Coincidence Theorems under Generalized Nonlinear Relational Contractions

Abstract

After the appearance of relation-theoretic contraction principle due to Alam and Imdad, the domain of fixed point theory applied to relational metric spaces has attracted much attention. Existence and uniqueness of fixed/coincidence points satisfying the different types of contractivity conditions in the framework of relational metric space have been studied in recent times. Such results have the great advantage to solve certain types of matrix equations and boundary value problems for ordinary differential equations, integral equations and fractional differential equations. This article is devoted to proving the coincidence and common fixed point theorems for a pair of mappings $(\mathcal{T},\mathcal{S})$ employing relation-theoretic $(\varphi ,\psi )$-contractions in a metric space equipped with a locally finitely $\mathcal{T}$-transitive relation. Our results improve, modify, enrich and unify several existing coincidence points as well as fixed point results. Several examples are provided to substantiate the utility of our results.

1. Introduction

The concepts of coincidence and common fixed points generalize the idea of fixed points, which are obtained by enhancing the number of involved mappings in the ambient space. Given a self-mapping $\mathcal{T}$ on a nonempty M, “r remains a fixed point of $\mathcal{T}$” which remains equivalent to saying that $\mathcal{T}\left(r\right)=I\left(r\right)$ (where I denotes identity mapping on M). This fact motivates whether the identity mapping can be replaced by another self-mapping $\mathcal{S}$ on M. Thus far, given two self-mappings $\mathcal{T}$ and $\mathcal{S}$ on a nonempty set M, consider the problem regarding finding $r,\overline{r}\in M$, such that

$$\mathcal{T}\left(r\right)=\mathcal{S}\left(r\right)=\overline{r}.$$

(1)

Then,

• r is referred to as a coincidence point of $\mathcal{T}$ and $\mathcal{S}$;
• $\overline{r}$ is referred to as a point of coincidence of $\mathcal{T}$ and $\mathcal{S}$;
• r is referred to as a common fixed point of $\mathcal{T}$ and $\mathcal{S}$ provided $\overline{r}=r$.

Clearly, every common fixed point of $\mathcal{T}$ and $\mathcal{S}$ remains also a coincidence point as well as point of coincidence. It is well known that the coincidence problem (1) is, under appropriate conditions, equivalent to a fixed point problem. Metrical coincidence theorems appeared with the works of Goebel [1] and Jungck [2], wherein they extended BCP (Banach contraction principle) for two mappings. Indeed, Goebel’s coincidence theorem involves the completeness of the range of one of the mappings. On the other hand, Jungck’s common fixed point theorem requires the completeness of whole metric space and commutativity of underlying mappings. There exists a considerable literature on theory of metrical coincidence and common fixed points, but we merely refer to [3,4,5,6,7,8,9,10,11,12] and references therein.

Alam and Imdad [13] presented a natural extension of the BCP in a metric space equipped with an arbitrary relation. For further study on relation-theoretic metric fixed point theory, one refers to [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. Such results involved relation-preserving contractions which are weaker than the usual contractions and are applied to solve the special types of matrix equations and boundary value problems, wherein fixed point results on abstract metric space cannot be applicable. In 2008, Dutta and Choudhury [33] initiated a new contraction condition involving a pair of auxiliary functions often called $(\varphi ,\psi )$-contractions and utilized the same to obtain a novel generalization of BCP. Alam et al. [34] enriched the results of Dutta and Choudhury [33] by enlarging the class of $(\varphi ,\psi )$-contractions. Sk et al. [35] established the relation-theoretic version of the results of Alam et al. [34], which extends the results of Harjani and Sadarangani [36].

The intent of this paper is to extend the recent fixed point theorems of Sk et al. [35] for a pair $(\mathcal{T},\mathcal{S})$ of self-mappings defined in a relational metric space and to prove the results on existence and uniqueness of coincidence and common fixed points satisfying the relation-theoretic $(\varphi ,\psi )$-contractivity conditions. In order to ascertain the existence of coincidence point of a pair $(\mathcal{T},\mathcal{S})$ of self-mappings satisfying linear contractivity condition, the given binary relation remains required to be merely $(\mathcal{T},\mathcal{S})$-closed (c.f. [14]). On the other hand, for $(\varphi ,\psi )$-contractions, transitivity of underlying relation remains additionally required. However, the transitivity condition remains very restrictive. Thus, we employed an optimal condition of transitivity, namely: locally finitely $\mathcal{T}$-transitive relation $\mathfrak{S}$. In the hypotheses of our main results, we supposed that either the entire metric space remains $\mathfrak{S}$-complete along with the $\mathfrak{S}$-compatibility requirement of underlying mappings or the range subspace of either of involved mappings remains $\mathfrak{S}$-complete. Thus far, our newly proved results are motivated by the classical approaches due to Goebel [1] and Jungck [2]. For universal relation, our results reduce to the variants of coincidence theorems of Goebel [1] and Jungck [2] employing $(\varphi ,\psi )$-contractions. In order to exhibit the worth of our results, we provide several illustrative examples.

2. Preliminaries

This section is comprised of relevant notions and essential results related to our results. $\mathbb{N}$ will stand for the set of natural numbers and ${\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$. A binary relation (or simply, a relation) $\mathfrak{S}$ on a set M means any subset of $M^2$. Thus far, in particular, $M^2$ is a subset of itself remains a relation on M, which is called ‘universal relation’.

In what follows, M remains a set, $\varrho$ is a metric on M, $\mathfrak{S}$ is a relation on M and $\mathcal{T},\mathcal{S}:M\to M$ remain two mappings.

Definition 1 

([37] (p. 66)). Inverse of $\mathfrak{S}$ is defined by ${\mathfrak{S}}^{-1}:=\{(r,t)\in M^2:(t,r)\in \mathfrak{S}\}$.

Definition 2 

([37] (p. 72)). Symmetric closure of $\mathfrak{S}$ is defined by ${\mathfrak{S}}^s:=\mathfrak{S}\cup {\mathfrak{S}}^{-1}$.

Definition 3 

([13]). A pair $r,t\in M$ is known as $\mathfrak{S}$-comparative, denoted by $[r,t]\in \mathfrak{S}$, if

$$(r,t)\in \mathfrak{S}or(t,r)\in \mathfrak{S}.$$

Proposition 1 

([13]). $[r,t]\in \mathfrak{S}⟺(r,t)\in {\mathfrak{S}}^s.$

Definition 4 

([38] (p. 114)). A relation on $A\subseteq M$ defined by

$${\mathfrak{S}|}_A:=\mathfrak{S}\cap A^2$$

is called a restriction of $\mathfrak{S}$ on A.

Definition 5 

([14]). $\mathfrak{S}$ is termed as $(\mathcal{T},\mathcal{S})$-closed if it satisfies

$$(\mathcal{T}r,\mathcal{T}t)\in \mathfrak{S},$$

$\forall r,t\in M$ verifying $(\mathcal{S}r,\mathcal{S}t)\in \mathfrak{S}$.

For identity map $\mathcal{S}=I$, Definition 5 reduces to the notion of “$\mathcal{T}$-closedness” (see [13]).

Proposition 2 

([14]). If $\mathfrak{S}$ remains $(\mathcal{T},\mathcal{S})$-closed, then ${\mathfrak{S}}^s$ also remains $(\mathcal{T},\mathcal{S})$-closed.

Definition 6 

([20]). $\mathfrak{S}$ is called $(\mathcal{T},\mathcal{S})$-compatible if it satisfies

$$\mathcal{T}\left(r\right)=\mathcal{T}\left(t\right),$$

$\forall r,t\in M$ verifying $(\mathcal{S}r,\mathcal{S}t)\in \mathfrak{S}$ and $\mathcal{S}\left(r\right)=\mathcal{S}\left(t\right)$.

Definition 7 

([13]). A sequence $\left\{r_n\right\}\subset M$ verifying $(r_n,r_{n+1})\in \mathfrak{S}$∀$n\in {\mathbb{N}}_0$, is called $\mathfrak{S}$-preserving.

Definition 8 

([14]). $(M,\varrho )$ is called $\mathfrak{S}$-complete metric space if every $\mathfrak{S}$-preserving Cauchy sequence in M remains convergent.

Clearly, completeness implies $\mathfrak{S}$-completeness but not conversely.

Definition 9 

([14]). $\mathcal{T}$ is referred to as $\mathfrak{S}$-continuous at $r\in M$ if it satisfies

$$\mathcal{T}\left(r_n\right)\stackrel{\varrho }{⟶}\mathcal{T}\left(r\right),$$

for any $\mathfrak{S}$-preserving sequence $\left\{r_n\right\}\subset M$ with $r_n\stackrel{\varrho }{⟶}r$. Naturally, if a map remains $\mathfrak{S}$-continuous at all points of M, then it is called $\mathfrak{S}$-continuous.

