A new approach to coincidence and common fixed points under a homotopy of families of mappings in $b$-metric spaces

TL;DR

This paper introduces a novel method for establishing coincidence and common fixed points for families of mappings in preordered b-metric spaces using order homotopies, expanding fixed point theory in generalized metric spaces.

Contribution

It presents new fixed point theorems under order homotopies in preordered b-metric spaces, a significant extension of existing fixed point results.

Findings

Established coincidence fixed point results under order homotopies.

Extended fixed point theory to preordered b-metric spaces.

Provided conditions for the existence of common fixed points.

Abstract

In this paper we derive coincidence and common fixed point results under order homotopies of families of mappings in preordered $b$-metric spaces.

Equations

$$x\preccurlyeq y\preccurlyeq z\text{mbox}implies\max \{d(x,y),d(y,z)\}\le s^{2}d(x,z).$$

$$d(x,y)=\left\{\begin{array}[]{cc}(x+y)^{2},&\mbox{if}\thinspace\thinspace x\neq y,\\ 0,&\mbox{if}\thinspace\thinspace x=y.\end{array}\right.$$

$$x\preccurlyeq y\text{mbox}impliesT(x)\preccurlyeq T(y).$$

$$O_X(x_0)$$

$$O_X^{\ast }(x_0)$$

$$\mathcal{C}(x_0,T,S,\preccurlyeq )$$

$${\mathcal{C}}^{\ast }(x_0,T,S,\preccurlyeq )$$

$$\text{mbox}Coin(T,S)$$

$$\text{mbox}Fix(T)$$

$$S^i(z)\preccurlyeq S(w)\preccurlyeq T(w)\text{mbox}foralli\in \mathbb{N},$$

$$d(T^i(w),S^i(z))\to 0\text{mbox}asi\to \infty .$$

$$V=\{x\in O_X(x_0):T(x)\preccurlyeq S(x),Sx\in T(O_X(x_0))\}.$$

$$x_1\preccurlyeq x_0\text{mbox}andT(x_0)=S(x_1).$$

$$d(T(w),S(w))$$

$$\le s^{2}d(T(w),S^i(z))+2s^{2}d(T^i(w),S^i(z))+s^{2}d(S^i(z),S(w)).$$

$$d(T(w),S(w))\le (s^4+s^2)d(S^i(z),T(w))+2s^{2}d(T^i(w),S^i(z)).$$

$$d(T(w),S(w))\le (s^6+s^4+2s^2)d(T^i(w),S^i(z)).$$

$$S^i(z)\succcurlyeq S(w)\succcurlyeq T(w)\text{mbox}foralli\in \mathbb{N},$$

$$d(T^i(w),S^i(z))\to 0\text{mbox}asi\to \infty .$$

$$T=H_0\preccurlyeq H_1\succcurlyeq H_2\preccurlyeq \dots \succcurlyeq H_n=S,$$

$$T=H_0\preccurlyeq H_1\succcurlyeq H_2\preccurlyeq \dots \succcurlyeq H_n=\tilde{T},$$

$$S=K_0\succcurlyeq K_1\preccurlyeq K_2\succcurlyeq \dots \preccurlyeq H_n=\tilde{S}.$$

$$H_t^i(z)\preccurlyeq H_t(w)\preccurlyeq K_t(w)\text{mbox}andd(K_t^i(w),H_t^i(z))\to \infty ,$$

$$K_t^i(z^{\prime })\preccurlyeq K_t(w^{\prime })\preccurlyeq H_t(w^{\prime })\text{mbox}andd(H_t^i(w^{\prime }),K_t^i(z^{\prime }))\to \infty .$$

$$x_0\succcurlyeq x_1\succcurlyeq x_2\succcurlyeq \dots \succcurlyeq x_n$$

$$x_0\succcurlyeq x_1\succcurlyeq x_2\succcurlyeq \dots \succcurlyeq x_n,$$

$$T=H_0\preccurlyeq H_1\succcurlyeq H_2\preccurlyeq \dots \succcurlyeq H_n=\tilde{T},$$

$$S=K_0\succcurlyeq K_1\preccurlyeq K_2\succcurlyeq \dots \preccurlyeq H_n=\tilde{S}.$$

$$K_t^i(z)\succcurlyeq K_t(w)\succcurlyeq H_t(w)\text{mbox}andd(H_t^i(w),K_t^i(z))\to \infty ,$$

$$K_t^i(z^{\prime })\preccurlyeq K_t(w^{\prime })\preccurlyeq H_t(w^{\prime })\text{mbox}andd(H_t^i(w^{\prime }),K_t^i(z^{\prime }))\to \infty .$$

$$x_0\preccurlyeq x_1\succcurlyeq x_2\preccurlyeq \dots \succcurlyeq x_n$$

$$T=H_0\preccurlyeq H_1\succcurlyeq H_2\preccurlyeq \dots \succcurlyeq H_n=\tilde{T}.$$

$$z\succcurlyeq w\succcurlyeq H_t(w)\text{mbox}andd(H_t^i(w),z)\to 0,$$

$$z^{\prime }\preccurlyeq w^{\prime }\preccurlyeq H_t(w^{\prime })\text{mbox}andd(H_t^i(w^{\prime }),z^{\prime })\to 0.$$

$$x_0\preccurlyeq x_1\succcurlyeq x_2\preccurlyeq \dots \succcurlyeq x_n$$

Full text

Mathematics Subject Classification: 54H25, 06A06.

Keywords: $b$-metric space, common fixed point, concordantly isotone mappings, coincidence point, preorder, order homotopy.

