Optimal pair of fixed points for a new class of noncyclic mappings under a $(\varphi, \mathcal{R}^t)$-enriched contraction condition

1.   Introduction

Let $F\neq \emptyset$ and $G\neq \emptyset$ be subsets of a metric space $(X, d)$. A self-mapping $\Gamma$ on $F\cup G$ is said to be noncyclic whenever $\Gamma (F)\subseteq F$ and $\Gamma (G)\subseteq G$. In this situation, a point $(p^*, q^*)\in F\times G$ is called an optimal pair of fixed points of $\Gamma$ provided that

where ${\rm Dist}(F, G) = \inf\{d(p, q):\; p\in F, \; q\in G\}$. We denote the set of all optimal pairs of fixed points of $\Gamma$ in $F\times G$ by $Fix(\Gamma\mid _{F\times G})$.

$\Gamma :F\cup G\to F\cup G$ is called a noncyclic contraction if there exists $\lambda\in [0, 1)$ such that

for all $(p, q)\in F\times G$.

In 2013, Espínola and Gabeleh proved that if $F$ and $G$ are nonempty, weakly compact, and convex subsets of a strictly convex Banach space $X$, then $Fix(\Gamma\mid _{F\times G})\neq \emptyset$ for every noncyclic contraction $\Gamma$ defined on $F\cup G$ (see Theorem 3.10 of ^[1]).

After that, Gabeleh used the projection operators and proved both existence and convergence of an optimal pair of fixed points for noncyclic contractions in the setting of uniformly convex Banach spaces (see Theorem 3.2 of ^[2]).

We refer to ^{[3,4,5,6,7,8,9,10]} to study the problem of the existence of an optimal pair of fixed points for various classes of noncyclic mappings.

Recently, the authors of ^[10] introduced a new class of noncyclic mappings called noncyclic Fisher quasi-contractions, which contains the class of noncyclic contractions as a subclass, and they surveyed the existence and convergence of an optimal pair of fixed points in metric spaces by using a geometric notion of property WUC (Definition 2.2) on a nonempty pair of subsets of a metric space.

In this article, we extend the main conclusion of the paper ^[10] by considering an appropriate control function and equipping the metric space $(X, d)$ with a transitive relation $\mathcal{R}^t$. Indeed, we introduce a new class of noncyclic mappings called noncyclic $(\varphi, \mathcal{R}^t)$-enriched quasi-contractions, which is a kind of contraction at a point defined first in ^[11] and generalized later on in ^[12,13]. We then study the existence, uniqueness, and convergence of an optimal pair of fixed points for such mappings in metric spaces equipped with a transitive relation $\mathcal{R}^t$. This idea to consider a contractive condition only for points in some transitive relations was first introduced in ^[14] in order to generalize the ideas of coupled fixed points in partially ordered spaces, and further developed in a sequence of articles ^[15,16,17]. We will also examine some other existence conclusions of an optimal pair of fixed points in the framework of reflexive and strictly convex Banach spaces.

2.   Preliminaries

In this section, we point out some definitions and notations, which will be used in our coming arguments.

In what follows, $\mathcal{B}_X$ and $\mathcal{S}_X$ denote the unit closed ball and the unit sphere in a Banach space $X$.

Definition 2.1. (^[18]) A Banach space $X$ is said to be

$({\rm{i}})$ uniformly convex provided that for every $\varepsilon\in(0, 2]$, one can find a corresponding $\delta = \delta(\varepsilon)$ with the property that, whenever $p, q\in \mathcal{B}_X$ with $\|p-q\|\geq\varepsilon$, it follows that

$({\rm{ii}})$ strictly convex if for any two distinct elements $p, q \in \mathcal{S}_X$, we have

It is evident that every uniformly convex Banach space $X$ is strictly convex. However, the reverse does not universally hold. For instance, the Banach space $\ell_1$, which is equipped with its standard norm

where $\|.\|_1$ and $\|.\|_2$ are the norms on $l^1$ and $l^2$, respectively, is strictly convex, which is not uniformly convex (see ^[19] for more details). Also, Hilbert spaces and $l^p$ spaces $(1 < p < \infty)$ are well-known examples of uniformly convex Banach spaces. It is worth noticing that by the Milman-Pettis theorem, every uniformly convex Banach space is reflexive, too.