Clearly, continuity implies $\mathfrak{S}$-continuity but not conversely.

Definition 10 

([14]). $\mathcal{T}$ is called $(\mathcal{S},\mathfrak{S})$-continuous at $r\in M$ if it satisfies

$$\mathcal{T}\left(r_n\right)\stackrel{\varrho }{⟶}\mathcal{T}\left(r\right),$$

for any sequence $\left\{r_n\right\}\subset M$, whereas $\left\{\mathcal{S}{r}_n\right\}$ remains $\mathfrak{S}$-preserving satisfying $\mathcal{S}\left(r_n\right)\stackrel{\varrho }{⟶}\mathcal{S}\left(r\right)$. Naturally, if a map remains $(\mathcal{S},\mathfrak{S})$-continuous at all points of M, then it is called $(\mathcal{S},\mathfrak{S})$-continuous.

In particular, if $\mathfrak{S}$ remains a universal relation, then $\mathcal{T}$ is referred to as $\mathcal{S}$-continuous. Clearly, $\mathcal{S}$-continuity implies $(\mathcal{S},\mathfrak{S})$-continuity but not conversely.

Definition 11 

([14]). $\mathcal{T}$ and $\mathcal{S}$ are referred to as $\mathfrak{S}$-compatible if they satisfy

$$\underset{n\to \infty }{lim}\varrho (\mathcal{S}\mathcal{T}{r}_n,\mathcal{T}\mathcal{S}{r}_n)=0,$$

for any sequence $\left\{r_n\right\}\subset M$, whereas $\left\{\mathcal{T}{r}_n\right\}$ and $\left\{\mathcal{S}{r}_n\right\}$ remain $\mathfrak{S}$-preserving and

$$\underset{n\to \infty }{lim}\mathcal{T}\left(r_n\right)=\underset{n\to \infty }{lim}\mathcal{S}\left(r_n\right).$$

In particular, if $\mathfrak{S}$ remains a universal relation, then one says that $\mathcal{T}$ and $\mathcal{S}$ are compatible. Clearly, compatibility implies $\mathfrak{S}$-compatibility but not conversely.

Definition 12 

([3]). $\mathcal{T}$ and $\mathcal{S}$ are known as weakly compatible if

$$\mathcal{T}\left(r\right)=\mathcal{S}\left(r\right)\Rightarrow \mathcal{T}\left(\mathcal{S}r\right)=\mathcal{S}\left(\mathcal{T}r\right),\forall r\in M.$$

Compatibility implies weak compatibility but not conversely.

Definition 13 

([14]). $\mathfrak{S}$ is called $(\mathcal{S},\varrho )$-self-closed if every $\mathfrak{S}$-preserving sequence $\left\{r_n\right\}\subset M$ satisfying $r_n\stackrel{\varrho }{⟶}r$ admits a subsequence $\left\{r_{n_k}\right\}$ verifying

$$[\mathcal{S}{r}_{n_k},\mathcal{S}r]\in \mathfrak{S}.$$

For identity map $\mathcal{S}=I$, Definition 13 reduces to the notion of “$\varrho$-self-closed relation” (see [13]).

Definition 14 

([38] (p. 116)). Given $r,t\in M$, a finite sequence $\{s_0,s_1,\dots ,s_l\}$⊂M is referred to as a path of length $l\in \mathbb{N}$ in $\mathfrak{S}$ from r to t if the following ones hold:

(i)

$s_0=r$ and $s_l=t$,

(ii)

$(s_i,s_{i+1})$ $\in \mathfrak{S},0\le i\le l-1$.

Definition 15 

([16]). A subset $A\subseteq M$ is said to be $\mathfrak{S}$-connected if, between any pair of elements of A, there exists some path in $\mathfrak{S}$.

Definition 16 

([39]). Given $K\in \mathbb{N}$, $K\ge 2$, $\mathfrak{S}$ is called K-transitive if, for any $r_0,r_1,\dots ,r_K\in M$,

$$(r_{j-1},r_j)\in \mathfrak{S},foreachj(1\le j\le K)\Rightarrow (r_0,r_K)\in \mathfrak{S}.$$

Clearly, the concept of 2-transitive relation coincides with that of transitive relation. $\mathfrak{S}$ is called finitely transitive relation if ∃ some $K\ge 2$ such that $\mathfrak{S}$ remains K-transitive (c.f. [40,41]).

Definition 17 

([19]). $\mathfrak{S}$ is known as locally finitely $\mathcal{T}$-transitive if, for every enumerable subset $A\subseteq \mathcal{T}\left(M\right)$, $\exists K=K\left(A\right)\ge 2$, such that ${\mathfrak{S}|}_A$ remains K-transitive.

Lemma 1 

([42]). Assume that $(M,\varrho )$ is a metric space and $\left\{r_n\right\}\subset M$ remains a sequence. If $\left\{r_n\right\}$ remains not Cauchy, then ∃$ϵ>0$ and a pair of subsequences $\left\{r_{n_k}\right\}$ and $\left\{r_{m_k}\right\}$ of $\left\{r_n\right\}$ verifying

(i)

$k\le m_k<n_k,\forall k\in \mathbb{N}$,

(ii)

$\varrho (r_{m_k},r_{n_k})\ge ϵ$,

(iii)

$\varrho (r_{m_k},r_{p_k})<ϵ,\forall p_k\in \{m_k+1,m_k+2,...,n_k-2,n_k-1\}$.

Moreover, if $\left\{r_n\right\}$ satisfies $\underset{n\to \infty }{lim}\varrho (r_n,r_{n+1})=0$, then

$$\underset{k\to \infty }{lim}\varrho (r_{m_k},r_{n_k+p})=ϵ,\forall p\in {\mathbb{N}}_0.$$

Lemma 2 

([40]). Suppose that M remains a nonempty set and $\left\{q_n\right\}\subset M$ remains an $\mathfrak{S}$-preserving sequence. If $\mathfrak{S}$ remains K-transitive on $A=\{q_n:n\in {\mathbb{N}}_0\}$ for some $K\ge 2$, then

$$(q_n,q_{n+1+N(K-1)})\in \mathfrak{S},\forall n,N\in {\mathbb{N}}_0.$$

Lemma 3 

([43]). If f is a self-mapping on a nonempty set M, then there exists a subset E of M such that $f\left(E\right)=f\left(M\right)$ and $f:E\to M$ is one-to-one.

Lemma 4 

([12]). Assume that $\mathcal{T}$ and $\mathcal{S}$ remain two self-mappings on a nonempty set M such that $\mathcal{T}$ and $\mathcal{S}$ admit a unique point of coincidence.

(i)

If $\mathcal{T}$ and $\mathcal{S}$ remain weakly compatible, then the point of coincidence is also a unique common fixed point.

(ii)

If anyone of $\mathcal{T}$ and $\mathcal{S}$ remains one-to-one, then $\mathcal{T}$ and $\mathcal{S}$ admit a unique coincidence point.

Alam et al. [34] introduced the following classes of auxiliary functions.

$$\Phi =\{\varphi :[0,\infty )\to [0,\infty ):\varphi \mathrm{i}\mathrm{s} \mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{m}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{c} \mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\}$$

and

$$\Psi =\{\psi :[0,\infty )\to [0,\infty ):\psi \left(a\right)>0,\forall a>0\mathrm{a}\mathrm{n}\mathrm{d}\underset{a\to l}{lim\; inf}\psi \left(a\right)>0,\forall l>0\}.$$

The following conclusion can be immediate using the symmetry of $\varrho$.

Proposition 3. 

Suppose that $(M,\varrho )$ remains a metric space endowed with a relation $\mathfrak{S}$ while $\mathcal{T}$ and $\mathcal{S}$ a pair of self-mappings on M. If $\varphi \in \Phi$ and $\psi \in \Psi$, then the following contractivity conditions are equivalent:

(I)

$\varphi \left(\varrho \right(\mathcal{T}r,\mathcal{T}t\left)\right)\le \varphi \left(\varrho \right(\mathcal{S}r,\mathcal{S}t\left)\right)-\psi \left(\varrho \right(\mathcal{S}r,\mathcal{S}t\left)\right),\forall r,t\in Mwith(\mathcal{S}r,\mathcal{S}t)\in \mathfrak{S}$,

(II)

$\varphi \left(\varrho \right(\mathcal{T}r,\mathcal{T}t\left)\right)\le \varphi \left(\varrho \right(\mathcal{S}r,\mathcal{S}t\left)\right)-\psi \left(\varrho \right(\mathcal{S}r,\mathcal{S}t\left)\right),\forall r,t\in Mwith[\mathcal{S}r,\mathcal{S}t]\in \mathfrak{S}$.

