A Common Fixed Point Result for Generalized Cyclic Contraction Pairs Involving Altering Distance and Control Functions in Partial Metric Spaces

Alemayehu Negash^*, Meaza Bogale

Department of Mathematics, Hampton University, Hampton, USA.

DOI: 10.4236/apm.2025.157024                

Abstract

In this paper, our purpose is to establish a common fixed point result for a pair of self-mappings satisfying some generalized cyclic contraction type conditions involving altering distance and control function with two variables in partial metric spaces. Moreover, we provide an example in support of our main result.

Keywords

Partial Metric, Cyclic Contraction, 0-Completeness, Common Fixed Point

1. Introduction

The Banach contraction principle, a cornerstone in fixed point theory, has been extensively generalized and applied across various branches of mathematics. Among these generalizations, the concept of cyclic contractions, introduced by Kirk et al. [1], has received significant attention. This framework, which allows for mappings defined over cyclically ordered subsets, differs from classical contractions by not requiring continuity. Numerous fixed point results in this setting have followed, see [2]-[9].

Parallel to these developments, partial metric spaces, introduced by Matthews [10] in the context of denotational semantics of computation, have emerged as a powerful generalization of metric spaces. In partial metric spaces, the self-distance of a point need not be zero, a feature well-suited for modeling in computer science and domain theory. Matthews also established a version of the Banach contraction principle in this framework and introduced a class of open p-balls generating a $T_0$ topology. Since then, numerous fixed point theorems have been obtained in this setting, see for examples [11]-[25].

The integration of cyclic contractions with partial metric spaces has further enriched the theory, yielding fixed point results under weaker or generalized contractive conditions [26]-[31].

A notable direction in this context is the use of altering distance functions, introduced by Khan et al. [32], which provide a flexible framework for defining contractions using control functions. These functions have been extensively utilized to establish fixed point theorems in both metric and generalized spaces [33]-[37].

Building upon these foundational ideas, recent works have explored fixed point results for cyclic mappings involving generalized control functions with two variables, particularly in 0-complete partial metric spaces [5] [38]-[48]. These results not only unify but also extend several classical and contemporary fixed point theorems.

In this paper, we contribute to this line of research by establishing a fixed point theorem for mappings satisfying cyclic weaker-type contraction conditions involving a two-variable control function in 0-complete partial metric spaces. We present to demonstrate the applicability and novelty of our results.

2. Preliminaries

In this section, we begin with some basic facts and properties of partial metric spaces.

Definition 2.1 ([10]). A partial metric on a nonempty set $X$ is a function $p:X\times X\to {ℝ}^{+}$ satisfying for all $x,y,z\in X$ :

(p1) $p\left(x,x\right)=p\left(y,y\right)=p\left(x,y\right)\iff x=y$ ;

(p2) $p\left(x,x\right)\le p\left(x,y\right)$ ;

(p3) $p\left(x,y\right)=p\left(y,x\right)$ ;

(p4) $p\left(x,y\right)\le p\left(x,z\right)+p\left(z,y\right)-p\left(z,z\right)$ .

The pair $\left(X,p\right)$ is called a partial metric space.

Remark 2.2. If $p\left(x,y\right)=0$ , then $x=y$ . However, $x=y$ does not necessarily imply $p\left(x,y\right)=0$ .

Example 2.3. ([10]) Let $X=\left[0,\infty \right)$ and $p\left(x,y\right)=\max \left\{x,y\right\}$ . Then $\left(X,p\right)$ is a partial metric space.

Example 2.4. ([10]) Let $X=\left\{\left[a,b\right]:a,b\in ℝ,a\le b\right\}$ and $p\left(\left[a,b\right],\left[c,d\right]\right)=\max \left\{b,d\right\}-\min \left\{a,c\right\}$ . Then $\left(X,p\right)$ is a partial metric space.

Remark 2.5. [29] Each partial metric $p$ on $X$ generates a $T_0$ topology ${\tau }_p$ on $X$ which has as a base the family of open $p$ -balls $\left\{B_p\left(x,ϵ\right):x\in X,ϵ>0\right\}$ , where $B_p\left(x,ϵ\right)=\left\{y\in X:p\left(x,y\right)<p\left(x,x\right)+ϵ\right\}$ for all $x\in X$ and $ϵ>0$ . If $U\in {\tau }_p$ and $x\in U$ , there exists $r>0$ such that $B_p\left(x,r\right)\subseteq U$ .

Remark 2.6. [29] A sequence $\left(x_n\right)$ converges to $x$ in ${\tau }_p$ if and only if ${\lim }_{n\to \infty }p\left(x_n,x\right)=p\left(x,x\right)$ .

Definition 2.7 ([10]) Let $\left(X,p\right)$ be a partial metric space.

• A sequence $\left(x_n\right)$ converges to $x$ if ${\lim }_{n\to \infty }p\left(x_n,x\right)=p\left(x,x\right)$ .