Definition 2.2. (^[20,21]) Let $F\neq \emptyset$ and $G\neq \emptyset$ be subsets of a metric space $(X, d)$, then $(F, G)$ is said to satisfy

$({\rm{i}})$ property ${\rm UC}$, if for all sequences $\{p_n\}, \{p'_n\}\subseteq F$ in $F$ and $\{q_n\}\subseteq G$, we have

$({\rm{ii}})$ property ${\rm WUC}$, if for any sequence $\{p_n\}\subseteq F$ such that

$\{p_n\}$ is Cauchy.

In ^[22], it was disclosed that each nonempty, closed, and convex pair in a uniformly convex Banach space $X$ possesses the property ${\rm UC}$. Additionally, if $F\neq \emptyset$ and $G\neq \emptyset$ are subsets in a metric space $(X, d)$, with $F$ being complete and the pair $(F, G)$ exhibiting the property ${\rm UC}$, then the pair $(F, G)$ is also endowed with the property ${\rm WUC}$ (see ^[20]). For more information and properties of the geometric notions of ${\rm UC}$, we refer to ^[23] and the most recent results in ^[24], where the authors have found a connection between the properties ${\rm UC}$ and uniform convexity and have introduced some generalizations of these properties.

Here, we sate the main result of ^[10].

Theorem 2.3. (^[10]) Given nonempty and complete subsets $F$ and $G$ of a metric space $(X, d)$, suppose that the pairs $(F, G)$ and $(G, F)$ have the property ${\rm WUC}$. Let noncyclic continuous self-mapping $\Gamma$ on $F\cup G$, be a noncyclic Fisher quasi-contraction, that is, for some $\alpha, \beta \in \mathbb{N}$, there exists $\lambda\in [0, 1)$ such that

where $C_n^{u}: = \{u, \Gamma u, \Gamma^2u, \cdots, \Gamma^{n}u \}$ for $u \in X$, $n \in\mathbb{N}$, and

There exists $(p^*, q^*)\in F \times G$ such that $Fix(\Gamma\mid _{F\times G}) = \{(p^*, q^*)\}$, $(\Gamma^np_0, \Gamma^nq_0)\to (p^*, q^*)$ as $n\to \infty$ for every $(p_0, q_0)\in F\times G$.

3.   Noncyclic $(\varphi, \mathcal{R}^t)$-enriched quasi-contractions

Throughout this section, we assume that $I$ is an identity function defined on $[0, +\infty)$ and $\varphi\in [\phi]$, such that

For instance, if we define $\varphi_1(t) = \lambda t$ for some $\lambda\in[0, 1)$ and $\varphi_2(t) = (t+2)-\ln(t+2)$ and $\varphi_3(t) = t-\sqrt{t+1}+3$, then $\varphi_j\in [\phi]$ for $j = 1, 2, 3$.

It is worth noticing that if $\varphi\in[\phi]$, then for all $t > 0$, we have

So, $(I-\varphi)(t) < t$ for all $t > 0$. Since $I-\varphi$ is increasing, it can be easily proven that $\varphi$ is continuous.

Also, for given nonempty subsets $F$ and $G$ of a metric space $(X, d)$, we set

Definition 3.1. Let $F\neq \emptyset$ and $G\neq \emptyset$ be subsets of a metric space $(X, d)$ and "$\mathcal{R}^t$" be a transitive relation on $F$. Let $\Gamma$ be a noncyclic mapping on $F\cup G$, then

${({\rm{i}})}$ we say that $\Gamma$ is $\mathcal{R}^t$-continuous at $p\in F$ if for every sequence $\{p_n\}$ in $F$ with $p_n\to p$ and $p_n\, \mathcal{R}^t\, p_{n+1}$, for all $n\in\mathbb{N}$, we have $\Gamma p_n\to \Gamma p$;

${ ({\rm{ii}})}$ we say that $\Gamma$ preserves "$\mathcal{R}^t$" on $F$ whenever $Tu\, \mathcal{R}^t\, Tv$ for every $u, v \in F$ with $u\, \mathcal{R}^t\, v$;

${ ({\rm{iii}})}$ we say that "$\mathcal{R}^t$" has a property $(*)$ on $F$, if for any sequence $\{p_n\}$ in $F$ with $p_n\to p\in F$ and $p_n\, \mathcal{R}^t \, p_{n+1}$ for all $n\in\mathbb{N}$, we have $p_n\, \mathcal{R}^t \, p$ for all $n\in\mathbb{N}$.

