic relatively nonexpansive mapping whenever $\mathcal{S}$ is continuous on E. On the other hand, if $\left\{e_n\right\}$ is the canonical basis of ${\ell }^1$ and we set 

$$E=\left\{e_1,e_1+e_2,e_1+e_3\right\},$$

then $M\subseteq E$ is compact and that for any $\mathbf{x}\in E$, the sequence $\left\{{\mathcal{S}}^n\mathbf{x}\right\}$ converges to an element in M. It now follows from Theorem 5 that ${\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right)\ne \varnothing$. Indeed,  $(e_1,2e_1)\in {\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right)\ne \varnothing$.

It is important to note that Theorem 4 does not guarantee the existence of an optimal fixed point pair for the mapping $\mathcal{S}$ due to the lack of weak compactness in the pair $(E,F)$.

4. Noncyclic Kannan Contractions

This section aims to establish existence and convergence results for an optimal pair of fixed points in strictly convex Banach spaces related to a new class of noncyclic mappings.

Definition 5. 

Let $(E,F)$ be a nonempty pair in a strictly convex Banach space X such that $E_0$ is nonempty. Consider the proximal projection mapping $\mathcal{P}:E_0\cup F_0\to E_0\cup F_0$ defined in (1). Then a mapping $\mathcal{S}:E\cup F\to E\cup F$ is called a noncyclic Kannan contraction if it is noncyclic, preserves the distance of $E,F$, that is,  $\parallel \mathcal{S}x-\mathcal{S}y\parallel =\parallel x-y\parallel$ whenever  $\parallel x-y\parallel =\mathrm{D}(E,F)$ and

$$\parallel \mathcal{S}x-\mathcal{S}y\parallel \le \beta \left(\parallel \mathcal{P}x-\mathcal{S}x\parallel +\parallel \mathcal{P}y-\mathcal{S}y\parallel \right)+(1-2\beta )\mathrm{D}(E,F),$$

for some $\beta \in (0,{\displaystyle \frac{1}{2}})$ and for all $(x,y)\in E_0\times F_0$.

Next is an example to clarify the concept mentioned above.

Example 3. 

Consider the Banach space  ${\mathbb{R}}^2$ with the Euclidean norm. Let

$$E=\left\{(x,0):0\le x\le 1\right\},F=\left\{(y,1):0\le y\le 1\right\}.$$

Then $\mathrm{D}(E,F)=1$ and $E_0=E,F_0=F$. Also, the proximal projection mapping $\mathcal{P}:E_0\cup F_0\to E_0\cup F_0$ is as follows:

$$\mathcal{P}(x,0)=(x,1),\mathcal{P}(y,1)=(y,0),\forall x,y\in [0,1].$$

Define $\mathcal{S}:E\cup F\to E\cup F$ with

$$\mathcal{S}(x,0)=\left\{\begin{array}{c}(0,0),\mathit{if}x=1,\hfill \\ ({\displaystyle \frac{1}{8}},0),\mathit{if}x\ne 1,\hfill \end{array}\right.\mathcal{S}(y,1)=\left\{\begin{array}{c}(0,1),\mathit{if}y=1,\hfill \\ ({\displaystyle \frac{1}{8}},1),\mathit{if}y\ne 1.\hfill \end{array}\right.$$

Then it is easy to see that  $\mathcal{S}$ is a noncyclic Kannan contraction with $\beta ={\displaystyle \frac{1}{4}}$.

In what follows, we present a convergence theorem by considering an appropriate geometric property and using a suitable iterative sequence. To this end, we recall the following notion.

Definition 6 

([15]). Let $(E,F)$ be a nonempty pair in a normed linear space X. Then  $(E,F)$ is said to satisfy property UC if the following holds:

If $\left\{x_n\right\}$ and  $\left\{z_n\right\}$ are sequences in E and $\left\{y_n\right\}$ is a sequence in F such that

$$\underset{n}{lim}\parallel x_n-y_n\parallel =\mathrm{D}(E,F)=\underset{n}{lim}\parallel z_n-y_n\parallel ,$$

then we have ${lim}_n\parallel x_n-z_n\parallel =0$.

It was proved in [16] that if $(E,F)$ is a nonempty and closed pair in a uniformly convex Banach space X such that E is convex, then $(E,F)$ has the property UC. Also, in the setting of strictly convex Banach space X if E is convex and relatively compact and the closure of a set F is weakly compact, then the pair $(E,F)$ has the property UC (see Proposition 5 of [15]). It is remarkable to note that UC is not symmetric, that is, if $(E,F)$ satisfies the property UC, we may not conclude that $(F,E)$ satisfies the property UC.

The upcoming lemma will play a crucial role in proving the main theorem of this section.

Lemma 3 

([15]). Let $(E,F)$ be a nonempty pair in a normed linear space X. Assume that $(E,F)$ satisfies the property UC. Let $\left\{x_n\right\}$ and $\left\{y_n\right\}$ be sequences in E and F, respectively, such that either of the following holds:

$$\underset{m\to \infty }{lim}\underset{n\ge m}{sup}\parallel x_m-y_n\parallel =\mathrm{D}(E,F)\mathit{or}\underset{n\to \infty }{lim}\underset{m\ge n}{sup}\parallel x_m-y_n\parallel =\mathrm{D}(E,F).$$

Then $\left\{x_n\right\}$ is a Cauchy sequence.

We are prepared to present the first key finding of this section.

Theorem 6. 

Let $(E,F)$ be a nonempty, closed and convex pair in a strictly convex Banach space X such that $E_0$ is nonempty and both $(E,F),(E,F)$ satisfies the property UC. Let  $\mathcal{S}:E\cup F\to E\cup F$ be a noncyclic Kannan contraction, and for an arbitrary element $x_0\in E_0$, define

$$\left\{\begin{array}{c}{x}_n={\mathcal{S}}^n{x}_0,\hfill \\ y_n=\mathcal{P}{x}_n,\hfill \end{array}\right.$$

(3)

for all $n\in \mathbb{N}$, where  $\mathcal{P}$ is the proximal projection mapping defined in (1). Then ${\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right)\ne \varnothing$ and the sequence $\left\{(x_n,y_n)\right\}\subseteq E_0\times F_0$ converges to an element of ${\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right)$.

Proof. 