Proposition 4 

([34]). If there exist $\varphi \in \Phi$ and $\psi \in \Psi$ satisfying

$$\varphi \left(s\right)\le \varphi \left(t\right)-\psi \left(t\right),s\in [0,\infty )andt\in (0,\infty ),$$

then

$$s<t.$$

3. Main Results

Firstly, we present the result on the existence of a coincidence point.

Theorem 1. 

Assume that $(M,\varrho )$ is a metric space, $\mathcal{T}$ and $\mathcal{S}$ remain self-mappings on M while $\mathfrak{S}$ remains a relation on M. In addition,

(a) 

$\mathcal{T}\left(M\right)\subseteq \mathcal{S}\left(M\right)$,

(b) 

$\mathfrak{S}$ remains $(\mathcal{T},\mathcal{S})$-closed and locally finitely $\mathcal{T}$-transitive,

(c) 

$\exists r_0\in M$ verifying $(\mathcal{S}{r}_0,\mathcal{T}{r}_0)\in \mathfrak{S}$,

(d) 

there exist $\varphi \in \Phi$ and $\psi \in \Psi$ verifying

$$\varphi \left(\varrho \right(\mathcal{T}r,\mathcal{T}t\left)\right)\le \varphi \left(\varrho \right(\mathcal{S}r,\mathcal{S}t\left)\right)-\psi \left(\varrho \right(\mathcal{S}r,\mathcal{S}t\left)\right),\forall r,t\in Mwith(\mathcal{S}r,\mathcal{S}t)\in \mathfrak{S},$$

(e) 

$\left(e1\right)$$(M,\varrho )$ remains $\mathfrak{S}$-complete,

$\left(e2\right)$$\mathcal{T}$ and $\mathcal{S}$ remain $\mathfrak{S}$-compatible,

$\left(e3\right)$$\mathcal{S}$ remains $\mathfrak{S}$-continuous,

$\left(e4\right)$$\mathcal{T}$ remains $\mathfrak{S}$-continuous or $\mathfrak{S}$ remains $(\mathcal{T},\mathcal{S})$-compatible as well as $(\mathcal{S},\varrho )$-self-closed, or, alternatively,

(e′) 

$\left(e^{\prime }1\right)$∃ an $\mathfrak{S}$-complete subspace A of M verifying $\mathcal{T}\left(M\right)\subseteq A\subseteq \mathcal{S}\left(M\right)$,

$\left(e^{\prime }2\right)$$\mathcal{T}$ remains $(\mathcal{S},\mathfrak{S})$-continuous or $\mathcal{T}$ and $\mathcal{S}$ remain continuous or $\mathfrak{S}$ and ${\mathfrak{S}|}_A$ remain $(\mathcal{T},\mathcal{S})$-compatible and ϱ-self-closed, respectively.

Then, $\mathcal{T}$ and $\mathcal{S}$ admit a coincidence point.

Proof. 

By assumption $\left(c\right)$, if $\mathcal{S}\left(r_0\right)=\mathcal{T}\left(r_0\right)$, then $r_0$ remains a coincidence point of $\mathcal{T}$ and $\mathcal{S}$ and hence we are finished. Otherwise, if $\mathcal{S}\left(r_0\right)\ne \mathcal{T}\left(r_0\right)$, then owing to $\mathcal{T}\left(M\right)\subseteq \mathcal{S}\left(M\right)$, we can choose $r_1\in M$ satisfying $\mathcal{S}\left(r_1\right)=\mathcal{T}\left(r_0\right)$. Again, using $\mathcal{T}\left(M\right)\subseteq \mathcal{S}\left(M\right)$, we can choose $r_2\in M$ satisfying $\mathcal{S}\left(r_2\right)=\mathcal{T}\left(r_1\right)$. Thus, one can construct a sequence $\left\{r_n\right\}\subset M$ verifying

$$\mathcal{S}\left(r_{n+1}\right)=\mathcal{T}\left(r_n\right),\forall n\in {\mathbb{N}}_0.$$

(2)

Now, by induction, one has to show that $\left\{\mathcal{S}{r}_n\right\}$ remains $\mathfrak{S}$-preserving sequence, $i.e.,$

$$(\mathcal{S}{r}_n,\mathcal{S}{r}_{n+1})\in \mathfrak{S},\forall n\in {\mathbb{N}}_0.$$

(3)

By assumption $\left(c\right)$ and (2) (for $n=0$), one obtains

$$(\mathcal{S}{r}_0,\mathcal{S}{r}_1)\in \mathfrak{S}.$$

Thus, (3) holds for $n=0.$ Assume that (3) holds for $n=k>0$, i.e.,

$$(\mathcal{S}{r}_k,\mathcal{S}{r}_{k+1})\in \mathfrak{S}.$$

By $(\mathcal{T},\mathcal{S})$-closedness of $\mathfrak{S}$, one has

$$(\mathcal{T}{r}_k,\mathcal{T}{r}_{k+1})\in \mathfrak{S},$$

which, using (2), gives rise to

$$(\mathcal{S}{r}_{k+1},\mathcal{S}{r}_{k+2})\in \mathfrak{S}.$$

Therefore, by induction, (3) holds $\forall n\in {\mathbb{N}}_0$.

In light of (2) and (3), $\left\{\mathcal{T}{r}_n\right\}$ remains also $\mathfrak{S}$-preserving so that

$$(\mathcal{T}{r}_n,\mathcal{T}{r}_{n+1})\in \mathfrak{S},\forall n\in {\mathbb{N}}_0.$$

(4)

If $\exists n_0\in {\mathbb{N}}_0$ satisfying $\varrho (\mathcal{S}{r}_{n_0},\mathcal{S}{r}_{n_0+1})=0$, then by (2), one concludes that $r_{n_0}$ remains a coincidence point of $\mathcal{T}$ and $\mathcal{S}$. Otherwise, one has

$${\varrho }_n:=\varrho (\mathcal{S}{r}_n,\mathcal{S}{r}_{n+1})>0,\forall n\in {\mathbb{N}}_0.$$

By (2), (3) and hypothesis $\left(d\right)$, one obtains

$$\begin{array}{c}\hfill \varphi \left(\varrho (\mathcal{S}{r}_{n+1},\mathcal{S}{r}_{n+2})\right)=\varphi \left(\varrho (\mathcal{T}{r}_n,\mathcal{T}{r}_{n+1})\right)\le \varphi \left(\varrho (\mathcal{S}{r}_n,\mathcal{S}{r}_{n+1})\right)-\psi \left(\varrho (\mathcal{S}{r}_n,\mathcal{S}{r}_{n+1})\right)\end{array}$$

so that

$$\varphi \left({\varrho }_{n+1}\right))\le \varphi \left({\varrho }_n\right)-\psi \left({\varrho }_n\right).$$

(5)

Using Proposition 4, one obtains

$$\begin{array}{ccc}\hfill {\varrho }_{n+1}& <& {\varrho }_n,\forall n\in {\mathbb{N}}_0,\hfill \end{array}$$

showing that the sequence $\left\{{\varrho }_n\right\}\subset (0,\infty )$ remains monotonic decreasing. Since $\left\{{\varrho }_n\right\}$ also remains bounded below, $\exists r\ge 0$ verifying

$$\underset{n\to \infty }{lim}{\varrho }_n=r.$$

Now, one has to prove that $r=0$. On the contrary, assume that $r>0$. Letting upper limit in (5), one obtains

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim\; sup}\varphi \left({\varrho }_{n+1}\right)& \le & \underset{n\to \infty }{lim\; sup}\varphi \left({\varrho }_n\right)+\underset{n\to \infty }{lim\; sup}[-\psi \left({\varrho }_n\right)]\hfill \\ \hfill & \le & \underset{n\to \infty }{lim\; sup}\varphi \left({\varrho }_n\right)-\underset{n\to \infty }{lim\; inf}\psi \left({\varrho }_n\right).\hfill \end{array}$$

Making use of right continuity of $\varphi$, one obtains

$$\begin{array}{ccc}\hfill \varphi \left(r\right)& \le & \varphi \left(r\right)-\underset{n\to \infty }{lim\; inf}\psi \left({\varrho }_n\right),\hfill \end{array}$$

yielding thereby,

$$\begin{array}{c}\hfill \underset{{\varrho }_n\to r>0}{lim\; inf}\psi \left({\varrho }_n\right)=\underset{n\to \infty }{lim\; inf}\psi \left({\varrho }_n\right)\le 0,\end{array}$$

which contradicts the property of $\psi$ so that one has

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\varrho }_n=\underset{n\to \infty }{lim}\varrho (\mathcal{S}{r}_n,\mathcal{S}{r}_{n+1})=0.\end{array}$$

(6)