Now, with these prerequisites and inspired by the main existence results of ^[10], we introduce the following new family of noncyclic mappings. Henceforth, we denote a metric space $(X, d)$ equipped with a transitive relation "$\mathcal{R}^t$" by $X^{d, t}$.

Definition 3.2. Let $\emptyset \neq F, G \subseteq X^{d, t}$. A mapping $\Gamma:F\cup G\to F\cup G$ is said to be a noncyclic $(\varphi, \mathcal{R}^t)$-enriched quasi-contraction if $\Gamma$ is noncyclic. For some $\alpha, \beta\in \mathbb{N}$,

for all $(p, q)\in F\times G$ that are comparable with respect to "$\mathcal{R}^t$".

Example 3.3. Let $F\neq \emptyset$ and $G\neq \emptyset$ be subsets of a metric space $(X, d)$ and let $\Gamma:F\cup G\to F\cup G$ be a noncyclic Fisher quasi-contraction in the sense of Theorem 2.3, then $\Gamma$ is a noncyclic $(\varphi, \mathcal{R}^t)$-enriched quasi-contraction with $\mathcal{R}^t: = X\times X$ and $\varphi(t): = (1-\lambda)t$ for $t \geq 0$ and $\lambda \in [0, 1)$.

Remark 3.4. Let $\emptyset \neq F, G \subseteq X^{d, t}$ and $\Gamma:F\cup G\to F\cup G$ be a noncyclic mapping. Set $\mathfrak{D}: = {\rm Dist}(F, G)$. If for any $(p, q)\in F\times G$, we have

then

Now, define $\varphi^*:[0, +\infty)\to [0, +\infty)$ with $\varphi^*(t): = \varphi(t+\mathfrak{D})-\varphi(\mathfrak{D})$ for all $t\geq 0$. In view of the fact that $(I-\varphi^*)(t) = (I-\varphi)(t+\mathfrak{D})-(I-\varphi)(\mathfrak{D})$, we can see that $\varphi^*$ is strictly increasing and $I-\varphi^*$ is increasing. So from (3.3), we get

Example 3.5. Given complete subsets $F\neq \emptyset$ and $G\neq \emptyset$ of a metric space $(X, d)$, let $\Gamma :F\cup G\to F\cup G$ be a noncyclic $\varphi$-contraction (^[8]), that is, $\Gamma$ is noncyclic on $F\cup G$ and

From Remark 3.4, $\Gamma$ is a noncyclic $(\varphi^*, \mathcal{R}^t)$-enriched quasi-contraction with $\mathcal{R}^t: = X\times X$.

The following lemmas play essential roles in proving our main result in this section.

Lemma 3.6. Let $\emptyset \neq F, G \subseteq X^{d, t}$ be complete. Let $\Gamma$ be a noncyclic $(\varphi, \mathcal{R}^t)$-enriched quasi-contraction mapping on $F\cup G$, and $\Gamma$ preserves "$\mathcal{R}^t$". Let $p_0\in F$ and $q_0 \in G$ be such that $p_0\, \mathcal{R}^t \, q_0 \, \mathcal{R}^t \, \Gamma p_0$. Define $p_{n+1}: = \Gamma p_n$ and $q_{n+1}: = \Gamma q_n$ for each $n \geq 0$, then for any $m, n \in \mathbb{N}$, we have

Proof. Since $\Gamma$ preserves "$\mathcal{R}^t$" on $F\cup G$ and $p_0\, \mathcal{R}^t \, q_0 \, \mathcal{R}^t \, p_1$, we get

So, from transitivity of $\mathcal{R}^t$, for all $i, j\in \mathbb{N}$, we have

Suppose that $\Delta^*[C_n^{p_0}, C_m^{q_0}] = d^*(\Gamma^{i}p_0, \Gamma^{j}q_0)$, where $\alpha \leq i \leq n$ and $\beta \leq j \leq m$. From (3.2) and (3.6), we have

Thus, we must have $\varphi\left(\Delta^*[C_n^{p_0}, C_m^{q_0}]\right) \leq 0$. Strictly increasing of the function $\varphi$ causes $\Delta^* [C_n^{p_0}, C_m^{q_0}] = 0$ and $\Delta^* [C_n^{p_0}, C_m^{q_0}] = d^*(p_0, q_0)$, which ensures that (3.4) holds.