Since the mapping $\mathcal{S}$ preserves the distance of the sets E and F, by a similar argument of the proof of Lemma 2, we can see that $\mathcal{S}$ is invariant on $E_0\cup F_0$, and also $\mathcal{P}$ and $\mathcal{S}$ commute on $E_0\cup F_0$. It now follows from the definition of the iterative sequence ${\left\{(x_n,y_n)\right\}}_{n\in \mathbb{N}\cup \left\{0\right\}}$ in $E_0\times F_0$ that

$$y_n=\mathcal{P}\left({\mathcal{S}}^n{x}_0\right)=\mathcal{S}\left(\mathcal{P}{\mathcal{S}}^{n-1}{x}_0\right)=\mathcal{S}\left(\mathcal{P}{x}_{n-1}\right)=\mathcal{S}\left(y_{n-1}\right),\forall n\in \mathbb{N}.$$

For the fixed element $x_0\in E_0$, we have

$$\begin{array}{c}\hfill \parallel \mathcal{S}{x}_0-{\mathcal{S}}^2\mathcal{P}{x}_0\parallel \le \beta (\parallel \mathcal{P}{x}_0-\mathcal{S}{x}_0\parallel +\parallel \underset{\mathcal{S}{x}_0}{\underbrace{\mathcal{P}\left(\mathcal{S}\mathcal{P}{x}_0\right)}}-{\mathcal{S}}^2\left(\mathcal{P}{x}_0\right)\parallel )+(1-2\beta )\mathrm{D}(E,F),\end{array}$$

where $\beta \in (0,{\displaystyle \frac{1}{2}})$ which implies that

$$\parallel \mathcal{S}{x}_0-{\mathcal{S}}^2\mathcal{P}{x}_0\parallel \le {\displaystyle \frac{\beta }{1-\beta }}\parallel \mathcal{P}{x}_0-\mathcal{S}{x}_0\parallel +({\displaystyle \frac{1-2\beta }{1-\beta }})\mathrm{D}(E,F).$$

Put $\gamma :={\displaystyle \frac{\beta }{1-\beta }}$, then $\gamma \in (0,1)$, and we have

$$\parallel \mathcal{S}{x}_0-{\mathcal{S}}^2\mathcal{P}{x}_0\parallel \le \gamma \parallel \mathcal{P}{x}_0-\mathcal{S}{x}_0\parallel +(1-\gamma )\mathrm{D}(E,F).$$

(4)

Similarly, we can see that

$$\parallel {\mathcal{S}}^2{x}_0-\mathcal{S}\mathcal{P}{x}_0\parallel \le \gamma \parallel x_0-\mathcal{SP}{x}_0\parallel +(1-\gamma )\mathrm{D}(E,F).$$

(5)

By the fact that ${\mathcal{P}}^2{|}_{E_0}=i_{E_0}$, we obtain

$$\begin{array}{c}\hfill \parallel {\mathcal{S}}^2{x}_0-{\mathcal{S}}^3\mathcal{P}{x}_0\parallel \le \beta (\parallel \mathcal{P}\left(\mathcal{S}{x}_0\right)-{\mathcal{S}}^2{x}_0\parallel +\parallel \underset{{\mathcal{S}}^2{x}_0}{\underbrace{\mathcal{P}\left({\mathcal{S}}^2\mathcal{P}{x}_0\right)}}-{\mathcal{S}}^3\left(\mathcal{P}{x}_0\right)\parallel )+(1-2\beta )\mathrm{D}(E,F).\end{array}$$

Using the inequality (5), we conclude that

$$\begin{array}{cc}\hfill \parallel {\mathcal{S}}^2{x}_0-{\mathcal{S}}^3\mathcal{P}{x}_0\parallel & \le \gamma \parallel \mathcal{P}\left(\mathcal{S}{x}_0\right)-{\mathcal{S}}^2{x}_0\parallel +(1-\gamma )\mathrm{D}(E,F)\hfill \\ \hfill & \le \gamma \left(\gamma \parallel x_0-\mathcal{S}\left(\mathcal{P}{x}_0\right)\parallel +(1-\gamma )\mathrm{D}(E,F)\right)+(1-\gamma )\mathrm{D}(E,F)\hfill \\ \hfill & ={\gamma }^2\parallel x_0-\mathcal{S}\left(\mathcal{P}{x}_0\right)\parallel +(1-{\gamma }^2)\mathrm{D}(E,F).\hfill \end{array}$$

By a similar argument and using the relation (4), we can see that

$$\parallel {\mathcal{S}}^2\left(\mathcal{P}{x}_0\right)-{\mathcal{S}}^3{x}_0\parallel \le {\gamma }^2\parallel \mathcal{P}{x}_0-\mathcal{S}{x}_0\parallel +(1-{\gamma }^2)\mathrm{D}(E,F),$$

In continuation, we obtain

$$\mathrm{D}(E,F)\le \parallel {\mathcal{S}}^n{x}_0-{\mathcal{S}}^{n+1}\mathcal{P}{x}_0\parallel \le {\gamma }^n\parallel x_0-\mathcal{S}\left(\mathcal{P}{x}_0\right)\parallel +(1-{\gamma }^n)\mathrm{D}(E,F),$$

(6)

for all $n\in \mathbb{N}\cup \left\{0\right\}$. In view of the fact that $\gamma \in (0,1)$, if we take $n\to \infty$ in (6), we deduce that

$$\underset{n\to \infty }{lim}\parallel x_n-y_{n+1}\parallel =\underset{n\to \infty }{lim}\parallel x_n-\mathcal{S}{y}_n\parallel =\underset{n\to \infty }{lim}\parallel {\mathcal{S}}^n{x}_0-{\mathcal{S}}^{n+1}\mathcal{P}{x}_0\parallel =\mathrm{D}(E,F).$$

(7)

Additionally, using the property of the projection mapping $\mathcal{P}$, we have

$$\parallel x_n-y_n\parallel =\parallel x_n-\mathcal{P}{x}_n\parallel =\mathrm{D}(E,F),\forall n\in \mathbb{N}\cup \left\{0\right\}.$$

Since $(E,F)$ satisfies the property UC, we must have

$$0=\underset{n\to \infty }{lim}\parallel y_n-\mathcal{S}{y}_n\parallel =\underset{n\to \infty }{lim}\parallel y_n-y_{n+1}\parallel =\underset{n\to \infty }{lim}\parallel \mathcal{P}{x}_n-\mathcal{P}{x}_{n+1}\parallel =\underset{n\to \infty }{lim}\parallel x_n-x_{n+1}\parallel .$$

Also, we have

$$\underset{n\to \infty }{lim}\parallel y_n-\mathcal{S}{x}_n\parallel =\underset{n\to \infty }{lim}\parallel \mathcal{P}{x}_n-x_{n+1}\parallel \le \underset{n\to \infty }{lim}\left(\parallel \mathcal{P}{x}_n-x_n\parallel +\parallel x_n-x_{n+1}\parallel \right)=\mathrm{D}(E,F).$$