• $\left(x_n\right)$ is Cauchy if ${\lim }_{n,m\to \infty }p\left(x_n,x_m\right)$ exists and is finite.

• $\left(X,p\right)$ is complete if every Cauchy sequence converges to some $x\in X$ with $p\left(x,x\right)={\lim }_{n,m\to \infty }p\left(x_n,x_m\right)$ .

Definition 2.8 ([24]).

(a) A sequence $\left(x_n\right)$ is 0-Cauchy if ${\lim }_{n,m\to \infty }p\left(x_n,x_m\right)=0$ .

(b) $\left(X,p\right)$ is 0-complete if every 0-Cauchy sequence converges to $x$ with $p\left(x,x\right)=0$ .

Lemma 2.9. Let $\left(X,p\right)$ be a partial metric space.

(a) [20] [49] If $p\left(x_n,z\right)\to p\left(z,z\right)=0$ , then $p\left(x_n,y\right)\to p\left(z,y\right)$ for all $y\in X$ .

(b) [24] If $\left(X,p\right)$ is complete, then it is 0-complete.

Example 2.10 ([24]). The space $X=\left[0,\infty \right)\cap \mathbb{Q}$ , equipped with the partial metric $p\left(x,y\right)=\max \left\{x,y\right\}$ , is 0-complete but not complete. The constant sequence $x_n=1$ is Cauchy but not 0-Cauchy.

Remark 2.11. [29] Every closed subset of a 0-complete partial metric space is 0-complete.

Definition 2.12 ([1]) Let $X$ be a nonempty set, $q\in \mathbb{N}$ , and $f:X\to X$ . A cyclic representation of $X$ w.r.t. $f$ is $X={\displaystyle {\cup }_{i=1}^q{A}_i}$ , where:

• $A_i$ are nonempty subsets of $X$ ,

• $f\left(A_1\right)\subseteq A_2,\cdots ,f\left(A_q\right)\subseteq A_1$ .

Definition 2.13 ([50] [51]).

• A coincidence point of $T$ and $S$ is $x\in X$ such that $Tx=Sx$ .

• $y=Tx=Sx$ is called a point of coincidence.

Definition 2.14 ([20] [51]). Mappings $T,S:X\to X$ are weakly compatible if $T\left(Sx\right)=S\left(Tx\right)$ whenever $Sx=Tx$ .

Proposition 2.15 ([50] [51]). If $T$ and $S$ are weakly compatible and have a unique point of coincidence $y$ , then $y$ is their unique common fixed point.

Definition 2.16 ([32]). A function $\gamma :\left[0,\infty \right)\to \left[0,\infty \right)$ is an altering distance function if:

• $\gamma$ is continuous and nondecreasing,

• $\gamma \left(t\right)=0\iff t=0$ .

We denote the set of altering distance functions by $\text{Γ}$ .

Definition 2.17 ([8]). Let $\left(X,d\right)$ be a metric space. An operator $f:X\to X$ is a cyclic weaker $\psi$ -contraction if:

1) $X={\displaystyle {\cup }_{i=1}^m{A}_i}$ is a cyclic representation w.r.t. $f$ ,

2) There exists a continuous, nondecreasing $\psi :\left[0,1\right]\to \left[0,1\right)$ with $\psi \left(t\right)>0$ for $t>0$ and $\psi \left(0\right)=0$ , such that

$d\left(fx,fy\right)\le d\left(x,y\right)-\psi \left(d\left(x,y\right)\right)\forall x\in A_i,y\in A_{i+1}.$

Theorem 2.18 ([8]). Every cyclic weaker $\psi$ -contraction on a complete metric space has a fixed point in ${\displaystyle {\cap }_{i=1}^m{A}_i}$ .

Definition 2.19 ([35]). A mapping $T:X\to X$ is a weak C-contraction if

$d\left(Tx,Ty\right)\le \frac{1}{2}\left[d\left(x,Ty\right)+d\left(y,Tx\right)\right]-\psi \left(d\left(x,Ty\right),d\left(y,Tx\right)\right),$

where $\psi :{\left[0,\infty \right)}^2\to \left[0,\infty \right)$ is continuous and $\psi \left(x,y\right)=0\iff x=y=0$ .

Theorem 2.20 ([35]). Every weak C-contraction on a complete metric space has a unique fixed point.

Example 2.21 ([24]). $X=\left[0,\infty \right)\cap \mathbb{Q}$ with $p\left(x,y\right)=\max \left\{x,y\right\}$ is 0-complete but not complete.