Lemma 3.7. Under the assumptions and notations of Lemma 3.6, for every $m, n \in \mathbb{N}$, we have

where

Proof. From Lemma 3.6, we have $\Delta^* [C_n^{p_0}, C_m^{q_0}] = d^*(\Gamma^{i}p_0, \Gamma^{j}q_0)$, for some $i, j\geq 0$ where $i < \alpha$ or $j < \beta$. In the case that $i < \alpha$ and $j < \beta$, (3.8) clearly holds. Therefore, without loss of generality, it can be assumed that $0 \leq i < \alpha$ and $\beta \leq j \leq m$. Using (3.7), we obtain

which deduces that

Since $\varphi \in [\phi]$, $\varphi^{-1}$ exists. Therefore,

and so (3.8) holds.

Lemma 3.8. Under the assumptions and notations of Lemma 3.6, for each $m, n, r, s \geq 0$ with $m, n \geq \max\{\alpha, \beta\}$, we have

Proof. It follows from the relation (3.7) that for some $0\leq r'\leq r, 0\leq s'\leq s$,

Hence, (3.9) holds.

Lemma 3.9. Under the assumptions and notations of Lemma 3.6,

Proof. From Lemma 3.8, for $n, m \geq \max\{2\alpha, 2\beta\}$, we have

Continuing this process and using Lemma 3.7, we get

where $k_{n, m} = \min\{\lfloor\frac{n}{\alpha}\rfloor, \lfloor\frac{m}{\beta}\rfloor\}.$ On the other hand, for the purposes of this discussion, it is permissible to presume that $M_{p_0, q_0} > 0$. Since $I-\varphi$ is increasing and $(I-\varphi)(t) < t$ for all $t > 0$, we obtain

Additionally, from (3.10), for every $i\in \mathbb{N}$ there exist $n_i, m_i\in \mathbb{N}$ such that $k_{n_i, m_i}\geq i$, and so (3.11) implies that

Thus,

which deduces that the sequence $\{(I-\varphi)^{k}\left(M_{p_0, q_0}\right)\}$ is decreasing. Since $\{(I-\varphi)^{k}\left(M_{p_0, q_0}\right)\}$ is bounded below, we assume that

for some $s\geq 0$. If $(I-\varphi)^{k_0}\left(M_{p_0, q_0}\right) = 0$ for some $k_0\geq 1$, then $s = 0$. Otherwise, if $(I-\varphi)^{k}\left(M_{p_0, q_0}\right) > 0$ for each $k\in \mathbb{N}$, from continuity of $I-\varphi$, we get

hence, $\varphi(s) = 0$, and from (3.1), we get $s = 0$. Therefore, from (3.10), we conclude that

and, in addition, the lemma.

The next result is a direct consequence of Lemma 3.9.

Corollary 3.10. Under the assumptions and notations of Lemma 3.6, if $(F, G)$ has the property ${\rm WUC}$, then the sequence $\{p_{n}\}$ is Cauchy.

We have now reached a level of preparedness that allows us to demonstrate the main existential finding of this segment, an expanded variant of Theorem 2.3.

Theorem 3.11. Under the assumptions and notations of Lemma 3.6, the following statements hold:

${({\rm{i}})}$ If the pair $(F, G)$ satisfies the property ${\rm WUC}$, the set $F$ is complete, and $\Gamma\mid _F :F \to F$ is $\mathcal{R}^t$-continuous on $F$, then there exists $p^*\in F$ such that $\Gamma p^* = p^*$;

${({\rm{ii}})}$ If the pair $(G, F)$ satisfies the property ${\rm WUC}$, the set $G$ is complete, and $\Gamma\mid _G:G \to G$ is $\mathcal{R}^t$-continuous on $G$, then there exists $q^*\in G$ such that $\Gamma q^* = q^*$;

${({\rm{iii}})}$ If, in addition to $({\rm{i}})$ and $({\rm{ii}})$, every pair of elements $(p, q)\in F\times G$ are comparable w.r.t. "$\mathcal{R}^t$", then $Fix(\Gamma\mid _{F\times G}) = \{(p^*, q^*)\}$.