(8)

We assert that for given $\epsilon >0$, there exists $F\in \mathbb{N}$ such that

$$\parallel y_m-\mathcal{S}{x}_n\parallel <\mathrm{D}(E,F)+\epsilon ,\forall m>n\ge F.$$

(9)

Suppose the contrary, then there exists ${\epsilon }_0>0$ such that for any $k\in \mathbb{N}$

$$\exists m_k>n_k\ge k;\parallel y_{m_k}-\mathcal{S}{x}_{n_k}\parallel >\mathrm{D}(E,F)+{\epsilon }_0,\parallel y_{m_k-1}-\mathcal{S}{x}_{n_k}\parallel \le \mathrm{D}(E,F)+{\epsilon }_0.$$

So we have

$$\mathrm{D}(E,F)+{\epsilon }_0<\parallel y_{m_k}-\mathcal{S}{x}_{n_k}\parallel \le \parallel y_{m_k}-y_{m_k-1}\parallel +\parallel y_{m_k-1}-\mathcal{S}{x}_{n_k}\parallel \to \mathrm{D}(E,F)+{\epsilon }_0,$$

which implies that $\parallel y_{m_k}-\mathcal{S}{x}_{n_k}\parallel \to \mathrm{D}(E,F)+{\epsilon }_0$. Additionally, from the inequalities (7) and (8), we obtain

$$\begin{array}{cc}\hfill & \parallel y_{m_k}-\mathcal{S}{x}_{n_k}\parallel \le \parallel y_{m_k}-y_{m_k+1}\parallel +\parallel \underset{\mathcal{S}{y}_{m_k}}{\underbrace{y_{m_k+1}}}-\mathcal{S}{x}_{n_k+1}\parallel +\parallel \mathcal{S}{x}_{n_k+1}-\mathcal{S}{x}_{n_k}\parallel \hfill \\ \hfill & \le \parallel y_{m_k}-y_{m_k+1}\parallel +\beta (\parallel \underset{x_{m_k}}{\underbrace{\mathcal{P}{y}_{m_k}}}-\underset{y_{m_k+1}}{\underbrace{\mathcal{S}{y}_{m_k}}}\parallel +\parallel \underset{y_{n_k+1}}{\underbrace{\mathcal{P}{x}_{n_{k+1}}}}-\mathcal{S}{x}_{n_{k+1}}\parallel )+(1-2\beta )\mathrm{D}(E,F)+\parallel \mathcal{S}{x}_{n_k+1}-\mathcal{S}{x}_{n_k}\parallel .\hfill \end{array}$$

Letting $k\to \infty$ in the above inequality, we get

$$\mathrm{D}(E,F)+{\epsilon }_0=\underset{k\to \infty }{lim}\parallel y_{m_k}-\mathcal{S}{x}_{n_k}\parallel \le 2\beta \mathrm{D}(E,F)+(1-2\beta )\mathrm{D}(E,F)=\mathrm{D}(E,F),$$

which is impossible and so, (9) is satisfied. Therefore,

$$\underset{m\to \infty }{lim}\underset{n\ge m}{sup}\parallel y_m-\mathcal{S}{x}_n\parallel =\mathrm{D}(E,F).$$

Since $(E,F)$ meets the condition of property UC, by Lemma 3, the sequence $\left\{y_n\right\}$ is Cauchy in $F_0$, and by the closedness of the set $F_0$ (see Lemma 3.1 of [17]), there exists an element $q\in F_0$ such that $y_n\to q$. The continuity of ${\mathcal{P}|}_{N_0}$ ensures that

$$x_n=\mathcal{P}{y}_n\to \mathcal{P}q.$$

Moreover, by the continuity of ${\mathcal{P}|}_{E_0}$,

$$\mathcal{S}{y}_n=\mathcal{S}\left(\mathcal{P}{x}_n\right)=\mathcal{P}\left(\mathcal{S}{x}_n\right)=\mathcal{P}\left(x_{n+1}\right)\to q.$$

Hence,

$$\parallel \mathcal{S}{y}_n-\mathcal{S}\mathcal{P}q\parallel \le \beta (\parallel \mathcal{P}{y}_n-\mathcal{S}{y}_n\parallel +\parallel \underset{q}{\underbrace{{\mathcal{P}}^{2}q}}-\mathcal{S}\mathcal{P}q\parallel )+(1-2\beta )\mathrm{D}(E,F),$$

and if $n\to \infty$, we have

$$\parallel q-\mathcal{S}\mathcal{P}q\parallel \le \beta (\underset{\mathrm{D}}{\underbrace{\parallel \mathcal{P}q-q\parallel }}(M,N)+\parallel q-\mathcal{S}\mathcal{P}q\parallel )+(1-2\beta )\mathrm{D}(E,F),$$

which helps us conclude that $\parallel q-\mathcal{S}\mathcal{P}q\parallel =\mathrm{D}=\parallel q-\mathcal{P}q\parallel$. Strict convexity assumption of X implies that $\mathcal{S}\mathcal{P}q=\mathcal{P}q$ and that

$$\mathcal{S}q=\mathcal{P}\left(\mathcal{S}\mathcal{P}q\right)={\mathcal{P}}^{2}q=q.$$

Therefore, $(\mathcal{P}q,q)\in {\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right)\ne \varnothing$ and we are finished. □

The subsequent corollary directly follows from Theorem 6 within the context of uniformly convex Banach spaces.

Corollary 1. 

Let $(E,F)$ be a nonempty, closed and convex pair in a uniformly convex Banach space X such that E is bounded. Let $\mathcal{S}:E\cup F\to E\cup F$ be a noncyclic Kannan contraction and for an arbitrary element $x_0\in E_0$ consider the iterative sequence $\left\{(x_n,y_n)\right\}$ defined in (3). Then ${\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right)\ne \varnothing$ and the sequence $\left\{(x_n,y_n)\right\}\subseteq E_0\times F_0$ converges to an optimal pair of fixed points of  $\mathcal{S}$.

Remark 1. 