Definition 2.22 ([21]) Let $\left(X,p\right)$ be a PMS, $C\subset X$ and $\phi :C\to {ℝ}^{+}$ a function on $C$ . Then, the function $\phi$ is called lower semi-continuous (l.s.c.) on $C$ whenever

$\underset{n\to \infty }{\lim }p\left(x_n,x\right)=p\left(x,x\right)\Rightarrow \phi \left(x\right)\le \underset{n\to \infty }{\lim \inf }\phi \left(x_n\right)=\underset{n\ge 1}{\sup }\underset{m\ge n}{\inf }\phi \left(x_m\right).$

In 2013, Nashine et al. [47] introduced a class of generalized control functions as follows:

Let $\Psi$ denote the class of all functions $\psi :{\left[0,\infty \right)}^2\to \left[0,\infty \right)$ satisfying the following conditions:

(a) $\psi$ is lower semicontinuous;

(b) $\psi \left(s,t\right)=0$ if and only if $s=t=0$ .

In 2021, Mohanta and Patra [29] established the following coincidence point and common fixed point result for a pair of self-mappings satisfying some generalized cyclic contraction type conditions involving a control function with two variables in partial metric spaces

Theorem 2.22. Let $\left(X,p\right)$ be a 0-complete partial metric space, $q\in \mathbb{N}$ and $A_1,A_2,\cdots ,A_q$ be nonempty subsets of $X$ . Suppose the mappings $T,f:X\to X$ are such that $f\left(A_1\right),f\left(A_2\right),\cdots ,f\left(A_q\right)$ are closed subsets of $\left(X,p\right)$ and satisfy the following conditions:

(C1) $T\left(A_i\right)\subseteq f\left(A_{i+1}\right)$ for $i=1,2,\cdots ,q$ , where $A_{q+1}=A_1$ ;

(C2) there exists $\psi \in \Psi$ such that

$p\left(Tx,Ty\right)\le M\left(fx,fy\right)-\psi \left(p\left(fx,fy\right),p\left(fx,Tx\right)\right)$

for any $\left(fx,fy\right)\in f\left(A_i\right)\times f\left(A_{i+1}\right)$ , $i=1,2,\cdots ,q$ with $A_{q+1}=A_1$ , where

$M\left(fx,fy\right)=\text{max}\left\{p\left(fx,fy\right),p\left(fx,Tx\right),p\left(fy,Ty\right),\frac{p\left(fx,Ty\right)+p\left(Tx,fy\right)}{2}\right\}.$ (2.1)

Then $T$ and $f$ have a unique point of coincidence $u$ in ${\displaystyle {\cap }_{i=1}^{q}f\left(A_i\right)}$ with $p\left(u,u\right)=0$ . Moreover, if $T$ and $f$ are weakly compatible, then $T$ and $f$ have a unique common fixed point in ${\displaystyle {\cap }_{i=1}^{q}f\left(A_i\right)}$ .

In the next section, we prove a coincidence point and common fixed point theorem for a pair of self-mappings on a 0-complete partial metric space, under a generalized contractive condition involving an altering distance function and a two-variable control function. This result generalizes Theorem 3.1 of [29].

3. Main Results

Theorem 3.1. Let $\left(X,p\right)$ be a 0-complete partial metric space, and let $q\in \mathbb{N}$ . Suppose $A_1,A_2,\cdots ,A_q$ are nonempty subsets of $X$ , and let $T,f:X\to X$ be two mappings such that the images $f\left(A_1\right),f\left(A_2\right),\cdots ,f\left(A_q\right)$ are closed subsets of $\left(X,p\right)$ . Assume the following conditions are satisfied:

(C1) $T\left(A_i\right)\subseteq f\left(A_{i+1}\right)$ for all $i=1,2,\cdots ,q$ , where $A_{q+1}:=A_1$ .

(C2) There exist functions $\psi \in \Psi$ and $\gamma \in \text{Γ}$ such that for all $\left(fx,fy\right)\in f\left(A_i\right)\times f\left(A_{i+1}\right)$ , $i=1,2,\cdots ,q$ with $A_{q+1}=A_1$ , the following inequality holds:

$\gamma \left(p\left(Tx,Ty\right)\right)\le \gamma \left(M\left(fx,fy\right)\right)-\psi \left(\gamma \left(p\left(fx,fy\right)\right),\gamma \left(p\left(fx,Tx\right)\right)\right),$

where

$M\left(fx,fy\right)=\text{max}\left\{p\left(fx,fy\right),p\left(fx,Tx\right),p\left(fy,Ty\right),\frac{p\left(fx,Ty\right)+p\left(Tx,fy\right)}{2}\right\}.$ (3.1)

Then $T$ and $f$ have a unique point of coincidence $u\in {\displaystyle {\cap }_{i=1}^{q}f\left(A_i\right)}$ with $p\left(u,u\right)=0$ . Furthermore, if $T$ and $f$ are weakly compatible, then $u$ is their unique common fixed point in ${\displaystyle {\cap }_{i=1}^{q}f\left(A_i\right)}$ .