Proof. (ⅰ) Let $p_{n+1}: = \Gamma p_n$ for each $n\geq 0$. From Corollary 3.10 and completeness of $F$, the sequence $\{p_n\}$ converges to some $p^*\in F$. Also from (3.5), we have $p_n\, \mathcal{R}^t \, p_{n+1}$ for each $n\geq 0$. Since $\Gamma\mid _F$ is $\mathcal{R}^t$-continuous, it follows that $\Gamma p^* = p^*$.

(ⅱ) By using a similar argument (ⅰ), the result is obtained.

(ⅲ) If $p^*\in F$ and $q^*\in G$ are the fixed points of $T$, then from Lemma 3.9 we have

that is, $(p^*, q^*)\in Fix(\Gamma\mid _{F\times G})$. Now, assume that each elements $p\in F$ and $q\in G$ are comparable with respect to "$\mathcal{R}^t$". Suppose $\overline{p}$ is another fixed point of $\Gamma$ in $F$ and let $q_0\in G$. From Lemma 3.9, we have

Since $(F, G)$ satisfies the property ${\rm WUC}$, we get $p^* = \overline{p}$. In a similar fashion, it becomes apparent that $q^*$ is a unique fixed point of $\Gamma$ in $G$.

Example 3.12. Consider $X : = \mathbb{R}$ with the usual metric and let

For $F = [0, 1]$ and $G = [-1, 0]$, define a noncyclic mapping $\Gamma:F\cup G\to F\cup G$ with

If $\varphi(t) = \frac{t^2}{1+2t}$ for $t\geq 0$, then $(I-\varphi)(t) = \frac{t+t^2}{1+2t}$ and $\varphi\in [\phi]$. Let $(p, q)\in F\times G$ be comparable w.r.t. "$\mathcal{R}^t$", then we must have $(p, q) = (\frac{1}{n+1}, -\frac{1}{m+1})$ for some $n, m\in \mathbb{N}$, which implies that

that is, $\Gamma$ is a noncyclic $(\varphi, \mathcal{R}^t)$-enriched quasi-contraction map, which is not a noncyclic $\varphi$-contraction. It is not difficult to see that all conditions of the part $({\rm{i}})$ of Theorem 3.11 are satisfied, and $p^* = 0$ is a fixed point of $\Gamma$ in $F$. Note that since every pair of elements $(p, q)\in F\times G$ are not comparable w.r.t. "$\mathcal{R}^t$", the fixed point of $\Gamma$ in $F$ is not unique.

Example 3.13. Again, consider $X : = \mathbb{R}$ with the usual metric and let $\mathcal{R}^t: = X\times X$. For $F = [0, 1]$ and $G = [-1, 0]$, define a noncyclic mapping $\Gamma:F\cup G\to F\cup G$ by

If $\varphi(t) = \frac{t^2}{1+2t}$ for $t\geq 0$, then $\varphi\in [\phi]$. A similar argument of the previous example shows that $d^*(\Gamma p, \Gamma q)\leq (I-\varphi)(d^*(p, q))$ for all $(p, q)\in F\times G$. Hence, $\Gamma$ is a noncyclic $(\varphi, \mathcal{R}^t)$-enriched quasi-contraction map. It now follows from Theorem 3.11 that $p^* = 0$ is a unique fixed point of $\Gamma$ in $F$.

The next theorem shows that if $\alpha = 1$ (resp., $\beta = 1$) in Definition 3.2, then we can drop the continuity of $T|_F$ (resp., $T|_G$) in Theorem 3.11. In this way, we obtain a real generalization of Theorem 3 in ^[6] as well as Theorem 2.7 in ^[10].

Theorem 3.14. Let $\emptyset \neq F, G \subseteq X^{d, t}$ be such that $F$ is complete and $(F, G)$ satisfies the property ${\rm WUC}$. Let "$\mathcal{R}^t$" be a transitive relation on $F\cup G$ with the property $(*)$ on $F$, and $\Gamma$ is a noncyclic $(\varphi, \mathcal{R}^t)$-enriched quasi-contraction mapping on $F\cup G$ with $\alpha = 1$, for which $\Gamma$ preserves "$\mathcal{R}^t$" on $F\cup G$. Let $p_0\in F$ and $q_0\in G$ be such that $p_0\, \mathcal{R}^t \, q_0 \, \mathcal{R}^t \, \Gamma p_0$, then there exists $p^*\in F$ such that $\Gamma p^* = p^*$. If every pair of elements $p\in F$ and $q\in G$ are comparable with respect to "$\mathcal{R}^t$", then $\Gamma$ has a unique fixed point in $F$.