Notably, Corollary 1 can also be derived from Theorem 5 in [18] . To show this, we recall that for a nonempty pair $(E,F)$ in a Banach space X, a mapping $T:E\cup F\to E\cup F$ is called a weak cyclic Kannan contraction in the sense of [18] whenever T is cyclic, i.e., $T\left(E\right)\subseteq F,T\left(F\right)\subseteq E$ and there exists $\beta \in (0,{\displaystyle \frac{1}{2}})$ for which

$$\parallel Tx-Ty\parallel \le \beta \left(\parallel x-Tx\parallel +\parallel y-Ty\parallel \right)+(1-2\beta )\mathrm{D}(E,F),\forall (x,y)\in E\times F.$$

(10)

Now under the assumptions of Corollary 1, if we define $\mathcal{T}:=\mathcal{SP}$, then we have

$$\begin{array}{cc}\hfill \mathcal{T}\left(E_0\right)& =\mathcal{SP}\left(E_0\right)\subseteq \mathcal{S}\left(F_0\right)\subseteq F_0,\hfill \\ \hfill \mathcal{T}\left(F_0\right)& =\mathcal{SP}\left(F_0\right)\subseteq \mathcal{S}\left(E_0\right)\subseteq E_0,\hfill \end{array}$$

which implies that $\mathcal{T}$ is cyclic on $E_0\cup F_0$. Also, for any $(x,y)\in E_0\times F_0$, we have

$$\begin{array}{cc}\hfill \parallel \mathcal{T}x-\mathcal{T}y\parallel & =\parallel \mathcal{S}\left(\mathcal{P}x\right)-\mathcal{S}\left(\mathcal{P}y\right)\parallel \hfill \\ \hfill & \le \beta \left(\parallel x-\mathcal{S}\left(\mathcal{P}x\right)\parallel +\parallel y-\mathcal{S}\left(\mathcal{P}y\right)\parallel \right)+(1-2\beta )\mathrm{D}(E,F)\hfill \\ \hfill & =\beta \left(\parallel x-\mathcal{T}x\parallel +\parallel y-\mathcal{T}y\parallel \right)+(1-2\beta )\mathrm{D}(E,F),\hfill \end{array}$$

which implies that $\mathcal{T}$ is a weak cyclic Kannan contraction. Now from Theorem 5 of [18] , for any $x_0\in E_0$, we have  ${\mathcal{T}}^{2n}{x}_0\to z\in E_0$ for which $\parallel z-\mathcal{T}z\parallel =\mathrm{D}(E,F)$. Thus  $\parallel z-\mathcal{SP}z\parallel =\mathrm{D}(E,F)=\parallel z-\mathcal{P}z\parallel$, which helps us deduce that $\mathcal{SP}z=\mathcal{P}z$ and so, $\mathcal{S}z=z$. Hence,  $(z,\mathcal{P}z)\in {\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right)$. Furthermore,

$$z=\underset{n\to \infty }{lim}{\mathcal{T}}^{2n}{x}_0=\underset{n\to \infty }{lim}{\left(\mathcal{SP}\right)}^{2n}{x}_0=\underset{n\to \infty }{lim}{\mathcal{S}}^{2n}{x}_0,\forall x_0\in E_0.$$

Additionally,

$$z=\underset{n\to \infty }{lim}{\mathcal{S}}^{2n}\left(\mathcal{S}{x}_0\right)=\underset{n\to \infty }{lim}{\mathcal{S}}^{2n+1}{x}_0,$$

which ensures that ${lim}_{n\to \infty }{\mathcal{S}}^n{x}_0=z$ and  ${lim}_{n\to \infty }\mathcal{P}{x}_n\to \mathcal{P}z$ and we are finished.

The next lemma is crucial for the subsequent results discussed in this paper.

Lemma 4. 

Let $(E,F)$ be a nonempty, weakly compact and convex pair in a Banach space X and $\mathcal{S}:E\cup F\to E\cup F$ be a noncyclic mapping which preserves the distance of $E,F$. Then there is a minimal, weakly compact, convex and  $\mathcal{S}$-invariant pair $(H_1,H_2)\subseteq (E_0,F_0)$ such that $\mathrm{D}(H_1,H_2)=\mathrm{D}(E,F)$. Moreover,  $(H_1,H_2)$ is proximinal and

$$\left(\overline{\mathrm{con}}\left(\mathcal{S}\left(H_1\right)\right),\overline{\mathrm{con}}\left(\mathcal{S}\left(H_2\right)\right)\right)=(H_1,H_2).$$

Proof. 

Let

$$\begin{array}{cc}\hfill \mathsf{\Lambda }:=\left\{\right(U,V)\subseteq (E,F);& (U,V)\mathrm{is}\mathrm{nonempty},\mathrm{weakly}\mathrm{compact},\mathrm{convex}\mathrm{and}\hfill \\ \hfill & \mathcal{S}-\mathrm{invariant}\mathrm{with}\mathrm{D}(U,V)=\mathrm{D}(E,F)\}.\hfill \end{array}$$

Clearly, $(E_0,F_0)\in \mathsf{\Lambda }\ne \varnothing$. Using Zorn’s lemma ([19]), we can see that $\mathsf{\Lambda }$ has a minimal element, say $(H_1,H_2)\in \mathsf{\Lambda }$. Note that $\left({\left(H_1\right)}_0,{\left(H_2\right)}_0\right)$ is nonempty, weakly compact and convex and since $\mathcal{S}$ is a noncyclic and preserves the distance of $E,F$, it is also $\mathcal{S}$-invariant. So, $\left({\left(H_1\right)}_0,{\left(H_2\right)}_0\right)\in \mathsf{\Lambda }$. The minimality of $(H_1,H_2)$ implies that ${\left(H_1\right)}_0=H_1$ and ${\left(H_2\right)}_0=H_2$, that is, $(H_1,H_2)$ is proximinal. Additionally, $\left(\mathcal{S}\left(H_1\right),\mathcal{S}\left(H_2\right)\right)\subseteq (H_1,H_2)$ and so, $\left(\overline{\mathrm{con}}\left(\mathcal{S}\left(H_1\right)\right),\overline{\mathrm{con}}\left(\mathcal{S}\left(H_2\right)\right)\right)\subseteq (H_1,H_2)$. Thus,

$$\left(\mathcal{S}\left(\overline{\mathrm{con}}\left(\mathcal{S}\left(H_1\right)\right)\right),\mathcal{S}\left(\overline{\mathrm{con}}\left(\mathcal{S}\left(H_2\right)\right)\right)\right)\subseteq \left(\mathcal{S}\left(H_1\right),\mathcal{S}\left(H_2\right)\right)\subseteq \left(\overline{\mathrm{con}}\left(\mathcal{S}\left(H_1\right)\right),\overline{\mathrm{con}}\left(\mathcal{S}\left(H_2\right)\right)\right),$$

which ensures that $\left(\overline{\mathrm{con}}\left(\mathcal{S}\left(H_1\right)\right),\overline{\mathrm{con}}\left(\mathcal{S}\left(H_2\right)\right)\right)$ is $\mathcal{S}$-invariant and it is easy to check that $\mathrm{D}\left(\overline{\mathrm{con}}\left(\mathcal{S}\left(H_1\right)\right),\overline{\mathrm{con}}\left(\mathcal{S}\left(H_2\right)\right)\right)=\mathrm{D}(E,F)$. Hence, $\left(\overline{\mathrm{con}}\left(\mathcal{S}\left(H_1\right)\right),\overline{\mathrm{con}}\left(\mathcal{S}\left(H_2\right)\right)\right)\in \mathsf{\Lambda }$ and again using the minimality of $(H_1,H_2)$, we must have

$$\left(\overline{\mathrm{con}}\left(\mathcal{S}\left(H_1\right)\right),\overline{\mathrm{con}}\left(\mathcal{S}\left(H_2\right)\right)\right)=(H_1,H_2),$$

and hence the lemma. □

Notation 1. 