Introduction and Preliminaries

Throughout this paper, $\mathbb{N}$ denotes the set of natural numbers. Bakhtin [4] and Czerwik [5] generalized the concept of metric spaces and introduced the notion of $b$-metric spaces as follows:

Definition 1.1**.**

Let $X$ be a nonempty set. A $b$-metric is a function $d:X\times X\rightarrow[0,\infty)$ such that for all $x,y,z\in X$ and a constant $s\geq 1$, the following conditions are satisfied:**

*(i) $d(x,y)=0$ if and only if $x=y$, ***

(ii) $d(x,y)=d(y,x)$,**

(iii) $d(x,y)\leq s[d(x,z)+d(z,y)]$.

The pair $(X,d)$ is called a $b$-metric space and the number $s$ is called the coefficient of $(X,d)$. If we take $s=1$, then $(X,d)$ reduces to a metric space.**

Let $X$ be a nonempty set equipped with a binary relation $\preccurlyeq$. Then the relation $\preccurlyeq$ is

(i) reflexive: if $x\preccurlyeq x$ for all $x\in X$,

(ii) antisymmetric: if $x\preccurlyeq y$ and $y\preccurlyeq x$, then $x=y$ for all $x,y\in X$,

(iii) transitive: if $x\preccurlyeq y$ and $y\preccurlyeq z$, then $x\preccurlyeq z$ for all $x,y,z\in X$.

A binary relation $\preccurlyeq$ is said to be a preorder if it is reflexive and transitive. A preorder is said to be a partial order if it is antisymmetric. Let $(X,d,\preccurlyeq)$ be a $b$-metric space with coefficient $s\geq 1$ equipped with a binary relation $\preccurlyeq$. Motivated by Kamihigashi and Stachurski [8] we define the following axiom:

$d$ is $s$-regular if for all $x,y,z\in$ we have

[TABLE]

Evidently, if $d$ is regular in the sense of Kamihigashi and Stachurski [8], then $d$ is $s$-regular but the converse is not true as illustrated by the following example:

Let $X=[0,\infty)$ and define $d:X\times X\rightarrow[0,\infty)$ by

[TABLE]

Then $(X,d)$ is a $b$-metric space with coefficient $s=2$. Suppose that $X$ is equipped with a preorder relation defined by: for $x,y\in X$, $x\preccurlyeq y$ if and only if $x\leq y$. It can be easily observed that if $x\preccurlyeq y\preccurlyeq z$, then $\max\{d(x,y),d(y,z)\}\leq 4d(x,z)$ for all $x,y,z\in X$. This implies that $d$ is $s$-regular. Also, $1\preccurlyeq 2\preccurlyeq 3$ but $\max\{d(1,2),d(2,3)\}=25\nleq d(1,3)$. Therefore, $d$ is not regular.

Kamihigashi and Stachurski [8] and Batsari et al. [3] established fixed point results of order-preserving self mappings in spaces equipped with a transitive binary relation and some distance measures. Recently, Fomenko and Podoprikhin [6, 10] generalized the results of Arutyunov [1, 2] to the case of families of multivalued mappings in partially ordered sets. They [7, 10] introduced the notion of order homotopy and proved that common fixed point and coincidence point properties are preserved under homotopies of families of mappings in ordered sets.

In this paper we establish results on preservation of coincidence points and common fixed points under order homotopies of families of mappings. The main objective of the paper is to develop a new approach and obtain sufficient conditions to gurantee the existence of coincidence and common fixed points under order homotopies of families of mappings in preordered $s$-regular $b$-metric spaces. We have generalized the results of [7, 9] to the case of preordered $s$-regular $b$-metric spaces and preordered regular metric spaces. This approach has enabled us to replace the classical approach of considering the underlying space to be complete by a preordered space and $b$-metric function satisfying the axiom of $s$-regularity. As a consequence we establish coincidence and common fixed points results under order homotopic families of mappings in the context of metric spaces and $b$-metric spaces from a unique point of view. An example demonstrating the utility of the results obtained is also provided.

Coincidence point results

In this section we prove some coincidence point results under a homotopy of families of mappings in preordered $s$-regular $b$-metric spaces. As a consequence, we formulate coincidence and fixed point results in preordered regular metric spaces.

For formulating the results, we recall some definitions and notions introduced by Fomenko and Podoprikhin [7, 9].

Definition 2.1**.**

Let $(X,\preccurlyeq)$ be a preordered set. A mapping $T:X\rightarrow X$* covers (covers from above) a mapping $S:X\rightarrow X$ on a set $A\subset X$ if and only if for any $x\in A$ such that $S(x)\preccurlyeq T(x)$ ($S(x)\succcurlyeq T(x)$) then there exists an element $y\in X$ satisfying $y\preccurlyeq x$ ($y\succcurlyeq x$) and $T(y)=S(x)$. If $A=X$, then we say that $T$ covers (covers from above) $S$.*

Let $(X,\preccurlyeq)$ be a preordered set. A mapping $T:X\rightarrow X$ is said to be isotone if for all $x,y\in X$

[TABLE]

Let $T,S:X\rightarrow X$ be a pair of self mappings on a preordered set $(X,\preccurlyeq)$. Let $\mathcal{C}(T,S,\preccurlyeq)$ denote the set of chains $C$ such that for all $x,y\in C$ we have

(i) $T(x)\preccurlyeq S(x)$,

(ii) $x\prec y$ implies $S(x)\preccurlyeq T(y)$,

(iii) $S(C)\subset T(X)$.