Proof. Let $Y={\displaystyle {\cup }_{i=1}^q{A}_i}$ , and let $x_0\in Y$ be arbitrary. Then there exists $i_0\in \left\{1,2,\cdots ,q\right\}$ such that $x_0\in A_{i_0}$ . Since $T\left(A_{i_0}\right)\subseteq f\left(A_{i_0+1}\right)$ , there exists $x_1\in A_{i_0+1}$ such that $u_1=T{x}_0$ . Continuing this process, we construct a sequence $\left(x_n\right)$ with $u_n=T{x}_{n-1}$ , for $n=1,2,3,\cdots$ , where $x_n\in A_{i_0+n}$ and $A_{q+k}:=A_k$ . Define $u_n:=f\left(x_n\right)$ . Then $u_n\in f\left(A_{j_n}\right)$ , and $T\left(x_n\right)=u_{n+1}$ . If $p\left(u_n,u_{n+1}\right)=0$ for some $n\in \mathbb{N}$ , then $u_n=u_{n+1}=T{x}_n$ , so $u_{n+1}$ is a point of coincidence of $T$ and $f$ . Assume $p\left(u_n,u_{n+1}\right)>0$ for all $n\in \mathbb{N}$ . Since $\psi \left(s,t\right)>0$ for $s+t>0$ and $\gamma \left(t\right)>0$ for $t>0$ , we have:

$\psi \left(\gamma \left(p\left(u_n,u_{n+1}\right)\right),\gamma \left(p\left(u_n,u_{n+1}\right)\right)\right)>0,\forall n\in \mathbb{N}.$ (3.2)

For each $n\in \mathbb{N}$ , there exists $i\in \left\{1,2,\cdots ,q\right\}$ such that $\left(x_n,x_{n+1}\right)\in A_i\times A_{i+1}$ , so $\left(u_n,u_{n+1}\right)\in f\left(A_i\right)\times f\left(A_{i+1}\right)$ . Applying (C2):

$\begin{array}{c}\gamma \left(p\left(u_{n+1},u_{n+2}\right)\right)=\gamma \left(p\left(T{x}_n,T{x}_{n+1}\right)\right)\\ \le \gamma \left(M\left(u_n,u_{n+1}\right)\right)-\psi \left(\gamma \left(p\left(u_n,u_{n+1}\right)\right),\gamma \left(p\left(u_n,T{x}_n\right)\right)\right)\\ =\gamma \left(M\left(u_n,u_{n+1}\right)\right)-\psi \left(\gamma \left(p\left(u_n,u_{n+1}\right)\right),\gamma \left(p\left(u_n,u_{n+1}\right)\right)\right),\end{array}$ (3.3)

where

$\begin{array}{l}M\left(u_n,u_{n+1}\right)\\ =\text{max}\left\{p\left(u_n,u_{n+1}\right),p\left(u_n,T{x}_n\right),p\left(u_{n+1},T{x}_{n+1}\right),\frac{p\left(u_n,T{x}_{n+1}\right)+p\left(T{x}_n,u_{n+1}\right)}{2}\right\}\\ \le \text{max}\left\{p\left(u_n,u_{n+1}\right),p\left(u_{n+1},u_{n+2}\right),\frac{p\left(u_n,u_{n+1}\right)+p\left(u_{n+1},u_{n+2}\right)}{2}\right\}\\ \le \text{max}\left\{p\left(u_n,u_{n+1}\right),p\left(u_{n+1},u_{n+2}\right)\right\}.\end{array}$

So,

$M\left(u_n,u_{n+1}\right)\le \text{max}\left\{p\left(u_n,u_{n+1}\right),p\left(u_{n+1},u_{n+2}\right)\right\}.$

Since $\gamma$ is non-decreasing, it preserves inequalities and commutes with the maximum operator due to its monotonicity. Thus, we obtain:

$\begin{array}{c}\gamma \left(M\left(u_n,u_{n+1}\right)\right)\le \gamma \left(\text{max}\left\{p\left(u_n,u_{n+1}\right),p\left(u_{n+1},u_{n+2}\right)\right\}\right)\\ =\text{max}\left\{\gamma \left(p\left(u_n,u_{n+1}\right)\right),\gamma \left(p\left(u_{n+1},u_{n+2}\right)\right)\right\}.\end{array}$ (3.4)

Let $a_n:=\gamma \left(p\left(u_n,u_{n+1}\right)\right)$ . Then from (3.3) using (3.4), we obtain:

$a_{n+1}\le \text{max}\left\{a_n,a_{n+1}\right\}-\psi \left(a_n,a_n\right).$ (3.5)

If $\text{max}\left\{a_n,a_{n+1}\right\}=a_{n+1}$ , then:

$a_{n+1}\le a_{n+1}-\psi \left(a_n,a_n\right)<a_{n+1},$

a contradiction. Thus $\text{max}\left\{a_n,a_{n+1}\right\}=a_n$ , and (3.5) becomes:

$a_{n+1}\le a_n-\psi \left(a_n,a_n\right)<a_n.$ (3.6)

Hence, the sequence $\left\{a_n\right\}$ is decreasing and bounded below by zero, and thus converges to some limit $L\ge 0$ . Taking the limit in (3.6) and using the continuity of $\psi$ , we obtain:

$L\le L-\underset{n\to \infty }{\text{lim}}\psi \left(a_n,a_n\right)\le L-\psi \left(L,L\right),$

which implies $\psi \left(L,L\right)=0$ , and hence $L=0$ . Therefore,

$\underset{n\to \infty }{\lim }\gamma \left(p\left(u_n,u_{n+1}\right)\right)=0.$ (3.7)

Since $\gamma$ is continuous and $\gamma \left(t\right)=0$ if and only if $t=0$ , it follows from (3.7) that

$\underset{n\to \infty }{\text{lim}}p\left(u_n,u_{n+1}\right)=0.$ (3.8)

To show that the sequence $\left(u_n\right)$ is 0-Cauchy, suppose the contrary. Then there exists $ϵ>0$ and subsequences $\left(u_{m_i}\right)$ and $\left(u_{n_i}\right)$ with $n_i>m_i>i$ such that:

$p\left(u_{m_i},u_{n_i}\right)\ge ϵ\text{and}p\left(u_{m_i},u_{n_i-1}\right)<ϵ.$ (3.9)

Using conditions (3.9), and property (p4), we obtain:

$\begin{array}{c}ϵ\le p\left(u_{m_i},u_{n_i}\right)\\ \le p\left(u_{m_i},u_{n_i-1}\right)+p\left(u_{n_i-1},u_{n_i}\right)-p\left(u_{n_i-1},u_{n_i-1}\right)\\ <ϵ+p\left(u_{n_i-1},u_{n_i}\right),\end{array}$

which implies:

$ϵ\le p\left(u_{m_i},u_{n_i}\right)<ϵ+p\left(u_{n_i-1},u_{n_i}\right).$

Taking the limit as $i\to \infty$ and using condition (3.8), we get:

$\underset{i\to \infty }{\text{lim}}p\left(u_{m_i},u_{n_i}\right)=ϵ.$ (3.10)

Note that for each $i$ , there exists $r_i\in \left\{1,2,\cdots ,q\right\}$ such that $n_i-m_i+r_i\equiv 1\left(\mathrm{mod}q\right)$ . Hence, for large $i$ , $x_{m_i-r_i}$ and $x_{n_i}$ lie in different, consecutively indexed sets $A_j$ and $A_{j+1}$ (modulo $q$ ). Thus,

$\left(u_{m_i-r_i},u_{n_i}\right)\in f\left(A_j\right)\times f\left(A_{j+1}\right).$

Using condition (C2), we get:

$\begin{array}{l}\gamma \left(p\left(u_{m_i-r_i+1},u_{n_i+1}\right)\right)=\gamma \left(p\left(T{x}_{m_i-r_i},T{x}_{n_i}\right)\right)\\ \le \gamma \left(M\left(u_{m_i-r_i},u_{n_i}\right)\right)-\psi \left(\gamma \left(p\left(u_{m_i-r_i},u_{n_i}\right)\right),\gamma \left(p\left(u_{m_i-r_i},T{x}_{m_i-r_i}\right)\right)\right),\end{array}$ (3.11)

where

$\begin{array}{c}M\left(u_{m_i-r_i},u_{n_i}\right)=\text{max}\{p\left(u_{m_i-r_i},u_{n_i}\right),p\left(u_{m_i-r_i},T{x}_{m_i-r_i}\right),p\left(u_{n_i},T{x}_{n_i}\right),\begin{array}{c}\\ \stackrel{}{}\end{array}\\ \frac{p\left(u_{m_i-r_i},T{x}_{n_i}\right)+p\left(T{x}_{m_i-r_i},u_{n_i}\right)}{2}\}.\end{array}$(3.12)

We now show:

$\underset{i\to \infty }{\text{lim}}p\left(u_{m_i-r_i},u_{m_i}\right)=0.$ (3.13)

By repeated use of (p4), we have:

$p\left(u_{m_i-r_i},u_{m_i}\right)\le {\displaystyle \sum }_{l=0}^{r_i-1}p\left(u_{m_i-r_i+l},u_{m_i-r_i+l+1}\right)\le {\displaystyle \sum }_{l=0}^{q-1}p\left(u_{m_i-r_i+l},u_{m_i-r_i+l+1}\right).$

Taking the limit and using (3.8), we obtain (3.13). From (p4), we also have:

$\begin{array}{c}p\left(u_{m_i-r_i},u_{n_i}\right)\le p\left(u_{m_i-r_i},u_{m_i}\right)+p\left(u_{m_i},u_{n_i}\right)-p\left(u_{m_i},u_{m_i}\right)\\ \le p\left(u_{m_i-r_i},u_{m_i}\right)+p\left(u_{m_i},u_{n_i}\right).\end{array}$

Hence,

$\underset{i\to \infty }{\lim \sup }p\left(u_{m_i-r_i},u_{n_i}\right)\le ϵ.$ (3.14)