Under the assumptions of Lemma 4, by ${\mathcal{E}}_{E\times F}\left(\mathcal{S}\right)$, we denote the set of all minimal, weakly compact, convex and $\mathcal{S}$-invariant pairs in $E_0\times F_0$. The second key finding of this section is structured as follows.

Theorem 7. 

Let $(E,F)$ be a weakly compact and convex pair in a strictly convex Banach space X and $\mathcal{S}:E\cup F\to E\cup F$ be a noncyclic Kannan contraction map. Then  ${\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right)\ne \varnothing$.

Proof. 

Let $(H_1,H_2)\in {\mathcal{E}}_{E\times F}\left(\mathcal{S}\right)$ and assume that $u\in H_1$. Then for any $v\in H_2$, we have

$$\begin{array}{cc}\hfill \parallel \mathcal{S}u-\mathcal{S}v\parallel & \le \beta \left(\parallel \mathcal{P}u-\mathcal{S}u\parallel +\parallel \mathcal{P}v-\mathcal{S}v\parallel \right)+(1-2\beta )\mathrm{D}(E,F)\hfill \\ \hfill & \le 2\beta \delta (H_1,H_2)+(1-2\beta )\mathrm{D}(E,F).\hfill \end{array}$$

Set $\rho :=2\beta \delta (H_1,H_2)+(1-2\beta )\mathrm{D}(E,F)$. Then for any $v\in H_2$ we obtain $\mathcal{S}v\in \mathcal{B}(\mathcal{S}u;\rho )$ and so, $\mathcal{S}\left(H_2\right)\subseteq \mathcal{B}(\mathcal{S}u;\rho )$ which helps us deduce that

$$H_2=\overline{\mathrm{con}}\left(\mathcal{S}\left(H_2\right)\right)\subseteq \mathcal{B}(\mathcal{S}u;\rho ).$$

Thereby, ${\delta }_{\mathcal{S}u}\left(H_2\right)\le \rho$ for all $u\in H_1$. Therefore, $\delta (\mathcal{S}\left(H_1\right),H_2)={sup}_{u\in H_1}{\delta }_{\mathcal{S}u}\left(H_2\right)\le \rho$. It follows from Lemma 1 that

$$\delta (H_1,H_2)=\delta \left(\overline{\mathrm{con}}\left(\mathcal{S}\left(H_1\right)\right),H_2\right)=\delta (\mathcal{S}\left(H_1\right),H_2)\le \rho ,$$

and this concludes that $\delta (H_1,H_2)=\mathrm{D}(E,F)$. Now for any $(p,q)\in H_1\times H_2$, we have

$$\parallel p-\mathcal{S}q\parallel =\parallel p-q\parallel =\parallel \mathcal{S}p-q\parallel =\mathrm{D}(E,F).$$

Strict convexity condition of the Banach space X ensures that $\mathcal{S}p=p$ and $\mathcal{S}q=q$. Therefore,

$$H_1\times H_2\subseteq {\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right),$$

and we are finished. □

5. Relatively Kannan Nonexpansive Mappings

In the final section of this article, we broaden the category of Kannan noncyclic contractions and outline additional sufficient conditions to guarantee the existence of optimal pairs of fixed points.

Definition 7. 

Let $(E,F)$ be a nonempty pair in a strictly convex Banach space X such that $E_0$ is nonempty. A mapping $\mathcal{S}:E\cup F\to E\cup F$ is called noncyclic relatively Kannan nonexpansive whenever $\mathcal{S}$ is noncyclic, and preserves the distance of $E,F$ and

$$\parallel \mathcal{S}x-\mathcal{S}y\parallel \le {\displaystyle \frac{1}{2}}\left(\parallel \mathcal{P}x-\mathcal{S}x\parallel +\parallel \mathcal{P}y-\mathcal{S}y\parallel \right),$$

for all $(x,y)\in E_0\times F_0$.

All noncyclic Kannan contractions are relatively Kannan nonexpansive, but the next example demonstrates that the converse is not necessarily true.

Example 4. 

Consider the Banach space  ${\mathbb{R}}^2$ with the Euclidean norm. Let

$$E=\left\{(s,0):0\le s\le 2\right\},F=\left\{(t,1):-1\le t\le 1\right\}.$$

Then $\mathrm{D}(E,F)=1$ and

$$E_0=\left\{(s,0):0\le s\le 1\right\},F_0=\left\{(t,1):0\le t\le 1\right\}.$$

Clearly, the proximal projection map $\mathcal{P}:E_0\cup F_0\to E_0\cup F_0$ is defined as

$$\mathcal{P}(s,0)=(s,1),\mathcal{P}(t,1)=(t,0),\forall s,t\in [0,1].$$

Let $\mathcal{S}:E\cup F\to E\cup F$ be defined by

$$\mathcal{S}(s,0)=\left\{\begin{array}{c}(1-s,0),\mathit{if}0\le s\le 1,\hfill \\ ({\displaystyle \frac{1}{2}},0),\mathit{if}1<s\le 2,\hfill \end{array}\right.\mathcal{S}(t,1)=\left\{\begin{array}{c}(1-t,1),\mathit{if}0\le t\le 1,\hfill \\ (-1,1),\mathit{if}1<t\le 2,\hfill \end{array}\right.$$