Let $\mathcal{C}^{*}(T,S,\preccurlyeq)$ denote the set of chains $C$ such that for all $x,y\in C$ we have

(i) $T(x)\succcurlyeq S(x)$,

(ii) $x\prec y$ implies $T(x)\preccurlyeq S(y)$,

(iii) $S(C)\subset T(X)$.

Recall that the dual order of $\preccurlyeq$ is the order $\preccurlyeq^{*}$ defined by: $x\preccurlyeq y$ if and only if $y\preccurlyeq^{*}x$. It can be easily observed that $\mathcal{C}^{*}(T,S,\preccurlyeq)=\mathcal{C}(T,S,\preccurlyeq^{*})$. For a fixed element $x_{0}\in X$ define the set

[TABLE]

Recall that an element $x\in X$ is said to a minimal (maximal) element of $X$ if and only if for every other element $y\in X$ we have $x\preccurlyeq y$ ($x\succcurlyeq y$).

Theorem 2.2**.**

Let $(X,d,\preccurlyeq)$ be a preordered $s$-regular $b$-metric space. Let $T,S:X\rightarrow X$ be self mappings and there exists $x_{0}\in X$ such that $T(x_{0})\preccurlyeq S(x_{0})$. Suppose that the following conditions are satisfied:

(i) the mapping $T$ is isotone,

(ii) the mapping $S$ covers the mapping $T$ on the set $O_{X}(x_{0})$,

(iii) any chain $C\in\mathcal{C}(x_{0},T,S,\preccurlyeq)$ has a lower bound $w\in X$ satisfying $w\preccurlyeq T(w)$ and there exists $z\in X$ such that

[TABLE]

Then the set $\mbox{Coin}(T,S)\cap O_{X}(x_{0})$ is nonempty and contains a minimal element.

Proof.

The proof is divided into four steps where the existence of an element in $\mbox{Coin}(T,S)\cap O_{X}(x_{0})$ is proven in Steps $1$ to Step $3$ and its minimality is established in Step $4$.

Step 1 Set

[TABLE]

We show that $V$ is a nonempty set. Since $\preccurlyeq$ is reflexive, $x_{0}\in O_{X}(x_{0})$. Also, $T(x_{0})\preccurlyeq S(x_{0})$. By (ii) there exists $x_{1}\in X$ satisfying

[TABLE]

Since $T$ is isotone, $T(x_{1})\preccurlyeq S(x_{1})$. Also, $S(x_{1})=T(x_{0})\in T(O_{X}(x_{0}))$ which implies that $x_{1}\in V$. Define an ordered relation on $V$ as follows: $v_{1}\sqsubseteq v_{2}$ if either $v_{1}=v_{2}$ or $v_{1}\prec v_{2}$ and $S(v_{1})\preccurlyeq T(v_{2})$. It is easily seen that $\sqsubseteq$ is reflexive and antisymmetric. Suppose that $v_{1},v_{2},v_{3}\in V$, $v_{1}\sqsubseteq v_{2}$ and $v_{2}\sqsubseteq v_{3}$. If $v_{1}=v_{2}$ or $v_{2}=v_{3}$, then evidently, $v_{1}\sqsubseteq v_{3}$. If $v_{1}\prec v_{2}$, $v_{2}\prec v_{3}$, $S(v_{1})\preccurlyeq T(v_{2})$ and $S(v_{2})\preccurlyeq T(v_{3})$, then $v_{1}\prec v_{3}$. Also, $v_{2}\in V$ gives $T(v_{2})\preccurlyeq S(v_{2})$ which implies that $S(v_{1})\preccurlyeq T(v_{3})$. Therefore, $v_{1}\sqsubseteq v_{3}$ which gives $\sqsubseteq$ is a partial order on $V$. By Hausdorff maximal principle, there exists a maximal chain $C\subset V$ with respect to the order $\sqsubseteq$.

Step 2 We show that $C\in\mathcal{C}(x_{0},T,S,\preccurlyeq)$. By the definition of V we have $C\subset O_{X}(x_{0})$, $S(C)\subset T(O_{X}(x_{0}))$ and $T(x)\preccurlyeq S(x)$ for all $x\in C$. Let $x,y\in C$ and $x\prec y$. Since $C$ is a chain, $x\sqsubseteq y$ which gives $S(x)\preccurlyeq T(y)$. Therefore, $C\in\mathcal{C}(x_{0},T,S,\preccurlyeq)$. By (iii), there exists a lower bound $w\in X$ of $C$ with respect to the order $\preccurlyeq$.

Step 3 It is to be shown that $w\in\mbox{Coin}(T,S)\cap O_{X}(x_{0})$. Since $w$ is a lower bound of $C$, $w\preccurlyeq x$ for all $x\in C$. Also, $C\subset O_{X}(x_{0})$ and $\preccurlyeq$ is transitive. Therefore, $w\preccurlyeq x_{0}$ which implies that $w\in O_{X}(x_{0})$. Consider

[TABLE]

Since $S^{i}(z)\preccurlyeq S(w)\preccurlyeq T(w)$ for all $i\in\mathbb{N}$ and $d$ is $s$-regular, $d(S^{i}(z),S(w))\leq s^{2}d(S^{i}(z),T(w))$. Therefore, (2) becomes

[TABLE]

Also, $w\preccurlyeq T(w)$ and $T$ is isotone gives $T(w)\preccurlyeq T^{2}(w)$. Continuing likewise we get $w\preccurlyeq T(w)\preccurlyeq T^{i}(w)$ for all $i\in\mathbb{N}$. Therefore, $S^{i}(z)\preccurlyeq T(w)\preccurlyeq T^{i}(w)$ which gives $d(S^{i}(z),T(w))\leq s^{2}d(S^{i}(z),T^{i}(w))$. Using (2.2) we have

[TABLE]

Letting $i\rightarrow\infty$ we get $T(w)=S(w)$. Therefore, $w\in\mbox{Coin}(T,S)\cap O_{X}(x_{0})$.