Proof. From the proof of Theorem 3.11, the sequence $\{\Gamma^np_0\}$ is convergent to some $p^*\in F$. By Lemma 3.9, $p_n\, \mathcal{R}^t \, p_{n+1}$ for each $n\geq 0$. By using property $(*)$, we get $p_n\, \mathcal{R}^t \, p^*$ for each $n\geq 0$. Now, from the relation (3.6), we obtain $q_n\, \mathcal{R}^t \, p_{n+1}\, \mathcal{R}^t \, p^*$ for each $n\geq 0$. Thus, $q_n\, \mathcal{R}^t \, p^*$ for each $n\geq 0$, and by the fact that $\Gamma$ is a noncyclic $(\varphi, \mathcal{R}^t)$-enriched quasi-contraction from (3.2), we have

Therefore,

By Lemma 3.9, we get

Hence,

So, from (3.1), we obtain

Since $\lim_{n\rightarrow \infty} d(p^*, \Gamma^nq_0) = {\rm Dist}(F, G)$, from (3.12) and by taking into account that $(F, G)$ has the property ${\rm WUC}$, we conclude that $\Gamma p^* = p^*$. The uniqueness of a fixed point of $\Gamma$ in $F$ follows from an equivalent discussion of Theorem 3.11.

Corollary 3.15. Let $F\neq \emptyset$ and $G\neq \emptyset$ be complete subsets of a metric space $(X, d)$ such that $(F, G)$ and $(G, F)$ satisfy the property ${\rm WUC}$. Let "$\mathcal{R}^t$" be a transitive relation on $F\cup G$ with the property $(*)$ on $F\cup G$. Assume that $\Gamma$ is a noncyclic mapping on $F\cup G$ satisfying

for each $(p, q)\in F\times G$ that are comparable with respect to "$\mathcal{R}^t$". Let $(p_0, q_0)\in F\times G$ be such that $p_0\, \mathcal{R}^t \, q_0 \, \mathcal{R}^t \, \Gamma p_0$ and $\Gamma$ preserves "$\mathcal{R}^t$" on $F\cup G$, then there exists $(p^*, q^*)\in Fix(\Gamma\mid _{F\times G})$. If every pair of elements $p\in F$ and $q\in G$ are comparable with respect to "$\mathcal{R}^t$", then $Fix(\Gamma\mid _{F\times G}) = \{(p^*, q^*)\}$.

Building upon the foundations laid by the preceding theorem, we arrive at a subsequent finding that serves as a generalization of Corollary 2.8 of ^[10].

Corollary 3.16. Let $F\neq \emptyset$ and $G\neq \emptyset$ be complete subsets of a metric space $(X, d)$ such that $(F, G)$ and $(G, F)$ satisfy the property ${\rm WUC}$. Assume that $\Gamma$ is a noncyclic mapping on $F\cup G$ satisfying

for each $p\in F$ and $q\in G$. There exists $(p^*, q^*)\in F\times G$ such that $Fix(\Gamma\mid _{F\times G}) = \{(p^*, q^*)\}$, and for every $p_0\in F$ and $q_0\in G$, the sequences $\{\Gamma^np_0\}$ and $\{\Gamma^nq_0\}$ converge to $p^*$ and $q^*$, respectively.

The following common fixed point results are obtained from Theorem 3.11 and Corollary 3.15, immediately. These results are extensions of Corollaries 2.10 and 2.11 of ^[10].

Corollary 3.17. Let $\Gamma$ and $\Lambda$ be two continuous self-mappings on a complete metric space $(X, d)$ such that for some $\alpha, \beta \in \mathbb{N}$,

for all $p, q \in X$, then $\Lambda$ and $\Gamma$ have a unique common fixed point $p^*\in X$ such that $\lim_{n \to \infty}\Gamma^np_0 = \lim_{n \to \infty}\Lambda^np_0 = p^*$ for every $p_0\in X$.

Corollary 3.18. Let $\Gamma$ and $\Lambda$ be two self-mappings on a complete metric space $(X, d)$ satisfying

for all $p, q\in X$, then $\Lambda$ and $\Gamma$ have a unique common fixed point in $X$.