Also,

$ϵ\le p\left(u_{m_i},u_{n_i}\right)\le p\left(u_{n_i},u_{m_i-r_i}\right)+p\left(u_{m_i-r_i},u_{m_i}\right)-p\left(u_{m_i-r_i},u_{m_i-r_i}\right),$

which implies

$ϵ\le \underset{i\to \infty }{\lim \sup }p\left(u_{m_i-r_i},u_{n_i}\right)\le ϵ,$

and hence

$\underset{i\to \infty }{\text{lim}}p\left(u_{m_i-r_i},u_{n_i}\right)=ϵ.$ (3.15)

Using similar arguments and (3.8), we also derive:

$\underset{i\to \infty }{\text{lim}}p\left(u_{n_i+1},u_{m_i-r_i}\right)=ϵ,$ (3.16)

$\underset{i\to \infty }{\text{lim}}p\left(u_{n_i},u_{m_i-r_i+1}\right)=ϵ,$ (3.17)

$\underset{i\to \infty }{\text{lim}}p\left(u_{n_i+1},u_{m_i-r_i+1}\right)=ϵ.$ (3.18)

From (3.12) and the limits above, we have:

$\underset{i\to \infty }{\text{lim}}M\left(u_{m_i-r_i},u_{n_i}\right)=ϵ.$ (3.19)

Now take the limit in (3.1), and apply the continuity of $\gamma$ and the lower semicontinuity of $\psi$ , we get:

$\gamma \left(ϵ\right)\le \gamma \left(ϵ\right)-\psi \left(\gamma \left(ϵ\right),0\right),$

which implies:

$\psi \left(\gamma \left(ϵ\right),0\right)\le 0.$

But by assumption, $\psi \left(s,t\right)>0$ for $s+t>0$ , so:

$\gamma \left(ϵ\right)=0\Rightarrow ϵ=0,$

a contradiction. Hence, $\left(u_n\right)$ is a 0-Cauchy sequence in $f\left(Y\right)$ . Since $f\left(Y\right)={\displaystyle {\cup }_{i=1}^{q}f\left(A_i\right)}$ and $f\left(Y\right)$ is closed in the 0-complete space $\left(X,p\right)$ , it follows that $f\left(Y\right)$ is 0-complete. So, $\left(u_n\right)$ converges to a point $u\in f\left(Y\right)$ such that:

$\underset{n\to \infty }{\text{lim}}f\left(g{x}_n,u\right)=p\left(u,u\right)=0.$ (3.20)

We now prove:

$u\in {\displaystyle \underset{i=1}{\overset{q}{\cap }}f}\left(A_i\right).$ (3.21)

As $x_0\in A_{i_0}$ , by (C1), the sequence ${(u_{nq})}_{n\ge 0}\subseteq f\left(A_{i_0}\right)$ . Since $f\left(A_{i_0}\right)$ is closed, by (3.20), $u\in f\left(A_{i_0}\right)$ . By (C1), ${\left(u_{nq+1}\right)}_{n\ge 0}\subseteq f\left(A_{i_0+1}\right)$ . Repeating this for $q$ steps (modulo $q$ ), we obtain:

$u\in f\left(A_{i_0}\right)\cap f\left(A_{i_0+1}\right)\cap \cdots \cap f\left(A_{i_0+q}\right)={\displaystyle \underset{i=1}{\overset{q}{\cap }}f}\left(A_i\right).$

Now we show that $u$ is a point of coincidence of $T$ and $f$ . Since $u\in f\left(Y\right)$ , there exists $t\in Y$ such that $u=f\left(t\right)$ . Let $x_n\in A_i$ for some $i\in \left\{1,2,\cdots ,q\right\}$ . Then, since $u\in {\displaystyle {\cap }_{i=1}^{q}f}\left(A_i\right)$ , we have

$\left(f\left(t\right),u_n\right)=\left(u,u_n\right)\in f\left(A_{i-1}\right)\times f\left(A_i\right),$

where $A_0:=A_q$ . Applying condition (C2), we get an inequality involving the mappings. Now choose $z\in X$ such that $f\left(z\right)=u$ . We claim that $T\left(z\right)=u$ . For each $n\in \mathbb{N}$ , apply condition (C2) with $x=z$ and $y=x_n$ :

$\begin{array}{c}\gamma \left(p\left(Tz,u_{n+1}\right)\right)=\gamma \left(p\left(Tz,T{x}_n\right)\right)\\ \le \gamma \left(M\left(fz,f{x}_n\right)\right)-\psi \left(\gamma \left(p\left(fz,f{x}_n\right)\right),\gamma \left(p\left(fz,Tz\right)\right)\right)\\ =\gamma \left(M\left(u,u_n\right)\right)-\psi \left(\gamma \left(p\left(u,u_n\right)\right),\gamma \left(p\left(u,Tz\right)\right)\right).\end{array}$ (3.22)

As $n\to \infty$ , we have $p\left(u,u_n\right)\to 0$ , so by continuity of $\gamma$ ,