Step 4 We claim that $w$ is a minimal element of the set $\mbox{Coin}(T,S)\cap O_{X}(x_{0})$. Assume on the contrary, there exists $u\in\mbox{Coin}(T,S)\cap O_{X}(x_{0})$ satisfying $u\prec w$. Since $T$ is isotone, $T(u)\preccurlyeq T(w)$ which implies that $S(u)\preccurlyeq T(w)$. Therefore, $u\sqsubseteq w$. As $w$ is a lower bound of $C$, $w\preccurlyeq x$ for all $x\in C$. Therefore, $T(w)\preccurlyeq T(x)$ which implies that $S(w)\preccurlyeq T(w)$. This gives $u\sqsubseteq w\sqsubseteq x$. As $C$ is a maximal chain with respect to $\sqsubseteq$, $u\in C$ which gives $w\preccurlyeq u$, a contradiction. Hence, $w$ is a minimal element of the set $\mbox{Coin}(T,S)\cap O_{X}(x_{0})$. ∎

The dual version of Theorem 2.2 can be stated as follows:

Theorem 2.3**.**

Let $(X,d,\preccurlyeq)$ be a preordered $s$-regular $b$-metric space. Let $T,S:X\rightarrow X$ be self mappings and there exists $x_{0}\in X$ such that $T(x_{0})\succcurlyeq S(x_{0})$. Suppose that the following conditions hold:

(i) the mapping $T$ is isotone,

(ii) the mapping $S$ covers the mapping $T$ from above on the set $O_{X}^{*}(x_{0})$,

(iii) any chain $C\in\mathcal{C}^{*}(x_{0},T,S,\preccurlyeq)$ has an upper bound $w\in X$ satisfying $w\succcurlyeq T(w)$ and there exists $z\in X$ such that

[TABLE]

Then the set $\mbox{Coin}(T,S)\cap O_{X}^{*}(x_{0})$ is nonempty and contains a maximal element.

Podoprikhin and Fomenko [9] generalized the notion of order homotopy of isotone mappings introduced by Walker [11] as follows:

Let $(X,\preccurlyeq)$ be a preordered set. A pair of mappings $T,S:X\rightarrow X$ are said to be order homotopic if there exists a finite sequence of mappings $H_{i}:X\rightarrow X$ $(i=1,2,\ldots,n)$ such that

[TABLE]

where $H_{i}\preccurlyeq H_{j}$ if and only if $H_{i}(x)\preccurlyeq H_{j}(x)$ for all $x\in X$.

Theorem 2.4**.**

Let $(X,\preccurlyeq)$ be a preordered $s$-regular $b$-metric space. Let $(T,S)$ and $(\tilde{T},\tilde{S})$ be a pair of self mappings on $X$. Suppose that there are homotopies $\{H_{t}\}_{0\leq t\leq n}$ and $\{K_{t}\}_{0\leq t\leq n}$ between these pair of mappings such that

[TABLE]

Suppose that $x_{0}\in\mbox{Coin}(T,S)$ and the following conditions are satisfied:

(i) for each odd $t$, $1\leq t\leq n$, $K_{t}$ is isotone, the mapping $H_{t}$ covers the mapping $K_{t}$ and any chain $C\in\mathcal{C}(x_{0},K_{t},H_{t},\preccurlyeq)$ has a lower bound $w\in X$ satisfying $w\preccurlyeq K_{t}(w)$ and there exists $z\in X$ such that for all $i\in\mathbb{N}$ we have

[TABLE]

(ii) for each even $t$, $1\leq t\leq n$, $H_{t}$ is isotone, the mapping $K_{t}$ covers the mapping $H_{t}$ and any chain $C^{\prime}\in\mathcal{C}(x_{0},H_{t},K_{t},\preccurlyeq)$ has a lower bound $w^{\prime}\in X$ satisfying $w^{\prime}\preccurlyeq H_{t}(w^{\prime})$ and there exists $z^{\prime}\in X$ such that for all $i\in\mathbb{N}$ we have

[TABLE]

Then there exists a chain

[TABLE]

such that for every $t$, $1\leq t\leq n$, $x_{t}\in\mbox{Coin}(H_{t},K_{t})\cap O_{X}(x_{t-1})$ and $x_{t}$ is a minimal element of the set $\mbox{Coin}(H_{t},K_{t})\cap O_{X}(x_{t-1})$.

Proof.

Since $x_{0}\in\mbox{Coin}(H_{0},K_{0})$, $K_{1}(x_{0})\preccurlyeq K_{0}(x_{0})=H_{0}(x_{0})\preccurlyeq H_{1}(x_{0})$ which implies that $K_{1}(x_{0})\preccurlyeq H_{1}(x_{0})$. Using (i) we infer that all the conditions of Theorem 2.2 are satisfied. Therefore, there exists $x_{1}\in\mbox{Coin}(H_{1},K_{1})\cap O_{X}(x_{0})$ such that $x_{1}$ is a minimal element of the set $\mbox{Coin}(H_{1},K_{1})\cap O_{X}(x_{0})$. Consider $H_{2}(x_{1})\preccurlyeq H_{1}(x_{1})=K_{1}(x_{1})\preccurlyeq K_{2}(x_{1})$ which gives $H_{2}(x_{1})\preccurlyeq K_{2}(x_{1})$. As $\mathcal{C}(x_{1},H_{2},K_{2},\preccurlyeq)\subset\mathcal{C}(x_{0},H_{2},K_{2},\preccurlyeq)$ and using (ii) we deduce that all the conditions of Theorem 2.2 are satisfied. Therefore, there exists $x_{2}\in\mbox{Coin}(H_{2},K_{2})\cap O_{X}(x_{1})$ such that $x_{2}$ is a minimal element of the set $\mbox{Coin}(H_{2},K_{2})\cap O_{X}(x_{1})$. Repeating this process we obtain a chain