4.   More results in reflexive and strictly convex Banach spaces

In the latest section of this article, motivated by the results of ^[25,26], we present some other existence, convergence, and uniqueness of an optimal pair of fixed points of noncyclic $\varphi$-quasi-contractions in the setting of reflexive and strictly convex Banach spaces. We also refer to ^[27,28,29] for different approaches to the same problems for cyclic mappings and some interesting applications in game theory.

Throughout this section, we assume that $\varphi\in [\phi]$. Also, by $"\xrightarrow[]{w}"$, we mean the weak convergence in a Banach space $X$.

Theorem 4.1. Suppose that $F\neq \emptyset$ and $G\neq \emptyset$ are weakly closed subsets of a reflexive Banach space $X$ and let $\Gamma :F\cup G\to F\cup G$ be a noncyclic $\varphi$-quasi-contraction map, that is,

for all $(p, q)\in F\times G$. There exists $(p^*, q^*) \in F \times G$ such that $\Vert p^*-q^* \Vert = {\rm Dist}(F, G)$.

Proof. In the case that ${\rm Dist}(F, G) = 0$, the result follows from Theorem 3.14. Otherwise, if ${\rm Dist}(F, G) > 0$, for an arbitrary element $(p_0, q_0)\in F\times G$, define

From Lemma 3.9, the sequence $\{(p_{n}, q_n)\}$ is bounded in $F\times G$. Since $F$ is weakly closed in a reflexive Banach space $X$, there exists a subsequence $\{p_{n_k}\}$ of $\{p_n\}$ with $p_{n_k}\xrightarrow[]{w} p^*\in F$. As $\{q_{n_k}\}$ is a bounded sequence in a weakly closed set $G$, without loss of generality, one may assume that $q_{n_k}\xrightarrow[]{w} q^*\in G$ as $k\to \infty$. Since $p_{n_k}-q_{n_k}\xrightarrow[]{w} p^*-q^*\neq 0$ as $k\to \infty$, one can find a bounded linear functional $f:X\to [0, +\infty)$ with the property that

It follows from Lemma 3.9 that

So, $\Vert p^*-q^*\Vert = {\rm Dist}(F, G)$.

Definition 4.2. Suppose that $F$ and $G$ are subsets of a normed linear space $X$ and $\Gamma$ is a noncyclic self-mapping on $F\cup G$. We say that $\Gamma$ satisfies the $D$-property on $F$ if $\{p_n\}$ is a sequence in $F$ and $\{q_n\}$ is a sequence in $G$, such that

then $\Gamma p^* = p^*$.

Note that if ${\rm Dist}(F, G) = 0$ or $(F, G)$ has the property ${\rm UC}$, then the conditions of the above definition require that

Therefore, in these cases, the $D$-property of $\Gamma$ on $F$ is equal to demiclosedness property of $I-\Gamma\mid_F$ at $0$.

Theorem 4.3. Suppose that $F\neq \emptyset$ and $G\neq \emptyset$ are weakly closed subsets of a reflexive and strictly convex Banach space $X$ and let $\Gamma :F\cup G\to F\cup G$ be a noncyclic $\varphi$-quasi-contraction map. Assume that one of the following conditions is satisfied:

(a) $F$ is convex and $\Gamma$ is weakly continuous on $F$;

(b) $\Gamma$ satisfies the $D$-property on $F$.

Thus $\Gamma$ has a fixed point in $F$.

Proof. In the case that ${\rm Dist}(F, G) = 0$, there is nothing to prove by Theorem 3.14, so assume that ${\rm Dist}(F, G) > 0$. Let $(p_0, q_0)\in F\times G$ be an arbitrary element and define

From Theorem 4.1, there exists a point $(p^*, q^*) \in F \times G$ and subsequences $\{p_{n_k}\}$ and $\{q_{n_k}\}$ such that $\Vert p^*-q^* \Vert = {\rm Dist}(F, G)$, $p_{n_k}\xrightarrow[]{w} p^*\in F$, and $q_{n_k}\xrightarrow[]{w} q^*\in G$ as $k\to \infty$.