Now, one will show that $\left\{\mathcal{S}{r}_n\right\}$ remains Cauchy. Let $\left\{\mathcal{S}{r}_n\right\}$ be not Cauchy. Using Lemma 1, ∃$ϵ>0$ and a pair of subsequences $\left\{\mathcal{S}{r}_{n_k}\right\}$ and $\left\{\mathcal{S}{r}_{m_k}\right\}$ of $\left\{\mathcal{S}{r}_n\right\}$ verifying $k\le m_k<n_k$, $\varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{n_k})\ge ϵ$ and $\varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{p_k})<ϵ$ where $p_k\in \{m_k+1,m_k+2,...,n_k-2,n_k-1\}$. In addition, in light of (6), one has

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{n_k+p})=ϵ,\forall p\in {\mathbb{N}}_0.\end{array}$$

(7)

Here, the range $D=\{\mathcal{S}{r}_n:n\in {\mathbb{N}}_0\}\subset \mathcal{T}\left(M\right)$ remains enumerable. Therefore, by locally finitely $\mathcal{T}$-transitivity of $\mathfrak{S}$, ∃$K=K\left(D\right)\ge 2$, such that ${\mathfrak{S}|}_D$ remains K-transitive. Applying division algorithm for $n_k-m_k>0$ and $K-1>0$, we obtain

$$\begin{gathered} n_k-m_k=(K-1)({\mu }_k-1)+(K-{\eta }_k), \\ {\mu }_k-1\ge 0,0\le K-{\eta }_k<K-1 \end{gathered}$$

$$\iff \left\{\begin{array}{cc}{n}_k+{\eta }_k=m_k+1+(K-1){\mu }_k,\hfill & \\ {\mu }_k\ge 1,1<{\eta }_k\le K.\hfill \end{array}\right.$$

Herein, ${\mu }_k$ and ${\eta }_k$ remain certain natural numbers such that ${\eta }_k\in (1,K]$ is chosen as a finite natural number. Therefore, one can choose subsequences $\left\{\mathcal{S}{r}_{n_k}\right\}$ and $\left\{\mathcal{S}{r}_{m_k}\right\}$ of $\left\{\mathcal{S}{r}_n\right\}$(satisfying (6)) such that ${\eta }_k=\eta$ remains constant. Now, one has

$$m_k^{\prime }=n_k+\eta =m_k+1+(K-1){\mu }_k.$$

(8)

By (7) and (8), one obtains

$$\underset{k\to \infty }{lim}\varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{m_k^{\prime }})=\underset{k\to \infty }{lim}\varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{n_k+\eta })=ϵ.$$

(9)

Using triangular inequality, one has

$$\varrho (\mathcal{S}{r}_{m_k+1},\mathcal{S}{r}_{m_k^{\prime }+1})\le \varrho (\mathcal{S}{r}_{m_k+1},\mathcal{S}{r}_{m_k})+\varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{m_k^{\prime }})+\varrho (\mathcal{S}{r}_{m_k^{\prime }},\mathcal{S}{r}_{m_k^{\prime }+1})$$

(10)

and

$$\varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{m_k^{\prime }})\le \varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{m_k+1})+\varrho (\mathcal{S}{r}_{m_k+1},\mathcal{S}{r}_{m_k^{\prime }+1})+\varrho (\mathcal{S}{r}_{m_k^{\prime }+1},\mathcal{S}{r}_{m_k^{\prime }})$$

or

$$\begin{array}{ccc}\hfill \varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{m_k^{\prime }})-\varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{m_k+1})-\varrho (\mathcal{S}{r}_{m_k^{\prime }+1},\mathcal{S}{r}_{m_k^{\prime }})& \le & \varrho (\mathcal{S}{r}_{m_k+1},\mathcal{S}{r}_{m_k^{\prime }+1}).\hfill \end{array}$$

(11)

Taking $k\to \infty$ in (10) and (11) and using (6) and (9), one obtains

$$\underset{k\to \infty }{lim}\varrho (\mathcal{S}{r}_{m_k+1},\mathcal{S}{r}_{m_k^{\prime }+1})=ϵ.$$

(12)

In light of (8) and Lemma 2, one has $\varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{m_k^{\prime }})\in \mathcal{R}$. By assumption $\left(d\right)$, one has

$$\begin{array}{ccc}\hfill \varphi \left(\varrho (\mathcal{S}{r}_{m_k+1},\mathcal{S}{r}_{m_k^{\prime }+1})\right)& =& \varphi \left(\varrho (T\mathcal{S}{r}_{m_k},T\mathcal{S}{r}_{m_k^{\prime }})\right)\hfill \\ \hfill & \le & \varphi \left(\varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{m_k^{\prime }})\right)-\psi \left(\varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{m_k^{\prime }})\right),\hfill \end{array}$$

yielding thereby

$$\begin{array}{ccc}\hfill \underset{k\to \infty }{lim\; sup}\varphi \left(\varrho (\mathcal{S}{r}_{m_k+1},\mathcal{S}{r}_{m_k^{\prime }+1})\right)& \le & \underset{k\to \infty }{lim\; sup}\varphi \left(\varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{m_k^{\prime }})\right)+\underset{k\to \infty }{lim\; sup}[-\psi \left(\varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{m_k^{\prime }})\right)].\hfill \end{array}$$

By right continuity of $\varphi$ and (9), one obtains

$$\begin{array}{ccc}\hfill \varphi \left(ϵ\right)& \le & \varphi (ϵ)-\underset{k\to \infty }{lim\; inf}\psi (\varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{m_k^{\prime }})),\hfill \end{array}$$

implying thereby

$$\begin{array}{ccc}\hfill \underset{k\to \infty }{lim\; inf}\psi (\varrho (\mathcal{S}{r}_{m_k},\mathcal{S}{r}_{m_k^{\prime }}))& \le & 0,\hfill \end{array}$$

which remains a contradiction so that $\left\{\mathcal{S}{r}_n\right\}$ remains Cauchy.

Finally, one has to use assumptions $\left(e\right)$ and $\left(e^{\prime }\right)$. Let $\left(e\right)$ hold. As $\left\{\mathcal{S}{r}_n\right\}$ remains a $\mathfrak{S}$-preserving Cauchy sequence in M and M remains $\mathfrak{S}$-complete, $\exists p\in M$ verifying

$$\underset{n\to \infty }{lim}\mathcal{S}\left(r_n\right)=p.$$

(13)

By (2) and (13), one obtains

$$\underset{n\to \infty }{lim}\mathcal{T}\left(r_n\right)=p.$$

(14)

By (3), (13) and assumption $\left(e2\right)$, one obtains

$$\underset{n\to \infty }{lim}\mathcal{S}\left(\mathcal{S}{r}_n\right)=\mathcal{S}\left(\underset{n\to \infty }{lim}\mathcal{S}{r}_n\right)=\mathcal{S}\left(p\right).$$

(15)

By (4), (14) and assumption $\left(e2\right)$, one obtains

$$\underset{n\to \infty }{lim}\mathcal{S}\left(\mathcal{T}{r}_n\right)=\mathcal{S}\left(\underset{n\to \infty }{lim}\mathcal{T}{r}_n\right)=\mathcal{S}\left(p\right).$$

(16)

Now, both $\left\{\mathcal{T}{r}_n\right\}$ and $\left\{\mathcal{S}{r}_n\right\}$ remain $\mathfrak{S}$-preserving (due to (3) and (4)) and $\underset{n\to \infty }{lim}\mathcal{T}\left(r_n\right)=\underset{n\to \infty }{lim}\mathcal{S}\left(r_n\right)=p$ (due to (13) and (14)). By assumption $\left(e1\right)$, one obtains

$$\underset{n\to \infty }{lim}\varrho (\mathcal{S}\mathcal{T}{r}_n,\mathcal{T}\mathcal{S}{r}_n)=0.$$

(17)

In light of hypothesis $\left(e4\right)$, firstly suppose that $\mathcal{T}$ remains $\mathfrak{S}$-continuous. By (3), (13) and $\mathfrak{S}$-continuity of $\mathcal{T}$, one obtains

$$\underset{n\to \infty }{lim}\mathcal{T}\left(\mathcal{S}{r}_n\right)=\mathcal{T}\left(\underset{n\to \infty }{lim}\mathcal{S}{r}_n\right)=\mathcal{T}\left(p\right).$$

(18)