[TABLE]

where $x_{t}\in\mbox{Coin}(H_{t},K_{t})\cap O_{X}(x_{t-1})$ and $x_{t}$ is a minimal element of the set $\mbox{Coin}(H_{t},K_{t})\cap O_{X}(x_{t-1})$, $1\leq t\leq n$. ∎

The following result is an immediate consequence of Theorem 2.2 and Theorem 2.3:

Theorem 2.5**.**

Let $(X,d,\preccurlyeq)$ be a preordered $s$-regular $b$-metric space. Let $(T,S)$ and $(\tilde{T},\tilde{S})$ be a pair of self mappings on $X$. Suppose that there are homotopies $\{H_{t}\}_{0\leq t\leq n}$ and $\{K_{t}\}_{0\leq t\leq n}$ between these pair of mappings such that

[TABLE]

Suppose that the mappings $H_{t}$ are isotone, $1\leq t\leq n$, $x_{0}\in\mbox{Coin}(T,S)$ and the following conditions are satisfied:

(i) for each odd $t$, $1\leq t\leq n$, the mapping $K_{t}$ covers the mapping $H_{t}$ from the above and any chain $C\in\mathcal{C}^{*}(H_{t},K_{t},\preccurlyeq)$ has an upper bound $w\in X$ satisfying $w\succcurlyeq H_{t}(w)$ and there exists $z\in X$ such that for all $i\in\mathbb{N}$ we have

[TABLE]

(ii) for each even $t$, $1\leq t\leq n$, the mapping $K_{t}$ covers the mapping $H_{t}$ and any chain $C\in\mathcal{C}(H_{t},K_{t},\preccurlyeq)$ has a lower bound $w^{\prime}\in X$ satisfying $w^{\prime}\preccurlyeq H_{t}(w^{\prime})$ and there exists $z^{\prime}\in X$ such that for all $i\in\mathbb{N}$ we have

[TABLE]

Then there exists a fence

[TABLE]

such that for each odd $t$, $1\leq t\leq n$, $x_{t}\in\mbox{Coin}(H_{t},K_{t})\cap O_{X}^{*}(x_{t-1})$ and $x_{t}$ is a maximal element of the set $\mbox{Coin}(H_{t},K_{t})\cap O_{X}^{*}(x_{t-1})$ and for each even $t$, $1\leq t\leq n$, $x_{t}\in\mbox{Coin}(H_{t},K_{t})\cap O_{X}(x_{t-1})$ and $x_{t}$ is a minimal element of the set $\mbox{Coin}(H_{t},K_{t})\cap O_{X}(x_{t-1})$.

Evidently, the identity mapping $I:X\rightarrow X$ covers (covers from above) any mapping $T:X\rightarrow X$. Therefore, we have the following fixed point result under a homotopy of self mappings:

Theorem 2.6**.**

Let $(X,d,\preccurlyeq)$ be a preordered $s$-regular $b$-metric space. Let $T,\tilde{T}:X\rightarrow X$ be isotone mappings and $\{H_{t}\}_{0\leq t\leq n}$ be homotopy between $T$ and $\tilde{T}$ such that

[TABLE]

Suppose that $x_{0}\in\mbox{Fix}(T)$ and the following conditions are satisfied:

(i) for each odd $t$, $1\leq t\leq n$, any chain $C\in\mathcal{C}^{*}(H_{t},I,\preccurlyeq)$ has an upper bound $w\in X$ and there exists $z\in X$ such that for all $i\in\mathbb{N}$ we have

[TABLE]

(ii) for each even $t$, $1\leq t\leq n$, any chain $C\in\mathcal{C}(H_{t},T,\preccurlyeq)$ has a lower bound $w^{\prime}\in X$ and there exists $z^{\prime}\in X$ such that for all $i\in\mathbb{N}$ we have

[TABLE]

Then there exists a fence

[TABLE]

such that for each odd $t$, $1\leq t\leq n$, $x_{t}\in\mbox{Fix}(H_{t})\cap O_{X}^{*}(x_{t-1})$ and $x_{t}$ is a maximal element of the set $\mbox{Fix}(H_{t})\cap O_{X}^{*}(x_{t-1})$ and for each even $t$, $1\leq t\leq n$, $x_{t}\in\mbox{Fix}(H_{t})\cap O_{X}(x_{t-1})$ and $x_{t}$ is a minimal element of the set $\mbox{Fix}(H_{t})\cap O_{X}(x_{t-1})$.