Obviously, $\mathcal{S}$ preserves the distance of $E,F$. Then for any $\left((s,0),(t,1)\right)\in E_0\times F_0$, we have

$$\begin{array}{cc}\hfill \parallel \mathcal{S}(s,0)-\mathcal{S}(t,1)\parallel & =\parallel (1-s,0)-(1-t,1)\parallel =\sqrt{{|s-t|}^2+1},\hfill \\ \hfill \parallel \mathcal{P}(s,0)-\mathcal{S}(s,0)\parallel & =\parallel (s,1)-(1-s,0)\parallel =\sqrt{{|2s-1|}^2+1},\hfill \\ \hfill \parallel \mathcal{P}(t,1)-\mathcal{S}(t,1)\parallel & =\parallel (t,0)-(1-t,1)\parallel =\sqrt{{|2t-1|}^2+1}.\hfill \end{array}$$

Now, it is easy to see that

$$\sqrt{{|s-t|}^2+1}\le {\displaystyle \frac{1}{2}}\left(\sqrt{{|2s-1|}^2+1}+\sqrt{{|2t-1|}^2+1}\right),$$

which implies that

$$\parallel \mathcal{S}(s,0)-\mathcal{S}(t,1)\parallel \le {\displaystyle \frac{1}{2}}\left(\parallel \mathcal{P}(s,0)-\mathcal{S}(s,0)\parallel +\parallel \mathcal{P}(t,1)-\mathcal{S}(t,1)\parallel \right),$$

that is, $\mathcal{S}$ is a noncyclic relatively Kannan nonexpansive map. Additionally, if we take $s=1,t=0$, then for any $\beta \in (0,{\displaystyle \frac{1}{2}})$, we have

$$\begin{array}{cc}\hfill \parallel \mathcal{S}(1,0)-\mathcal{S}(0,1)\parallel & =\sqrt{2}>2\sqrt{2}\beta +(1-2\beta )\hfill \\ \hfill & =\beta \left(\parallel \mathcal{P}(1,0)-\mathcal{S}(1,0)\parallel +\parallel \mathcal{P}(0,1)-\mathcal{S}(0,1)\parallel \right)+(1-2\beta )\mathrm{D}(E,F),\hfill \end{array}$$

which implies that $\mathcal{S}$ is not a noncyclic Kannan contraction.

We now establish the next existence theorem.

Theorem 8. 

Let $(E,F)$ be a nonempty weakly compact and convex pair in a strictly convex Banach space X. Suppose that  $\mathcal{S}:E\cup F\to E\cup F$ is noncyclic relatively Kannan nonexpansive. If

$$\parallel {\mathcal{S}}^{2}x-\mathcal{S}\mathcal{P}x\parallel <\parallel \mathcal{S}x-\mathcal{P}x\parallel ,\forall x\in E_0\cup F_0,\mathrm{with}\parallel \mathcal{S}x-\mathcal{P}x\parallel >\mathrm{D}(E,F),$$

(11)

then ${\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right)\ne \varnothing$.

Proof. 

Using Lemma 4, let $(H_1,H_2)\in {\mathcal{E}}_{E\times F}\left(\mathcal{S}\right)$. Let $z\in H_1$ be an arbitrary element and put $s:=\parallel \mathcal{S}z-\mathcal{P}z\parallel$. If $s=\mathrm{D}(E,F)$, then by the fact that $\parallel z-\mathcal{P}z\parallel =\mathrm{D}(E,F)$ and X is strictly convex, we obtain $\mathcal{S}z=z$. We also have

$$\parallel \mathcal{S}\mathcal{P}z-\mathcal{S}z\parallel =\mathrm{D}(E,F)=\parallel \mathcal{P}z-\mathcal{S}z\parallel ,$$

which implies that $\mathcal{S}\mathcal{P}z=\mathcal{P}z$. Thus $(z,\mathcal{P}z)\in {\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right)\ne \varnothing$. So by contradiction, assume that $s>\mathrm{D}(A,B)$. Set

$$\begin{array}{cc}\hfill K_1:& =\left\{x\in H_1:max\{\parallel \mathcal{S}x-\mathcal{P}x\parallel ,\parallel x-\mathcal{SP}x\}\le s\right\}\hfill \\ \hfill K_2:& =\left\{y\in H_2:max\{\parallel \mathcal{S}y-\mathcal{P}y\parallel ,\parallel y-\mathcal{SP}y\parallel \le s\right\}.\hfill \end{array}$$

Note that $z\in K_1\ne \varnothing$. Also, if $w:=\mathcal{S}\mathcal{P}z$, then $w\in H_2$, and by this reality, $\mathcal{S},\mathcal{P}$ commute and ${\mathcal{P}}^2{|}_{E_0}=i_{E_0}$,

$$\begin{array}{cc}\hfill \parallel \mathcal{S}w-\mathcal{P}w\parallel & =\parallel {\mathcal{S}}^2\mathcal{P}z-\mathcal{S}z\parallel \le {\displaystyle \frac{1}{2}}\left(\parallel \mathcal{P}\left(\mathcal{S}\mathcal{P}z\right)-{\mathcal{S}}^2\left(\mathcal{P}z\right)\parallel +\parallel \mathcal{P}z-\mathcal{S}z\parallel \right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left(\parallel \mathcal{S}z-{\mathcal{S}}^2\mathcal{P}z\parallel +\parallel \mathcal{P}z-\mathcal{S}z\parallel \right),\hfill \end{array}$$

which implies that

$$\parallel \mathcal{S}w-\mathcal{P}w\parallel \le \parallel \mathcal{P}z-\mathcal{S}z\parallel \le s.$$

(12)

Also,

$$\begin{array}{cc}\hfill \parallel w-\mathcal{S}\mathcal{P}w\parallel & =\parallel \mathcal{S}\mathcal{P}z-\mathcal{SP}\left(\mathcal{SP}z\right)\parallel =\parallel \mathcal{S}\mathcal{P}z-{\mathcal{S}}^{2}z\parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}\left(\parallel {\mathcal{P}}^{2}z-\mathcal{S}\mathcal{P}z\parallel +\parallel \mathcal{PS}z-{\mathcal{S}}^{2}z\parallel \right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left(\parallel z-\mathcal{S}\mathcal{P}z\parallel +\parallel \mathcal{PS}z-{\mathcal{S}}^{2}z\parallel \right),\hfill \end{array}$$

which ensures that

$$\parallel w-\mathcal{SP}w\parallel \le s.$$

(13)