(a) Since $\Gamma$ is weakly continuous on $F$ and $\Gamma(F)\subseteq F$, we have $p_{n_k+1}\xrightarrow[]{w} \Gamma p^*\in F$ as $k\to \infty$. Since $p_{n_k+1}-q_{n_k}\xrightarrow[]{w} \Gamma p^*-q^*\neq 0$ as $k\to \infty$, one can find a bounded linear functional $f:X\to [0, +\infty)$ with the property that

It follows from Lemma 3.9 that

So, $\Vert \Gamma p^*-q^*\Vert = {\rm Dist}(F, G)$. We assume the contrary, $\Gamma p^*\neq p^*$, and it follows from the strict convexity of $X$ that

Since $F$ is convex, $\frac{p^*+\Gamma p^*}{2}\in F$, so (4.1) is a contradiction.

(b) It follows from Lemma 3.9 that

and by the $D$-property of $\Gamma$ on $F$, we get $\Gamma p^* = p^*$.

Theorem 4.4. Suppose that $F\neq \emptyset$ and $G\neq \emptyset$ are weakly closed and convex subsets of a reflexive and strictly convex Banach space $X$, and let $\Gamma :F\cup G\to F\cup G$ be a noncyclic $\varphi$-quasi-contraction map. Let one of the following conditions be satisfied:

(a) $\Gamma$ is weakly continuous on $F\cup G$;

(b) $\Gamma$ satisfies the $D$-property on $F\cup G$.

Thus, $Fix(\Gamma\mid _{F\times G})\neq \emptyset$. Also, if $(F-F)\cap (G-G) = \{0\}$, then $Fix(\Gamma\mid _{F\times G}) = \{(p^*, q^*)\}$ for some $(p^*, q^*)\in F\times G$.

Proof. According to Theorems 4.1 and 4.3, it is enough to prove the uniqueness of an optimal pair of fixed points $(p^*, q^*)\in F\times G$. Suppose that there exists another point $(p', q')\in F\times G$ for which $\Vert p'-q' \Vert = {\rm Dist}(F, G)$. As $(F-F)\cap (G-G) = \{0\}$, we obtain that $p'-p^*\neq q'-q^*$ (since $(p', q')\neq (p^*, q^*)$, we have $p'\neq p^*$ or $q'\neq q^*$. Hence, $p'-p^*\neq 0$ or $q'-q^*\neq 0$), so $p^*-q^*\neq p'-q'$. From the strict convexity of $X$, we have

which is a contradiction.

The next result guarantees the uniqueness of an optimal pair of fixed points in Theorem 3.5 of ^[5].

Theorem 4.5. Suppose that $F\neq \emptyset$ and $G\neq \emptyset$ are closed and convex subsets of a reflexive and strictly convex Banach space $X$ and let $\Gamma :F\cup G\to F\cup G$ be a noncyclic $\varphi$-contraction map, that is,

for all $(p, q)\in F\times G$. If $(F-F)\cap (G-G) = \{0\}$, then there exists $(p^*, q^*)\in F\times G$ such that $Fix(\Gamma\mid _{F\times G}) = \{(p^*, q^*)\}$.

Proof. In the case that ${\rm Dist}(F, G) = 0$, the result concludes from Theorem 3.14 directly. Otherwise, if ${\rm Dist}(F, G) > 0$, since $F$ is closed and convex, it is weakly closed. It follows from Theorem 4.1 that there exists $(p^*, q^*)\in F\times G$ such that $\Vert p^*-q^* \Vert = {\rm Dist}(F, G)$. The proof of uniqueness of $(p^*, q^*)\in F\times G$ with $\Vert p^*-q^* \Vert = {\rm Dist}(F, G)$ is concluded from a similar discussion of Theorem 4.4. It follows from (4.3) that

which ensures that $(\Gamma p^*, \Gamma q^*) = (p^*, q^*)$. Thus, $\Gamma p^* = p^*$ and $\Gamma q^* = q^*$, and we are finished.

5.   Conclusions

In this paper, we defined a new class of noncyclic mappings and investigated the existence, uniqueness, and convergence of an optimal pair fixed point for such maps in the framework of metric spaces equipped with a transitive relation. We also presented the counterpart results under some other sufficient conditions in strictly convex and reflexive Banach spaces. In this way, we obtained some real extensions of previous results that appeared in ^[2,10,22,25].

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Manuel De La Sen is thankful for the support of Basque Government (Grant No. IT1555-22).

Conflict of interest

The authors declare there are no conflicts of interest.