Remark 2.7**.**

In Theorems 2.2-2.6 if we assume $(X,d,\preccurlyeq)$ is a preordered regular metric space, then also the results hold.**

We conclude this section with the following example to illustrate the efficiency of our results:

Example 2.8**.**

Let $X=[0,2]$ and define $d:X\times X\rightarrow[0,\infty)$ by $d(x,y)=(x-y)^{2}$. Then $(X,d)$ is a $b$-metric space with coefficient $s=2$ and $d$ is $2$-regular. Suppose that $X$ is equipped with a preorder given by: for $x,y\in X$, $x\preccurlyeq y$ if and only if $x\leq y$. Define $T,H,\tilde{T}:X\rightarrow X$ by

[TABLE]

Then $T\preccurlyeq H\succcurlyeq\tilde{T}$. Clearly, $x_{0}=0\in\mbox{Fix}(T)$ and $O_{X}^{*}(x_{0})=[0,2]$. Also, every chain $C\in\mathcal{C}^{*}(H,I,\preccurlyeq)$ has an upper bound $w=1$ and there exists $z=1+\frac{1}{i}\in X$, $i\in\mathbb{N}$ such that

[TABLE]

and

[TABLE]

Also, every chain $C\in\mathcal{C}(\tilde{T},I,\preccurlyeq)$ has a lower bound $w^{\prime}=0.2$ and $z^{\prime}=0.2-\frac{i}{2i^{2}+4}\in X$, $i\in\mathbb{N}$ such that

[TABLE]

and

[TABLE]

Then all the conditions of Theorem 2.6 are satisfied for a preordered $2$-regular $b$-metric space. Therefore, we get a fence $x_{0}\preccurlyeq x_{1}\succcurlyeq x_{2}$, where $x_{0}=0$, $x_{1}=1$ and $x_{2}=0.2$. It is observed that $x_{1}\in\mbox{Fix}(H)\cap O_{X}^{*}(x_{0})$ and $x_{1}$ is a maximal element of the set $\mbox{Fix}(H)\cap O_{X}^{*}(x_{0})$. Also, $O_{X}(x_{1})=[0,1]$, $x_{2}\in\mbox{Fix}(\tilde{T})\cap O_{X}(x_{1})$ and $x_{2}$ is a minimal element of the set $\mbox{Fix}(\tilde{T})\cap O_{X}(x_{1})$.**

Common Fixed Point Results

In this section we prove some common fixed point results under a homotopy of families of mappings in preordered $s$-regular $b$-metric spaces.

Fomenko and Podoprikhin [6] introduced the notion of concordantly isotone mappings as follows:

Let $(X,\preccurlyeq)$ be a preordered set, $\mathcal{I}$ be a nonempty set and a family of mappings $\mathcal{F}=\{f_{\alpha}\}_{\alpha\in\mathcal{I}}$, where $f_{\alpha}:X\rightarrow X$ for all $\alpha\in\mathcal{I}$. The family $\mathcal{F}$ is said to be condordantly isotone if for all $x,y\in X$

[TABLE]

Let $\mathcal{C}_{1}(\mathcal{F},\preccurlyeq)$ denote the set of chains $C\subset\bigcup\limits_{\alpha\in\mathcal{I}}f_{\alpha}(X)$ such that

(i) $f_{\alpha}(x)\preccurlyeq x$ for all $x\in C$ and $\alpha\in\mathcal{I}$,

(ii) $x\prec y$ implies $x\preccurlyeq f_{\alpha}(y)$ for all $x,y\in C$ and $\alpha\in\mathcal{I}$.

Let $\mathcal{C}_{1}^{*}(\mathcal{F},\preccurlyeq)$ denote the set of chains $C\subset\bigcup\limits_{\alpha\in\mathcal{I}}f_{\alpha}(X)$ such that

(i) $f_{\alpha}(x)\succcurlyeq x$ for all $x\in C$ and $\alpha\in\mathcal{I}$,

(ii) $x\prec y$ implies $f_{\alpha}(x)\preccurlyeq y$ for all $x,y\in C$ and $\alpha\in\mathcal{I}$.

Observe that $\mathcal{C}_{1}^{*}(\mathcal{F},\preccurlyeq)=\mathcal{C}_{1}(\mathcal{F},\preccurlyeq^{*})$. For a fixed element $x_{0}\in X$ define the set

[TABLE]

Theorem 3.1**.**

Let $(X,d,\preccurlyeq)$ be a preordered $s$-regular $b$-metric space in which a point $x_{0}$ is fixed and $\mathcal{F}=\{f_{\alpha}\}_{\alpha\in\mathcal{I}}$ be a concordantly isotone family of mappings verifying $f_{\alpha}(x_{0})\preccurlyeq x_{0}$ for all $\alpha\in\mathcal{I}$. Suppose that for any chain $C\in\mathcal{C}_{1}(x_{0},\mathcal{F},\preccurlyeq)$ there exists $w\in X$ such that $w$ is a common lower bound of the chains $f_{\alpha}(C)$ for each $\alpha\in\mathcal{I}$ and there exists $z\in X$ and $\beta\in\mathcal{I}$ satisfying

[TABLE]

Then the set $\mbox{ComFix}(\mathcal{F})\cap O_{X}(x_{0})$ is nonempty and contains a minimal element.

Proof.

The proof is divided into three steps where the existence of an element in $\mbox{ComFix}(\mathcal{F})\cap O_{X}(x_{0})$ is estalished in Steps $1$ and $2$ and its minimality is proven in Step $3$.