It follows from (12) and (13) that $w\in K_2\ne \varnothing$. Now assume that $u\in K_1$. Then $u\in H_1$ and by the fact that $(H_1,H_2)$ is proximinal, $\mathcal{P}u\in H_2$. Additionally,

$$\parallel \mathcal{S}\left(\mathcal{P}u\right)-\mathcal{P}\left(\mathcal{P}\right)u\parallel =\parallel \mathcal{SP}u-u\parallel \le s(\mathrm{since}u\in K_1),$$

and

$$\parallel \mathcal{P}u-\mathcal{SP}\left(\mathcal{P}u\right)\parallel =\parallel \mathcal{P}u-\mathcal{S}u\parallel \le s(\mathrm{since}u\in K_1),$$

which ensures that $\mathcal{P}u\in K_2$. Hence, $(u,\mathcal{P}u)\in K_1\times K_2$ and so, $\mathrm{D}(K_1,K_2)=\mathrm{D}(H_1,H_2)$ (=$\mathrm{D}(E,F))$. Since for any $u\in K_1$, we have $\mathcal{P}u\in K_2$, then $\mathcal{P}\left(K_1\right)\subseteq K_2$. By a similar discussion, we can see that $\mathcal{P}\left(K_2\right)\subseteq K_1$. Put

$$L_1:=\overline{\mathrm{con}}\left(\mathcal{S}\left(K_1\right)\right),L_2:=\overline{\mathrm{con}}\left(\mathcal{S}\left(K_2\right)\right).$$

Then $(L_1,L_2)\subseteq (K_1,K_2)$ is a nonempty, weakly compact and convex pair. It is worth noticing that $(u,\mathcal{P}u)\in K_1\times K_2$ for any $u\in K_1$ and so by the fact that $\mathcal{S}$ preserves the distance of $E,F$, we have

$$\mathrm{D}(E,F)\le \mathrm{D}(L_1,L_2)\le \parallel \mathcal{S}u-\mathcal{S}\left(\mathcal{P}u\right)\parallel =\mathrm{D}(E,F),$$

which helps us deduce that $\mathrm{D}(L_1,L_2)=\mathrm{D}(H_1,H_2)$. We claim that $\mathcal{P}\left(L_1\right)\subseteq K_2$. Suppose $\mathbf{x}\in L_1$ and let $\epsilon >0$ be given. Then there exist elements $x_1,x_2,\dots ,x_n\in K_1$ and $t_1,t_2,\dots ,t_n\in [0,1]$ such that

$${\displaystyle \sum _{j=1}^n}{t}_j=1,\parallel \mathbf{x}-{\displaystyle \sum _{j=1}^n}{t}_j\mathcal{S}\left(x_j\right)\parallel <\epsilon .$$

Thus

$$\begin{array}{cc}\hfill & \parallel \mathcal{SP}\mathbf{x}-\mathcal{P}\left(\mathcal{P}\mathbf{x}\right)\parallel =\parallel \mathcal{SP}\mathbf{x}-\mathbf{x}\parallel \hfill \\ \hfill & \le \parallel \mathcal{SP}\mathbf{x}-{\displaystyle \sum _{j=1}^n}{t}_j\mathcal{S}\left(x_j\right)\parallel +\parallel {\displaystyle \sum _{j=1}^n}{t}_j\mathcal{S}\left(x_j\right)-\mathbf{x}\parallel \hfill \\ \hfill & <\epsilon +\parallel {\displaystyle \sum _{j=1}^n}{t}_j(\mathcal{SP}\mathbf{x}-\mathcal{S}\left(x_j\right)\parallel \hfill \\ \hfill & \le \epsilon +{\displaystyle \sum _{j=1}^n}{\displaystyle \frac{t_j}{2}}(\parallel \underset{\mathbf{x}}{\underbrace{{\mathcal{P}}^2\mathbf{x}}}-\mathcal{SP}\mathbf{x}\parallel +\underset{\le s}{\underbrace{\parallel \mathcal{P}{x}_j-\mathcal{S}{x}_j\parallel }})\hfill \\ \hfill & \le \epsilon +{\displaystyle \frac{1}{2}}\left(\parallel \mathbf{x}-\mathcal{SP}\mathbf{x}\parallel +s\right),\hfill \end{array}$$

which helps us conclude that $\parallel \mathbf{x}-\mathcal{SP}\mathbf{x}\parallel <2\epsilon +s$. Since $\epsilon >0$ is arbitrary, we obtain

$$\parallel \mathcal{S}\left(\mathcal{P}\mathbf{x}\right)-\mathcal{P}\left(\mathcal{P}\mathbf{x}\right)\parallel \le s.$$

(14)

Moreover, by this reality that ${\mathcal{P}|}_{E_0}$ is affine and isometry,

$$\begin{array}{cc}\hfill & \parallel \mathcal{P}\mathbf{x}-\mathcal{SP}\left(\mathcal{P}\mathbf{x}\right)\parallel =\parallel \mathcal{P}\mathbf{x}-\mathcal{S}\mathbf{x}\parallel \hfill \\ \hfill & \le \parallel \mathcal{P}\mathbf{x}-\mathcal{P}({\displaystyle \sum _{j=1}^n}{t}_j\mathcal{S}\left(x_j\right))\parallel +\parallel \mathcal{P}({\displaystyle \sum _{j=1}^n}{t}_j\mathcal{S}\left(x_j\right))-\mathcal{S}\mathbf{x}\parallel \hfill \\ \hfill & =\parallel \mathbf{x}-{\displaystyle \sum _{j=1}^n}{t}_j\mathcal{S}\left(x_j\right)\parallel +\parallel {\displaystyle \sum _{j=1}^n}{t}_j\left(\mathcal{S}\left(\mathcal{P}{x}_j\right)-\mathcal{S}\mathbf{x}\right)\parallel \hfill \\ \hfill & <\epsilon +{\displaystyle \sum _{j=1}^n}{\displaystyle \frac{t_j}{2}}(\underset{\le s}{\underbrace{\parallel x_j-\mathcal{S}\left(\mathcal{P}{x}_j\right)\parallel }}+\parallel \mathcal{P}\mathbf{x}-\mathcal{S}\mathbf{x}\parallel )\hfill \\ \hfill & \le \epsilon +{\displaystyle \frac{1}{2}}\left(s+\parallel \mathcal{P}\mathbf{x}-\mathcal{S}\mathbf{x}\parallel \right),\hfill \end{array}$$

and so, $\parallel \mathcal{P}\mathbf{x}-\mathcal{S}\mathbf{x}\parallel <2\epsilon +s$. Again, since $\epsilon >0$ is arbitrary, we must have

$$\parallel \mathcal{P}\mathbf{x}-\mathcal{SP}\left(\mathcal{P}\mathbf{x}\right)\parallel \le s.$$

(15)