Using (16)–(18) and continuity of $\varrho$, one obtains

$$\begin{array}{ccc}\hfill \varrho (\mathcal{S}p,\mathcal{T}p)& =& \varrho (\underset{n\to \infty }{lim}\mathcal{S}\mathcal{T}{r}_n,\underset{n\to \infty }{lim}\mathcal{T}\mathcal{S}{r}_n)\hfill \\ & =& \underset{n\to \infty }{lim}\varrho (\mathcal{S}\mathcal{T}{r}_n,\mathcal{T}\mathcal{S}{r}_n)\hfill \\ & =& 0\hfill \end{array}$$

so that

$$\mathcal{S}\left(p\right)=\mathcal{T}\left(p\right).$$

Hence, we are finished. Secondly, assume that $\mathfrak{S}$ remains $(\mathcal{T},\mathcal{S})$-compatible as well as $(\mathcal{S},\varrho )$-self-closed. Since $\left\{\mathcal{S}{r}_n\right\}$ remains $\mathfrak{S}$-preserving (due to (3)) and $\mathcal{S}\left(r_n\right)\stackrel{\varrho }{⟶}p$ (due to (13)), therefore, using $(\mathcal{S},\varrho )$-self-closedness of $\mathfrak{S}$, one can find a subsequence $\left\{\mathcal{S}{r}_{n_k}\right\}$ of $\left\{\mathcal{S}{r}_n\right\}$ satisfying

$$[\mathcal{S}\mathcal{S}{r}_{n_k},\mathcal{S}p]\in \mathfrak{S},\forall k\in \mathbb{N}.$$

(19)

As $\mathcal{S}{r}_{n_k}\stackrel{\varrho }{⟶}p,$ Equations (13)–(17) also hold for $\left\{r_{n_k}\right\}$ instead of $\left\{r_n\right\}$. One claims that

$$\varrho (\mathcal{T}\mathcal{S}{r}_{n_k},\mathcal{T}p)\le \varrho (\mathcal{S}\mathcal{S}{r}_{n_k},\mathcal{S}p),\forall k\in \mathbb{N}.$$

(20)

Let $\{{\mathbb{N}}^0,{\mathbb{N}}^{+}\}$ be a partition of $\mathbb{N}$ satisfying the following cases:

(i)

$\varrho (\mathcal{S}\mathcal{S}{r}_{n_k},\mathcal{S}p)=0,\forall k\in {\mathbb{N}}^0,$

(ii)

$\varrho (\mathcal{S}\mathcal{S}{r}_{n_k},\mathcal{S}p)>0,\forall k\in {\mathbb{N}}^{+}.$

In case (i), making use of (19) and $(\mathcal{T},\mathcal{S})$-compatibility of $\mathcal{R}$, one obtains $\varrho (\mathcal{T}\mathcal{S}{r}_{n_k},\mathcal{T}p)=0,\forall k\in {\mathbb{N}}^0$ so that (20) holds $\forall k\in {\mathbb{N}}^0.$ In case (ii), by (19), assumption $\left(d\right)$ and Proposition 3, one obtains

$$\varphi \left(\varrho (\mathcal{T}\mathcal{S}{r}_{n_k},\mathcal{T}p)\right)\le \varphi \left(\varrho (\mathcal{S}\mathcal{S}{r}_{n_k},\mathcal{S}p)\right)-\psi \left(\varrho (\mathcal{S}\mathcal{S}{r}_{n_k},\mathcal{S}p)\right),\forall k\in {\mathbb{N}}_0,$$

which in view of Proposition 4, gives rise to $\varrho (\mathcal{T}\mathcal{S}{r}_{n_k},\mathcal{T}p)<\varrho (\mathcal{S}\mathcal{S}{r}_{n_k},\mathcal{S}p),\forall k\in {\mathbb{N}}^{+}$ so that (20) holds $\forall k\in {\mathbb{N}}^{+}$. Thus, (20) holds $\forall k\in \mathbb{N}.$ By triangular inequality, (15)–(17) and (20), one obtains

$$\begin{array}{ccc}\hfill \varrho (\mathcal{S}p,\mathcal{T}p)& \le & \varrho (\mathcal{S}p,\mathcal{S}\mathcal{T}{r}_{n_k})+\varrho (\mathcal{S}\mathcal{T}{r}_{n_k},\mathcal{T}\mathcal{S}{r}_{n_k})+\varrho (\mathcal{T}\mathcal{S}{r}_{n_k},\mathcal{T}p)\hfill \\ & \le & \varrho (\mathcal{S}p,\mathcal{S}\mathcal{T}{r}_{n_k})+\varrho (\mathcal{S}\mathcal{T}{r}_{n_k},\mathcal{T}\mathcal{S}{r}_{n_k})+\varrho (\mathcal{S}\mathcal{S}{r}_{n_k},\mathcal{S}p)\hfill \\ & \to & 0\mathrm{as}k\to \infty \hfill \end{array}$$

so that

$$\mathcal{S}\left(p\right)=\mathcal{T}\left(p\right).$$

Thus, p remains a coincidence point of $\mathcal{T}$ and $\mathcal{S}$.

Alternately, we assume that $\left(e^{\prime }\right)$ holds. By assumption $A\subseteq \mathcal{S}\left(M\right)$, $\exists r\in M$ such that $p=\mathcal{S}\left(r\right).$ Therefore, (13) and (14) respectively become

$$\underset{n\to \infty }{lim}\mathcal{S}\left(r_n\right)=\mathcal{S}\left(r\right).$$

(21)

$$\underset{n\to \infty }{lim}\mathcal{T}\left(r_n\right)=\mathcal{S}\left(r\right).$$

(22)

Now, one has to prove that r remains a coincidence point of $\mathcal{T}$ and $\mathcal{S}$ by using assumption $\left(e^{\prime }2\right)$. Firstly, suppose that $\mathcal{T}$ remains $(\mathcal{S},\mathfrak{S})$-continuous. By (21), one obtains

$$\underset{n\to \infty }{lim}\mathcal{T}\left(r_n\right)=\mathcal{T}\left(r\right).$$

(23)

On using (22) and (23), one obtains

$$\mathcal{S}\left(r\right)=\mathcal{T}\left(r\right).$$

Therefore, we are finished. Secondly, suppose that $\mathcal{T}$ and $\mathcal{S}$ remain continuous. Using Lemma 2, ∃ a subset $E\subseteq M$ verifying $\mathcal{S}\left(E\right)=\mathcal{S}\left(M\right)$ and $\mathcal{S}:E\to M$ remains one-to-one. Define a function $f:\mathcal{S}\left(E\right)\to \mathcal{S}\left(M\right)$ by

$$f\left(\mathcal{S}a\right)=\mathcal{T}\left(a\right),\forall \mathcal{S}\left(a\right)\in \mathcal{S}\left(E\right).$$

(24)

As $\mathcal{S}:E\to M$ remains one-to-one and $\mathcal{T}\left(M\right)\subseteq \mathcal{S}\left(M\right)$, f remains well defined. Continuities of $\mathcal{T}$ and $\mathcal{S}$ guarantee that f remains continuous. By $\mathcal{S}\left(M\right)=\mathcal{S}\left(E\right)$, assumption $\left(a\right)$ reduces to $\mathcal{T}\left(M\right)\subseteq \mathcal{S}\left(E\right)$. Thus, one can construct ${\left\{r_n\right\}}_{n=1}^{\infty }\subset E$ satisfying (2) and enabling us to choose $r\in E$. By (21), (22), (24) and continuity of f, one has

$$\mathcal{T}\left(r\right)=\mathcal{S}\left(\mathcal{S}r\right)=f\left(\underset{n\to \infty }{lim}\mathcal{S}{r}_n\right)=\underset{n\to \infty }{lim}f\left(\mathcal{S}{r}_n\right)=\underset{n\to \infty }{lim}\mathcal{T}\left(r_n\right)=\mathcal{S}\left(r\right).$$

Therefore, r remains a coincidence point of $\mathcal{T}$ and $\mathcal{S}$ and hence we are finished. Finally, assume that $\mathfrak{S}$ and ${\mathfrak{S}|}_A$ remain $(\mathcal{T},\mathcal{S})$-compatible and $\varrho$-self-closed, respectively. As $\left\{\mathcal{S}{r}_n\right\}$ remains ${\mathfrak{S}|}_A$-preserving (due to (3)) and $\mathcal{S}\left(r_n\right)\stackrel{d}{⟶}\mathcal{S}\left(r\right)\in A$ (due to (21)), by $\varrho$-self-closeness of ${\mathfrak{S}|}_A$, ∃ a subsequence $\left\{\mathcal{S}{r}_{n_k}\right\}$ of $\left\{\mathcal{S}{r}_n\right\}$ satisfying

$$[\mathcal{S}{r}_{n_k},\mathcal{S}r]{\in \mathfrak{S}|}_A,\forall k\in {\mathbb{N}}_0.$$

(25)

One claims that

$$\varrho (\mathcal{S}{r}_{n_k+1},\mathcal{T}r)\le \varrho (\mathcal{S}{r}_{n_k},\mathcal{S}r),\forall k\in \mathbb{N}.$$

(26)

Let $\{{\mathbb{N}}^0,{\mathbb{N}}^{+}\}$ be a partition of $\mathbb{N}$ satisfying the following cases:

(i)

$\varrho (\mathcal{S}{r}_{n_k},\mathcal{S}r)=0,\forall k\in {\mathbb{N}}^0,$

(ii)

$\varrho (\mathcal{S}{r}_{n_k},\mathcal{S}r)>0,\forall k\in {\mathbb{N}}^{+}.$

If case (i) holds, then by (25) and $(\mathcal{T},\mathcal{S})$-compatibility of $\mathfrak{S}$, one obtains $\varrho (\mathcal{T}{r}_{n_k},\mathcal{T}r)=0,$$\forall k\in {\mathbb{N}}^0$, which in light of (2), gives rise to $\varrho (\mathcal{S}{r}_{n_k+1},\mathcal{T}r)=0,\forall k\in {\mathbb{N}}^0$ and hence (27) holds $\forall k\in {\mathbb{N}}^0.$ In case (ii), by (2), (25), assumption $\left(d\right)$ and Proposition 3, one obtains

$$\varphi \left(\varrho (\mathcal{S}{r}_{n_k+1},\mathcal{T}r)\right)=\varphi \left(\varrho (\mathcal{T}{r}_{n_k},\mathcal{T}r)\right)\le \varphi \left(\varrho (\mathcal{S}{r}_{n_k},\mathcal{S}r)\right)-\psi \left(\varrho (\mathcal{S}{r}_{n_k},\mathcal{S}r)\right),\forall k\in {\mathbb{N}}_0,$$

which, in view of Proposition 4, gives rise to $\varrho (\mathcal{S}{r}_{n_k+1},\mathcal{T}r)<\varrho (\mathcal{S}{r}_{n_k},\mathcal{S}r),\forall k\in {\mathbb{N}}^{+}$ and hence (27) holds $\forall k\in {\mathbb{N}}^{+}$. Therefore, (27) holds $\forall k\in \mathbb{N}$.

By (21), (27) and continuity of $\varrho$, one obtains

$$\begin{array}{ccc}\hfill \varrho (\mathcal{S}r,\mathcal{T}r)& =& \varrho (\underset{k\to \infty }{lim}\mathcal{S}{r}_{n_k+1},\mathcal{T}r)\hfill \\ & =& \underset{k\to \infty }{lim}\varrho (\mathcal{S}{r}_{n_k+1},\mathcal{T}r)\hfill \\ & \le & \underset{k\to \infty }{lim}\varrho (\mathcal{S}{r}_{n_k},\mathcal{S}r)\hfill \\ & =& 0\hfill \end{array}$$

so that

$$\mathcal{S}\left(r\right)=\mathcal{T}\left(r\right).$$

Therefore, r remains a coincidence point of $\mathcal{T}$ and $\mathcal{S}$. □

Now, the uniqueness result corresponding to Theorem 1 runs as follows:

Theorem 2. 

Under the hypotheses of Theorem 1, if the following assumptions hold:

$\left(f_1\right)$

$\mathcal{T}\left(M\right)$ remains ${\mathfrak{S}|}_{\mathcal{S}\left(M\right)}^s$-connected,

$\left(f_2\right)$

$\mathfrak{S}$ remains $(\mathcal{T},\mathcal{S})$-compatible,

then $\mathcal{T}$ and $\mathcal{S}$ admit a unique point of coincidence. Moreover,

(i)

if $\mathcal{T}$ and $\mathcal{S}$ remain weakly compatible, then the point of coincidence is also a unique common fixed point and

(ii)

if anyone of $\mathcal{T}$ and $\mathcal{S}$ remains one-to-one, then $\mathcal{T}$ and $\mathcal{S}$ admit a unique coincidence point.

Proof. 

By Theorem 1, if $\overline{p}$ and $\overline{q}$ remain two points of coincidence of $\mathcal{T}$ and $\mathcal{S}$, then $\exists p,q\in M$ satisfying

$$\overline{p}=\mathcal{T}\left(p\right)=\mathcal{S}\left(p\right)\mathrm{and}\overline{q}=\mathcal{T}\left(q\right)=\mathcal{S}\left(q\right).$$

(27)

One has to show that $\overline{p}=\overline{q}.$ As $\mathcal{T}\left(p\right),\mathcal{T}\left(q\right)\in \mathcal{T}\left(M\right)\subseteq \mathcal{S}\left(M\right)$, by assumption $\left(f_1\right)$, ∃ a path $\{\mathcal{S}{s}_0,\mathcal{S}{s}_1,\mathcal{S}{s}_2,...,\mathcal{S}{s}_l\}$ in ${\mathfrak{S}|}_{\mathcal{S}\left(M\right)}^s$ from $\mathcal{T}\left(p\right)$ to $\mathcal{T}\left(q\right)$, whereas $s_0,s_1,s_2,...,s_l\in M$. By (27), without loss of generality, one can set $s_0=p$ and $s_l=q$. Therefore, one obtains

$$[\mathcal{S}{s}_j,\mathcal{S}{s}_{j+1}]{\in \mathfrak{S}|}_{\mathcal{S}\left(M\right)},\forall j(0\le j\le l-1).$$

(28)

Define the constant sequences $s_n^0=p$ and $s_n^l=q.$ Using (27), one has $\mathcal{S}\left(s_{n+1}^0\right)=\mathcal{T}\left(s_n^0\right)=\overline{p}$ and $\mathcal{S}\left(s_{n+1}^l\right)=\mathcal{T}\left(s_n^l\right)=\overline{q},\forall n\in {\mathbb{N}}_0$. Put $s_0^1=s_1,s_0^2=s_2,...,s_0^{l-1}=s_{l-1}$. Since $\mathcal{T}\left(M\right)\subseteq \mathcal{S}\left(M\right)$, therefore similar to Theorem 1, one can construct sequences $\left\{s_n^1\right\},\left\{s_n^2\right\},...,\left\{s_n^{l-1}\right\}$ in M verifying $\mathcal{S}\left(s_{n+1}^1\right)=\mathcal{T}\left(s_n^1\right),\mathcal{S}\left(s_{n+1}^2\right)=\mathcal{T}\left(s_n^2\right),...,\mathcal{S}\left(s_{n+1}^{l-1}\right)=\mathcal{T}\left(s_n^{l-1}\right),\forall n\in {\mathbb{N}}_0$. Hence, one obtains

$$\mathcal{S}\left(s_{n+1}^j\right)=\mathcal{T}\left(s_n^j\right),\forall n\in {\mathbb{N}}_0\mathrm{and}\forall j(0\le j\le l).$$

(29)

Now, one claims that

$$[\mathcal{S}{s}_n^j,\mathcal{S}{s}_n^{j+1}]\in \mathfrak{S},\forall n\in {\mathbb{N}}_0\mathrm{and}\forall j(0\le j\le l-1).$$

(30)

This claim will be verified by induction. Using (28), (30) holds for $n=0.$ Assume that (30) holds for $n=k>0$ so that

$$[\mathcal{S}{s}_k^j,\mathcal{S}{s}_k^{j+1}]\in \mathfrak{S},\forall j(0\le j\le l-1).$$

As $\mathfrak{S}$ remains $(\mathcal{T},\mathcal{S})$-closed, by Proposition 2, one obtains

$$[\mathcal{T}{s}_k^j,\mathcal{T}{s}_k^{j+1}]\in \mathfrak{S},\forall j(0\le j\le l-1),$$

which making use of (29), gives rise to

$$[\mathcal{S}{s}_{k+1}^j,\mathcal{S}{s}_{k+1}^{j+1}]\in \mathfrak{S},\forall j(0\le j\le l-1).$$

Therefore, by induction, (30) holds $\forall n\in {\mathbb{N}}_0$. Now, $\forall n\in {\mathbb{N}}_0$ and $\forall j(0\le j\le l-1)$, define ${\eta }_n^j:=\varrho (\mathcal{S}{s}_n^j,\mathcal{S}{s}_n^{j+1})$. One claims that

$$\underset{n\to \infty }{lim}{\eta }_n^j=0,\forall j(0\le j\le l-1).$$

(31)

Fix j and then two cases arise. Firstly, assume that ${\eta }_{n_0}^j=\varrho (\mathcal{S}{s}_{n_0}^j,\mathcal{S}{s}_{n_0}^{j+1})=0$ for some $n_0\in {\mathbb{N}}_0$, then by assumption $\left(f_2\right)$, one obtains $\varrho (\mathcal{T}{s}_{n_0}^j,\mathcal{T}{s}_{n_0}^{j+1})=0$. Consequently, by (29), one obtains ${\eta }_{n_0+1}^j=\varrho (\mathcal{S}{s}_{n_0+1}^j,\mathcal{S}{s}_{n_0+1}^{j+1})=\varrho (\mathcal{T}{s}_{n_0}^j,\mathcal{T}{s}_{n_0}^{j+1})=0$. Hence, by induction, one obtains ${\eta }_n^j=0,\forall n\ge n_0$ implying thereby $\underset{n\to \infty }{lim}{\eta }_n^j=0$.