$\gamma \left(p\left(u,u_n\right)\right)\to 0\text{and}\gamma \left(M\left(u,u_n\right)\right)\to \gamma \left(p\left(u,fz\right)\right)=\gamma \left(p\left(u,u\right)\right)=0.$

Taking the upper limit as $n\to \infty$ in inequality (3.22), and using Lemma 2.9 and the lower semicontinuity of $\gamma$ , we obtain:

$\gamma \left(p\left(Tz,u\right)\right)\le \gamma \left(p\left(u,Tz\right)\right)-\psi \left(0,\gamma \left(p\left(u,Tz\right)\right)\right).$

Suppose $p\left(Tz,u\right)>0$ . Then $\gamma \left(p\left(u,Tz\right)\right)>0$ , and by the assumption on $\psi$ , we have $\psi \left(0,\gamma \left(p\left(u,Tz\right)\right)\right)>0$ . This leads to:

$\gamma \left(p\left(Tz,u\right)\right)<\gamma \left(p\left(u,Tz\right)\right)=\gamma \left(p\left(Tz,u\right)\right),$

a contradiction. Hence, $p\left(Tz,u\right)=0$ , so $T\left(z\right)=u=f\left(z\right)$ . Therefore, $u$ is a point of coincidence of $T$ and $f$ , with $u\in {\displaystyle {\cap }_{i=1}^{q}f}\left(A_i\right)$ and $p\left(u,u\right)=0$ . To prove uniqueness, assume that there exists another point of coincidence $v\in {\displaystyle {\cap }_{i=1}^{q}f}\left(A_i\right)$ with $p\left(v,v\right)=0$ . Then there exists $w\in X$ such that $v=f\left(w\right)=T\left(w\right)$ . Since both $u\in f\left(A_i\right)$ and $v\in f\left(A_{i+1}\right)$ for some $i$ , applying (C2) yields:

$\begin{array}{c}\gamma \left(p\left(u,v\right)\right)=\gamma \left(p\left(Tz,Tw\right)\right)\\ \le \gamma \left(M\left(fz,fw\right)\right)-\psi \left(\gamma \left(p\left(fz,fw\right)\right),\gamma \left(p\left(fz,Tz\right)\right)\right)\\ =\gamma \left(M\left(u,v\right)\right)-\psi \left(\gamma \left(p\left(u,v\right)\right),\gamma \left(p\left(u,u\right)\right)\right)\\ =\gamma \left(p\left(u,v\right)\right)-\psi \left(\gamma \left(p\left(u,v\right)\right),0\right).\end{array}$

Hence, $\psi \left(\gamma \left(p\left(u,v\right)\right),0\right)\le 0$ , which implies $\psi \left(\gamma \left(p\left(u,v\right)\right),0\right)=0$ . By the properties of $\psi$ , this gives $\gamma \left(p\left(u,v\right)\right)=0$ , and thus $p\left(u,v\right)=0$ , implying $u=v$ . Therefore, $T$ and $f$ have a unique point of coincidence $u\in {\displaystyle {\cap }_{i=1}^{q}f}\left(A_i\right)$ with $p\left(u,u\right)=0$ . Finally, if $T$ and $f$ are weakly compatible, then by Proposition (2.15), they have a unique common fixed point in ${\displaystyle {\cap }_{i=1}^{q}f}\left(A_i\right)$ . □

Remark 3.2. If we take $\gamma \left(t\right)=t$ in Theorem 3.1, we recover Theorem 3.1 of [29]. Moreover, Corollaries 3.2 through 3.7 follow directly as special cases of Theorem (0.23).

Remark 3.3. If we set $g=I$ and $\gamma \left(t\right)=t$ in Theorem 3.1, we obtain Theorem 13 of [47]. Furthermore, as a special case of Corollary 3.6, several classical fixed point results in partial metric spaces can be deduced, including the Matthews version of Banach’s contraction principle [10].

The following example illustrates the importance of using an altering distance function $\gamma$ in fixed point theory within partial metric spaces. Specifically, we show that a fixed point result may fail under a standard contraction but holds when modified with a suitable altering function.

Example 3.4. Let $X=\left[0,1\right]$ be equipped with the partial metric $p\left(x,y\right)=\max \left\{x,y\right\}$ . Define the subsets:

$A_1=\left[0,0.6\right],A_2=\left[0.4,1\right],$

and the mappings $T,f:X\to X$ as follows:

$Tx=\{\begin{array}{ll}0.4+\frac{x}{10},\hfill & x\in A_1,\hfill \\ \frac{x}{15},\hfill & x\in A_2,\hfill \end{array}fx=x.$

Let the altering distance functions be $\gamma \left(t\right)=t^3$ and $\psi \left(s,t\right)=\frac{s+t}{8}$ .