Using the inequalities (14) and (15), we conclude that

$$max\left\{\parallel \mathcal{S}\left(\mathcal{P}\mathbf{x}\right)-\mathcal{P}\left(\mathcal{P}\mathbf{x}\right)\parallel ,\parallel \mathcal{P}\mathbf{x}-\mathcal{SP}\left(\mathcal{P}\mathbf{x}\right)\parallel \right\}\le s,$$

which yields that $\mathcal{P}\mathbf{x}\in K_2$ for any $\mathbf{x}\in L_1$. Thereby, $\mathcal{P}\left(L_1\right)\subseteq K_2$. A similar argument shows that $\mathcal{P}\left(L_2\right)\subseteq K_1$. We now have

$$\left\{\begin{array}{c}{L}_1\subseteq \mathcal{P}\left(K_2\right)\subseteq K_1,\hfill \\ L_2\subseteq \mathcal{P}\left(K_1\right)\subseteq K_2,\hfill \end{array}\right.$$

and so,

$$\left\{\begin{array}{c}\mathcal{S}\left(L_1\right)\subseteq \mathcal{S}\left(K_1\right)\subseteq L_1,\hfill \\ \mathcal{S}\left(L_2\right)\subseteq \mathcal{S}\left(K_2\right)\subseteq L_2,\hfill \end{array}\right.$$

that is, $(L_1,L_2)$ is $\mathcal{S}$-invariant. It now follows from the minimality of the pair $(H_1,H_2)$ that $L_1=H_1,L_2=H_2$. Thus $K_1=H_1$ and $K_2=H_2$, and by the fact that the element $z\in H_1$ is arbitrary, we must have $\parallel \mathcal{S}x-\mathcal{P}x\parallel =s$ for any $x\in H_1$. Equivalently, $\parallel \mathcal{S}y-\mathcal{P}y\parallel =s$ for all $y\in H_2$. Since $\parallel \mathcal{S}z-\mathcal{P}z\parallel >\mathrm{D}(E,F)$, it follows from the inequality (11) that

$$s=\parallel \mathcal{S}\left(\mathcal{S}z\right)-\mathcal{P}\left(\mathcal{S}z\right)\parallel =\parallel \mathcal{S}\left(\mathcal{S}z\right)-\mathcal{S}\left(\mathcal{P}z\right)\parallel <\parallel \mathcal{S}z-\mathcal{P}z\parallel =s,$$

which is impossible and this completes the proof. □

Motivated by Kannan’s fixed point theorem (Theorem 3), we give the next existence result.

Theorem 9. 

Let $(E,F)$ be a nonempty, weakly compact and convex pair in a strictly convex Banach space X and $\mathcal{S}:E\cup F\to E\cup F$ be noncyclic relatively Kannan nonexpansive such that for any nonempty, closed, convex and $\mathcal{S}$-invariant pair $(E,F)\subseteq (E_0,F_0)$ with  $\mathrm{D}(M,N)=\mathrm{D}(E,F)<\delta (M,N)$, we have

$$sup\left\{\parallel \mathcal{P}w-\mathcal{S}w\parallel :w\in M\cup N\right\}<\delta (M,N).$$

(16)

Then ${\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right)\ne \varnothing$.

Proof. 

Applying Lemma 4 to find an element $(H_1,H_2)\in {\mathcal{E}}_{E\times F}\left(\mathcal{S}\right)$ for which

$$\left(\overline{\mathrm{con}}\left(\mathcal{S}\left(H_1\right)\right),\overline{\mathrm{con}}\left(\mathcal{S}\left(H_2\right)\right)\right)=(H_1,H_2).$$

We have the following cases:

Case (1): If $\mathrm{D}(H_1,H_2)=\delta (H_1,H_2)$, then by a similar proof of Theorem 7, we conclude that

$$H_1\times H_2\subseteq {\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right),$$

and we are finished.

Case (2): Let $\left(\mathrm{D}(E,F)=\right)\mathrm{D}(H_1,H_2)<\delta (H_1,H_2)$. For an arbitrary element $z\in H_1$, set $s:=\parallel \mathcal{S}z-\mathcal{P}z\parallel$. If $s=\mathrm{D}(E,F)$, then by a similar argument of the proof of Theorem 8, $(z,\mathcal{P}z)\in {\mathrm{OPF}}_{E\times F}\left(\mathcal{S}\right)$ and the proof is completed. So, assume that $\mathrm{D}(E,F)<s$. Again, using an equivalent discussion of the proof of Theorem 8, we can see that

$$\parallel \mathcal{P}w-\mathcal{S}w\parallel =s,\forall w\in H_1\cup H_2.$$

So, from the inequality (16), we have

$$s=sup\left\{\parallel \mathcal{P}w-\mathcal{S}w\parallel :w\in H_1\cup H_2\right\}<\delta (H_1,H_2).$$

Additionally, for any $(x,y)\in H_1\times H_2$, we have

$$\parallel \mathcal{S}x-\mathcal{S}y\parallel \le {\displaystyle \frac{1}{2}}\left(\parallel \mathcal{P}x-\mathcal{S}x\parallel +\parallel \mathcal{P}y-\mathcal{S}y\parallel \right)=s,$$

and so, $\mathcal{S}\left(H_2\right)\subseteq \mathcal{B}(\mathcal{S}x;s)$. Therefore,

$$H_2=\overline{\mathrm{con}}\left(\mathcal{S}\left(H_2\right)\right)\subseteq \mathcal{B}(\mathcal{S}x;s),$$

which gives $\parallel y-\mathcal{S}x\parallel \le s$ for all $y\in H_2$. Hence,

$$\underset{y\in H_2}{sup}\parallel y-\mathcal{S}x\parallel \le s,\forall x\in H_1.$$

It now follows from Lemma 1 that

$$\begin{array}{c}\hfill s\ge \delta \left(H_2,\mathcal{S}\left(H_1\right)\right)=\delta \left(H_2,\overline{\mathrm{con}}\left(\mathcal{S}\left(H_1\right)\right)\right)=\delta (H_2,H_1),\end{array}$$

which is a contradiction and this completes the proof. □

6. Conclusions and Future Works

This work elicited some existence theorems for an optimal pair of fixed points for noncyclic relatively nonexpansive mappings and noncyclic relatively Kannan nonexpansive maps in the framework of strictly convex Banach spaces which led to us obtaining real extensions of two well-known fixed point theorems due to Göhde and Kannan (Theorems 5, 7 and 8). We also presented a convergence theorem for an optimal pair of fixed points for noncyclic Kannan contractions by considering a geometric notion of property UC (Theorem 6). It will be interesting to generalize the existence theorems of this article to metric spaces equipped with a convex structure, for example, strictly convex geodesic metric spaces, Busemann convex spaces, and on a smooth projective connected curve (see [20,21,22,23,24] for more information about such spaces).