If ${\eta }_n>0,\forall n\in {\mathbb{N}}_0$, then by (29), (30), assumption $\left(d\right)$ and Proposition 3, one obtains

$$\begin{array}{ccc}\hfill \varphi \left({\eta }_{n+1}^j\right)& =& \varphi \left(\varrho (\mathcal{S}{s}_{n+1}^j,\mathcal{S}{s}_{n+1}^{j+1})\right)\hfill \\ \hfill & =& \varphi \left(\varrho (\mathcal{T}{s}_n^j,\mathcal{T}{s}_n^{j+1})\right)\hfill \\ \hfill & \le & \varphi \left(\varrho (\mathcal{S}{s}_n^j,s_n^{j+1})\right)-\psi \left(\varrho (\mathcal{S}{s}_n^j,s_n^{j+1})\right)\hfill \end{array}$$

so that

$$\begin{array}{ccc}\hfill \varphi \left({\eta }_{n+1}^j\right)& \le & \varphi \left({\eta }_n^j\right)-\psi \left({\eta }_n^j\right).\hfill \end{array}$$

(32)

Using Proposition 4, one has

$$\begin{array}{ccc}\hfill {\eta }_{n+1}^j& <& {\eta }_n^j.\hfill \end{array}$$

Hence, $\left\{{\eta }_n^j\right\}\subset [0,\infty )$ remains a decreasing sequence. Consequently, $\exists \eta \ge 0$ verifying

$$\underset{n\to \infty }{lim}{\eta }_n^j=\eta .$$

One has to show that $\eta =0$. On the contrary, assume that $\eta >0$. Letting lower limit in (32), one obtains

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim\; inf}\varphi \left({\eta }_{n+1}^j\right)& \le & \underset{n\to \infty }{lim\; inf}\varphi \left({\eta }_n^j\right)-\underset{n\to \infty }{lim\; inf}\psi \left({\eta }_n^j\right).\hfill \end{array}$$

Using right continuity of $\varphi$, one obtains $\underset{n\to \infty }{lim\; inf}\psi \left({\eta }_n^j\right)\le 0,$ which remains a contradiction yielding thereby $\underset{n\to \infty }{lim}{\eta }_n^j=0.$ Thus, in both cases, (31) is proved $\forall j$$(0\le j\le l-1)$. By triangular inequality and (31), one obtains

$$\varrho (\overline{p},\overline{q})\le {\eta }_n^0+{\eta }_n^1+\cdots +{\eta }_n^{l-1}\to 0,asn\to \infty$$

$$⟹\overline{p}=\overline{q}.$$

Hence, $\mathcal{T}$ and $\mathcal{S}$ admit a unique point of coincidence. Finally, conclusions (i) and (ii) are immediate in the light of Lemma 4. □

Remark 1. 

We can derive several well known existing coincidence and common fixed point theorems as consequences of our results as follows:

•

For $\mathcal{S}=I,$ the identity mapping, Theorems 1 and 2 reduce to Theorems 4.1 and 4.2 of Sk et al. [35];

•

Taking $\varphi \left(a\right)=a$ and $\psi \left(a\right)=(1-k)a$ where $0\le k<1$ in Theorems 1 and 2, one obtains Theorems 4.1, 4.5 and 4.7 and 4.8 of Alam and Imdad [14];

•

Setting $\varphi \left(a\right)=a$ and $\psi \left(a\right)=a-\phi \left(a\right)$ Theorems 1 and 2, whereas φ is the Boyd–Wong function, one obtains Theorems 5, 6, 7 and 8 of Alam et al. [20];

•

For $\mathcal{S}=I,$ the identity mapping and the universal $\mathfrak{S}=M^2$, Theorem 2 reduces the following result of Alam et al. [34]:

Corollary 1. 

[34] Assume that $(M,\varrho )$ is a complete metric space, while $\mathcal{T}:M\to M$ is a mapping. If there exist $\varphi \in \Phi$ and $\psi \in \Psi$ verifying

$$\varphi \left(\varrho \right(\mathcal{T}r,\mathcal{T}t\left)\right)\le \varphi \left(\varrho \right(r,t\left)\right)-\psi \left(\varrho \right(r,t\left)\right),\forall r,t\in M,$$

then $\mathcal{T}$ admits a unique fixed point.

•

Taking $\mathcal{S}=I,$ the identity mapping and $\mathfrak{S}=⪯$, the partial order Theorems 1 and 2, one obtains the sharpened versions of the results of Harjani and Sadarangani [36] in the form of the following corollary:

Corollary 2 

([36]). Assume that $(M,\varrho ,⪯)$ remains a partially ordered metric space, while $\mathcal{T}:M\to M$ is a mapping. In addition,

(i)

∃⪯-complete subspace A of M verifying $\mathcal{T}\left(M\right)\subseteq A\subseteq M$;

(ii)

$\mathcal{T}$ remains ⪯-increasing;

(iii)

$\exists r_0\in M$ verifying $r_0⪯\mathcal{T}\left(r_0\right)$;

(iv)

there exist $\varphi \in \Phi$ and $\psi \in \Psi$ satisfying

$$\varphi \left(\varrho \right(\mathcal{T}r,\mathcal{T}t\left)\right)\le \varphi \left(\varrho \right(r,t\left)\right)-\psi \left(\varrho \right(r,t\left)\right),\forall r,t\in Mwithr⪯t;$$

(v)

either $\mathcal{T}$ remains ⪯-continuous or ⪯ is ϱ-self-closed.

Then, $\mathcal{T}$ admits a fixed point. Furthermore, if $\mathcal{T}\left(M\right)$ remains ${⪯}^s$-connected, then $\mathcal{T}$ admits a unique fixed point.

4. Illustrative Examples

In order to highlight the worth and utility of our results, we present the following examples.

Example 1. 

Consider $M=\mathbb{R}$ endowed with the Euclidean metric ϱ and the relation $\mathfrak{S}=\{(r,t)\in {\mathbb{R}}^2:|r|-|t|\ge 0\}$. Then, $\mathfrak{S}$ remains locally finitely $\mathcal{T}$-transitive. In addition, $(M,\varrho )$ remains $\mathfrak{S}$-complete.

Define the mappings $\mathcal{T},\mathcal{S}:M\to M$ by

$$\mathcal{T}\left(r\right)=\frac{r^2}{3}and\mathcal{S}\left(r\right)=\frac{r^2}{2},\forall r\in M.$$

Then, $\mathfrak{S}$ remains $(\mathcal{T},\mathcal{S})$-closed while $\mathcal{T}$ and $\mathcal{S}$ remain $\mathfrak{S}$-compatible. In addition, $\mathcal{T}$ and $\mathcal{S}$ remain $\mathfrak{S}$-continuous.

Let $\varphi ,\psi :[0,\infty )\to [0,\infty )$ be auxiliary functions defined by

$$\varphi \left(a\right)=\frac{a}{2}.and\psi \left(a\right)=\frac{a}{10}.$$

Clearly, $\varphi \in \Phi$ and $\psi \in \Psi$.

Let $r,t\in M$ verify $(\mathcal{S}r,\mathcal{S}t)\in \mathfrak{S}$; then, we have

$$\varphi \left(\varrho (\mathcal{T}r,\mathcal{T}t)\right)=\frac{1}{2}\left|\frac{r^2}{3}-\frac{t^2}{3}\right|=\frac{1}{3}\left|\frac{r^2}{2}-\frac{t^2}{2}\right|<\frac{2}{5}\varrho (\mathcal{S}r,\mathcal{S}t))=\varphi \left(\varrho (\mathcal{S}r,\mathcal{S}t)\right)-\psi \left(\varrho (\mathcal{S}r,\mathcal{S}t)\right).$$

Thus, $\mathcal{T}$ and $\mathcal{S}$ verify the contractivity condition $\left(d\right)$ of Theorem 1. Therefore, all the conditions mentioned in $\left(a\right)$–$\left(e\right)$ of Theorem 1 are satisfied. Consequently, $\mathcal{T}$ and $\mathcal{S}$ admit a coincidence point. Similarly, one can verify all the hypotheses of Theorem 2. Consequently, $\mathcal{T}$ and $\mathcal{S}$ admit a unique common fixed point $r=0$.

Example 2. 

Consider $M=[-1,1]$ endowed with the Euclidean metric ϱ and the relation $\mathfrak{S}=\{(r,t)\in M^2:r>t\}.$ Then, $\mathfrak{S}$ remains locally finitely $\mathcal{T}$-transitive. In addition, $(M,\varrho )$ remains $\mathfrak{S}$-complete.