Verification of Conditions. We observe that Condition (C1) (Cyclic Representation) is satisfied. The images of the sets under $T$ satisfy:

$T\left(A_1\right)=\left[0.4,0.46\right]\subseteq A_2=\left[0.4,1\right],$

$T\left(A_2\right)=\left[\frac{0.4}{15},\frac{1}{15}\right]\approx \left[0.0267,0.0667\right]\subseteq A_1=\left[0,0.6\right].$

Next we show that for all $x\in A_1$ , $y\in A_2$ , the contractive condition

$\gamma \left(p\left(Tx,Ty\right)\right)\le \gamma \left(M\left(fx,fy\right)\right)-\psi \left(\gamma \left(p\left(fx,fy\right)\right),\gamma \left(p\left(fx,Tx\right)\right)\right)$ (3.23)

is satisfied, where

$M\left(fx,fy\right)=\text{max}\left\{p\left(fx,fy\right),p\left(fx,Tx\right),p\left(fy,Ty\right),\frac{p\left(fx,Ty\right)+p\left(Tx,fy\right)}{2}\right\}.$

Key Observations.

1) For $x\in A_1$ , $Tx\in \left[0.4,0.46\right]$ ; for $y\in A_2$ , $Ty\in \left[0.0267,0.0667\right]$ .

2) Since $Tx>Ty$ , we have:

$p\left(Tx,Ty\right)=Tx\Rightarrow \gamma \left(p\left(Tx,Ty\right)\right)={\left(Tx\right)}^3.$

3) Lower bound estimate for $M$ :

$M\ge \text{max}\left\{\text{max}\left\{x,Tx\right\},y\right\},\text{as}y>Ty.$

Analytical Verification.

Case 1: If $x\le 0.4+\frac{x}{10}$ (i.e., $x\le \frac{4}{9}\approx 0.4444$ ), then $\text{max}\left\{x,Tx\right\}=Tx$ , so:

$M\ge \text{max}\left\{Tx,y\right\}\text{and}\text{RHS}\ge {\left(\text{max}\left\{Tx,y\right\}\right)}^3-\frac{{\left[p\left(x,y\right)\right]}^3+{\left(Tx\right)}^3}{8}.$

Case 2: If $x>\frac{4}{9}$ , then $\text{max}\left\{x,Tx\right\}=x$ , and:

$M\ge \text{max}\left\{x,y\right\},\text{RHS}\ge {\left(\text{max}\left\{x,y\right\}\right)}^3-\frac{{\left[p\left(x,y\right)\right]}^3+x^3}{8}.$

Case 3: Take critical values $x=y=0.5$ :

$Tx=0.4+0.05=0.45,Ty=\frac{0.5}{15}\approx 0.0333,$

$p\left(Tx,Ty\right)=0.45,\gamma \left(p\left(Tx,Ty\right)\right)={0.45}^3=0.091125,$

$M=\text{max}\left\{0.5,0.5,0.5,0.5\right\}=0.5,\gamma \left(M\right)={0.5}^3=0.125,$

$\psi =\frac{0.125+0.125}{8}=0.03125,\text{RHS}=0.125-0.03125=0.09375,$

$\Rightarrow \gamma \left(p\left(Tx,Ty\right)\right)=0.091125<0.09375\left(\text{inequality satisfied}\right).$

Next we show that the contractive condition (3.23) fails to hold at $x=y=0.6$ with $\gamma \left(t\right)=t$ . Let $x=y=0.6$ . Then:

$Tx=0.4+\frac{0.6}{10}=0.46,Ty=\frac{0.6}{15}=0.04,$

$p\left(Tx,Ty\right)=0.46,\gamma \left(p\left(Tx,Ty\right)\right)=0.46,$

$M=\text{max}\left\{0.6,0.6,0.6,0.6\right\}=0.6,\gamma \left(M\right)=0.6,$

$\psi =\frac{0.6+0.6}{8}=0.15,\text{RHS}=0.6-0.15=0.45,$

$\Rightarrow \gamma \left(p\left(Tx,Ty\right)\right)=0.46>0.45\left(\text{inequality fails}\right).$

Finally, the unique common fixed point of $T$ and $f$ is $z=\frac{4}{9}$ .

4. Conclusions and Open Questions

In this paper, we established a common fixed point theorem for generalized cyclic contraction pairs in 0-complete partial metric spaces, incorporating altering distance functions ($\gamma$ ) and control functions ($\psi$ ). Our results extend and unify several existing theorems, including those of [29] and [47]. The introduction of nonlinear altering distance functions (e.g., $\gamma \left(t\right)=t^3$ ) allows for fixed point results in cases where traditional linear contractions fail, as demonstrated in Example 3.4.

Future research could explore:

1) Weaker Contraction Conditions: Can the assumptions on $\psi$ or $\gamma$ be relaxed?

2) Multivalued Mappings: Do analogous results hold for set-valued cyclic contractions?

This study contributes to the broader landscape of fixed point theory, offering a more flexible framework for analyzing cyclic mappings in generalized metric spaces.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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