Step 1 We show that $\mathcal{C}_{1}(x_{0},\mathcal{F},\preccurlyeq)$ is nonempty. Since $f_{\alpha}(x_{0})\preccurlyeq x_{0}$,

[TABLE]

Also, $f_{\alpha}(x_{0})\preccurlyeq x_{0}$ and $\mathcal{F}$ is concordantly isotone implies that $f_{\beta}(f_{\alpha}(x_{0}))\preccurlyeq f_{\alpha}(x_{0})$ for all $\beta\in\mathcal{I}$. Therefore, $\{f_{\alpha}(x_{0})\}\in\mathcal{C}_{1}(x_{0},\mathcal{F},\preccurlyeq)$ which gives $\mathcal{C}_{1}(x_{0},\mathcal{F},\preccurlyeq)$ is nonempty. Define an ordered relation on $\mathcal{C}_{1}(x_{0},\mathcal{F},\preccurlyeq)$ as follows: $C_{1}\trianglerighteq C_{2}$ if and only if $C_{1}\subset C_{2}$. It is easily seen that $\trianglerighteq$ is a partial order on $C_{1}(x_{0},\mathcal{F},\preccurlyeq)$. Then by Hausdorff maximal principle, there exists a maximal chain $C$ in $\mathcal{C}_{(}x_{0},\mathcal{F},\preccurlyeq)$ containing $\{f_{\alpha}(x_{0})\}$ with respect to the order $\trianglerighteq$. By our assumption there exists $w\in X$ such that $w$ is a common lower bound of the chains $f_{\alpha}(C)$ for each $\alpha\in\mathcal{I}$ and there exists $z\in X$ and $\beta\in\mathcal{I}$ satisfying

[TABLE]

Step 2 In this step we show that $w\in\mbox{ComFix}(\mathcal{F})\cap O_{X}(x_{0})$. Since $w$ is a common lower bound of the chains $f_{\alpha}(C)$ for all $\alpha\in\mathcal{I}$. Then $w\preccurlyeq f_{\alpha}(x)$ for all $x\in C$ and $\alpha\in\mathcal{I}$. As $f_{\alpha}(x_{0})\in C$,

[TABLE]

Then transitivity of $\preccurlyeq$ implies that $w\in O_{X}(x_{0})$. Consider

[TABLE]

Since $f_{\alpha}(w)\preccurlyeq w$ and $\mathcal{F}$ is concordantly isotone, $f_{\alpha}(f_{\alpha}(w))\preccurlyeq f_{\alpha}(w)$. Proceeding likewise we have $f_{\alpha}^{i}(w)\preccurlyeq f_{\alpha}(w)\preccurlyeq w$ for all $i\in\mathbb{N}$. Therefore, for all $i\in\mathbb{N}$

[TABLE]

As $d$ is $s$-regular, $d(f_{\alpha}(w),f_{\beta}^{i}(z))\leq s^{2}d(f_{\alpha}^{i}(w),f_{\beta}^{i}(z))$. Also, $f_{\alpha}^{i}(w)\preccurlyeq w\preccurlyeq f_{\beta}^{i}(z)$ and $d$ is $s$-regular gives $d(w,f_{\beta}^{i}(z))\leq s^{2}d(f_{\alpha}^{i}(w),f_{\beta}^{i}(z))$. Therefore, (3.1) becomes

[TABLE]

Letting $i\rightarrow\infty$ we get $f_{\alpha}(w)=w$ for all $\alpha\in\mathcal{I}$. Therefore, $w\in\mbox{ComFix}(\mathcal{F})\cap O_{X}(x_{0})$.

Step 3 We claim that $w$ is a minimal element of the set $\mbox{ComFix}(\mathcal{F})\cap O_{X}(x_{0})$. Assume on the contrary, there exists $v\in\mbox{ComFix}(\mathcal{F})\cap O_{X}(x_{0})$ satisfying $v\prec w$. As $v=f_{\alpha}(v)\in\bigcup\limits_{\alpha\in\mathcal{I}}f_{\alpha}(O_{X}(x_{0}))$, $v\in O_{X}(x_{0})\cap\bigcup\limits_{\alpha\in\mathcal{I}}f_{\alpha}(O_{X}(x_{0}))$. Since $\preccurlyeq$ is reflexive, $f_{\alpha}(v)\preccurlyeq v$ for all $\alpha\in\mathcal{I}$. Therefore, $f_{\alpha}(x)\preccurlyeq x$ for all $x\in C\cup\{v\}$ and $\alpha\in\mathcal{I}$. As $w$ is a common lower bound of the chains $f_{\alpha}(C)$ and $v\prec w$, $v\preccurlyeq w\preccurlyeq f_{\alpha}(x)\preccurlyeq x$ for all $x\in C$ and $\alpha\in\mathcal{I}$. Then transitivity of $\preccurlyeq$ implies that $v\preccurlyeq x$ for all $x\in C$. Since $\mathcal{F}$ is concordantly isotone, $f_{\beta}(v)\preccurlyeq f_{\alpha}(x)$ which gives $v\preccurlyeq f_{\alpha}(x)$ for all $x\in C$ and $\alpha\in\mathcal{I}$. Therefore, $C\cup\{v\}$ is a chain in $\mathcal{C}_{1}(x_{0},\mathcal{F},\preccurlyeq)$ but this contradicts the maximality of $C$. Hence, $w$ is a minimal element of the set $\mbox{ComFix}(\mathcal{F})\cap O_{X}(x_{0})$.‘ ∎

The dual version of Theorem 3.1 can be stated as follows:

Theorem 3.2**.**

Let $(X,d,\preccurlyeq)$ be a preordered $s$-regular $b$-metric space in which a point $x_{0}$ is fixed and $\mathcal{F}=\{f_{\alpha}\}_{\alpha\in\mathcal{I}}$ be a concordantly isotone family of mappings verifying $f_{\alpha}(x_{0})\succcurlyeq x_{0}$ for all $\alpha\in\mathcal{I}$. Suppose that for any chain $C\in\mathcal{C}_{1}^{*}(x_{0}\mathcal{F},\preccurlyeq)$ there exists $w\in X$ such that $w$ is a common upper bound of the chains $f_{\alpha}(C)$ for each $\alpha\in\mathcal{I}$ and there exists $z\in X$ and $\beta\in\mathcal{I}$ satisfying

[TABLE]

Then the set $\mbox{ComFix}(\mathcal{F})\cap O_{X}^{*}(x_{0})$ is nonempty and contains a maximal element.

Theorem 3.3**.**

Let $(X,d,\preccurlyeq)$ be a preordered $s$-regular $b$-metric space. Let $\mathcal{F}=\{f_{\alpha}\}_{\alpha\in\mathcal{I}}$ and $\mathcal{G}=\{g_{\alpha}\}_{\alpha\in\mathcal{I}}$ be a pair of families of self mappings on $X$. Suppose that $\{H_{t,\alpha}\}_{0\leq t\leq n}$ be a homotopy between $f_{\alpha}$ and $g_{\alpha}$ such that

[TABLE]

Suppose that $x_{0}\in\mbox{ComFix}(\mathcal{F})$ and the following conditions are satisfied:

(i) for each $t$, $1\leq t\leq n$, the family $\mathcal{H}_{t}=\{H_{t,\alpha}\}_{\alpha\in\mathcal{I}}$ is concordantly isotone,

(ii) for each odd $t$, $1\leq t\leq n$, for any chain $C\in\mathcal{C}_{1}^{*}(\mathcal{H}_{t},\preccurlyeq)$ there exists $w\in X$ such that $w$ is a common upper bound of the chains $H_{t,\alpha}(C)$ for all $\alpha\in\mathcal{I}$ and there exists $z\in X$ and $\beta\in\mathcal{I}$ satisfying

[TABLE]

(iii) for each even $t$, $1\leq t\leq n$, for any chain $C\in\mathcal{C}_{1}(\mathcal{H}_{t},\preccurlyeq)$ there exists $w^{\prime}\in X$ such that $w^{\prime}$ is a common lower bound of the chains $H_{t,\alpha}(C)$ for all $\alpha\in\mathcal{I}$ and there exists $z^{\prime}\in X$ and $\gamma\in\mathcal{I}$ satisfying

[TABLE]

Then there exists a fence

[TABLE]

such that for each odd $t$, $1\leq t\leq n$, $x_{t}\in\mbox{ComFix}(\mathcal{H}_{t})\cap O_{X}^{*}(x_{t-1})$ and $x_{t}$ is a maximal element of the set $\mbox{ComFix}(\mathcal{H}_{t})\cap O_{X}^{*}(x_{t-1})$ and for each even $t$, $1\leq t\leq n$, $x_{t}\in\mbox{ComFix}(\mathcal{H}_{t})\cap O_{X}(x_{t-1})$ and $x_{t}$ is a minimal element of the set $\mbox{ComFix}(\mathcal{H}_{t})\cap O_{X}(x_{t-1})$.

Proof.

Since $x_{0}\in\mbox{ComFix}(\mathcal{F})$, $H_{1,\alpha}(x_{0})\succcurlyeq H_{0,\alpha}(x_{0})=f_{\alpha}(x_{0})=x_{0}$ for all $\alpha\in\mathcal{I}$. This gives $H_{1,\alpha}(x_{0})\succcurlyeq x_{0}$ for all $\alpha\in\mathcal{I}$. Using $\mathcal{C}_{1}^{*}(x_{0},\mathcal{H}_{1},\preccurlyeq)\subset\mathcal{C}_{1}^{*}(\mathcal{H}_{1},\preccurlyeq)$, (i) and (ii) we deduce that all the conditions of Theorem 3.2 are satisfied. Therefore, there exists $x_{1}\in\mbox{ComFix}(\mathcal{H}_{1})\cap O_{X}^{*}(x_{0})$ such that $x_{1}$ is a maximal element of the set $\mbox{ComFix}(\mathcal{H}_{1})\cap O_{X}^{*}(x_{0})$. Also, $H_{2,\alpha}(x_{1})\preccurlyeq H_{1,\alpha}(x_{1})=x_{1}$ for all $\alpha\in\mathcal{I}$ and $\mathcal{C}_{1}(x_{1},\mathcal{H}_{2},\preccurlyeq)\subset\mathcal{C}_{1}(\mathcal{H}_{2},\preccurlyeq)$. Using (i) and (iii) we conclude that all the conditions of Theorem 3.1 are satisfied. Therefore, there exists $x_{2}\in\mbox{ComFix}(\mathcal{H}_{2})\cap O_{X}(x_{1})$ such that $x_{2}$ is a minimal element of the set $\mbox{ComFix}(\mathcal{H}_{2})\cap O_{X}(x_{1})$. Proceeding likewise we get a fence

[TABLE]

where for each odd $t$, $1\leq t\leq n$, $x_{t}\in\mbox{ComFix}(\mathcal{H}_{t})\cap O_{X}^{*}(x_{t-1})$ and $x_{t}$ is a maximal element of the set $\mbox{ComFix}(\mathcal{H}_{t})\cap O_{X}^{*}(x_{t-1})$ and for each even $t$, $1\leq t\leq n$, $x_{t}\in\mbox{ComFix}(\mathcal{H}_{t})\cap O_{X}(x_{t-1})$ and $x_{t}$ is a minimal element of the set $\mbox{ComFix}(\mathcal{H}_{t})\cap O_{X}(x_{t-1})$. ∎

Remark 3.4**.**

It is observed that Theorems 3.1-3.3 hold if we take $(X,d,\preccurlyeq)$ to be a preordered regular metric space.**

Acknowledgements

∗corresponding author

The corresponding author is supported by UGC Non-NET fellowship (Ref.No. Sch/139/Non-NET/ Math./Ph.D./2017-18/1028).