Define the mappings $\mathcal{T},\mathcal{S}:M\to M$ by

$$\mathcal{T}\left(r\right)=\left\{\begin{array}{cc}r+1,\hfill & if-1\le r<0,\hfill \\ r-\frac{r^2}{2},\hfill & if0\le r\le 1\hfill \end{array}\right.$$

and

$$\mathcal{S}\left(r\right)=r.$$

Then, $\mathfrak{S}$ remains $(\mathcal{T},\mathcal{S})$-closed while $\mathcal{T}$ and $\mathcal{S}$ remain $\mathfrak{S}$-compatible. In addition, $\mathcal{S}$ remains $\mathfrak{S}$-continuous. Notice that here $\mathcal{T}$ remains not $\mathfrak{S}$-continuous at $r=0$. However, in view of assumption $\left(e4\right)$, $\mathfrak{S}$ remains $(\mathcal{T},\mathcal{S})$-compatible as well as $(\mathcal{S},\varrho )$-self-closed.

Let $\varphi ,\psi :[0,\infty )\to [0,\infty )$ be auxiliary functions defined by

$$\varphi \left(a\right)=\left\{\begin{array}{cc}a,\hfill & if0\le a\le 1\hfill \\ a^2,\hfill & ifa>1\hfill \end{array}\right.$$

and

$$\psi \left(a\right)=\left\{\begin{array}{cc}\frac{a^2}{2},\hfill & \mathrm{if}0\le a\le 1\hfill \\ 4,\hfill & \mathrm{if}a>1.\hfill \end{array}\right.$$

Clearly, $\varphi \in \Phi$ and $\psi \in \Psi$.

Let $r,t\in M$ verify $(\mathcal{S}r,\mathcal{S}t)\in \mathfrak{S}$; then, $\mathcal{S}\left(r\right)>\mathcal{S}\left(t\right)\ge 0$. Thus, one has

$$\begin{array}{ccc}\hfill \varphi \left(\varrho \right(\mathcal{T}r,\mathcal{T}t\left)\right)& =& \left(r-\frac{1}{2}{r}^2\right)-\left(t-\frac{1}{2}{t}^2\right)\hfill \\ \hfill & =& (r-t)-\frac{1}{2}(r-t)(r+t)\hfill \\ \hfill & \le & (\mathcal{S}r-\mathcal{S}t)-\frac{1}{2}{(\mathcal{S}r-\mathcal{S}t)}^2\hfill \\ \hfill & =& \varrho (\mathcal{S}r,\mathcal{S}t)-\frac{1}{2}{\left(\varrho (\mathcal{S}r,\mathcal{S}t)\right)}^2\hfill \\ \hfill & =& \varphi \left(\varrho \right(\mathcal{S}r,\mathcal{S}t\left)\right)-\psi \left(\varrho \right(\mathcal{S}r,\mathcal{S}t\left)\right).\hfill \end{array}$$

Therefore, $\mathcal{T}$ and $\mathcal{S}$ verify the contractivity condition $\left(d\right)$ of Theorem 1. Similarly, the rest of the conditions of Theorems 1 and 2 can be verified. Consequently, $\mathcal{T}$ and $\mathcal{S}$ admit a unique common fixed point $r=0$.

Example 3. 

Consider $M=[0,\infty )$ endowed with Euclidean metric ϱ and the relation $\mathfrak{S}$ defined by

$$\mathfrak{S}=\left\{\right(0,0),(0,1),(1,0),(1,1),(3,0\left)\right\}.$$

Then, $\mathfrak{S}$ remains locally finitely $\mathcal{T}$-transitive.

Define the mappings $\mathcal{T},\mathcal{S}:M\to M$ by

$$\mathcal{T}\left(r\right)=\left\{\begin{array}{cc}0,\hfill & ifr\in [0,1]\hfill \\ 1,\hfill & ifr\in (1,\infty )\hfill \end{array}\right.$$

and

$$\mathcal{S}\left(r\right)=\left\{\begin{array}{cc}\left[r\right],\hfill & ifr\in [0,1]\hfill \\ 3,\hfill & ifr\in (1,\infty ).\hfill \end{array}\right.$$

Then, $\mathfrak{S}$ remains $(\mathcal{T},\mathcal{S})$-closed. In addition, one can verify the contractivity condition $\left(d\right)$ of Theorem 1 for the auxiliary functions $\varphi \left(a\right)=a$ and $\psi \left(a\right)=\frac{a}{2}$.

In addition, here $\mathcal{T}$ and $\mathcal{S}$ are not $\mathfrak{S}$-compatible and hence $\left(e\right)$ does not hold. However, the subspace $A=\{0,1\}$ remains $\mathfrak{S}$-complete and $\mathcal{T}\left(M\right)=\{0,1\}\subseteq A\subseteq \mathcal{S}\left(M\right)=\{0,1,3\}$. Let $\left\{r_n\right\}\subset M$ be any ${\mathfrak{S}|}_A$-preserving sequence satisfying $r_n\stackrel{\varrho }{⟶}r.$ Since $(r_n,r_{n+1}){\in \mathfrak{S}|}_A,\forall n\in \mathbb{N},$∃$N\in \mathbb{N}$ verifying $r_n=r\in \{0,1\},\forall n\ge N$. Consequently, one can choose a subsequence $\left\{r_{n_k}\right\}$ of the sequence $\left\{r_n\right\}$ verifying $r_{n_k}=r,\forall k\in \mathbb{N}$ implying thereby $[r_{n_k},r]{\in \mathfrak{S}|}_A,\forall k\in \mathbb{N}$. Therefore, in view of assumption $\left(e^{\prime }2\right)$, ${\mathfrak{S}|}_A$ remains ϱ-self-closed.

Similarly, the rest of the hypotheses of Theorems 1 and 2 can be verified. Consequently, $\mathcal{T}$ and $\mathcal{S}$ admit a unique common fixed $r=0$.

Example 4. 

Consider $M=\mathbb{R}$ endowed with Euclidean metric ϱ and the relation $\mathfrak{S}$ defined by

$$\mathfrak{S}=\{(r,t)\in {\mathbb{R}}^2:r\ge 0,t\in \mathbb{Q}\}.$$

Define the mappings $\mathcal{T},\mathcal{S}:M\to M$ by

$$\mathcal{T}\left(r\right)=1and\mathcal{S}\left(r\right)=r^2-3,\forall r\in M.$$

Thus, $\mathfrak{S}$ remains $(\mathcal{T},\mathcal{S})$-closed.

Let $\varphi ,\psi :[0,\infty )\to [0,\infty )$ be auxiliary functions defined by

$$\varphi \left(a\right)=2aand\psi \left(a\right)=\frac{a}{3}.$$

Clearly, $\varphi \in \Phi$ and $\psi \in \Psi$.

Now, for $r,t\in M$ with $(\mathcal{S}r,\mathcal{S}t)\in \mathfrak{S}$, we have

$$\varphi \left(\varrho (\mathcal{T}r,\mathcal{T}t)\right)=2|1-1|=0\le \frac{5}{3}|r^2-t^2|=\varphi \left(\varrho (\mathcal{S}r,\mathcal{S}t)\right)-\psi \left(\varrho (\mathcal{S}r,\mathcal{S}t)\right).$$

Therefore, $\mathcal{T}$ and $\mathcal{S}$ verify the contractivity condition $\left(d\right)$ of Theorem 1.

In addition, here $\left(e\right)$ does not hold (as $\mathcal{T}$ and $\mathcal{S}$ are not $\mathfrak{S}$-compatible). However, the subspace $A:=\mathcal{S}\left(M\right)=[-3,\infty )$ remains $\mathfrak{S}$-complete. In view of assumption $\left(e^{\prime }2\right)$, $\mathcal{T}$ and $\mathcal{S}$ remain continuous. Therefore, in the light of Theorem 1, $\mathcal{T}$ and $\mathcal{S}$ admitting a coincidence point.

Furthermore, herein assumptions $\left(f_1\right)$ and $\left(f_2\right)$ also hold and hence, owing to Theorems 2, $\mathcal{T}$ and $\mathcal{S}$, admit a unique point of coincidence $\overline{r}=1$. Furthermore, neither $\mathcal{T}$ nor $\mathcal{S}$ remains one-to-one, and hence there is no guarantee of the uniqueness of coincidence point. Indeed, there are two coincidence points, r=2 and r=-2. Here, $\mathcal{T}$ and $\mathcal{S}$ remain not weakly compatible and hence there is no guarantee of uniqueness of common fixed point. Indeed, there is no common fixed point of $\mathcal{T}$ and $\mathcal{S}$.

5. Conclusions

In this manuscript, we established some coincidence and common fixed point theorems in the sense of Goebel [1] as well as Jungck [2] for a pair of self-mappings on a metric space endowed with a weaker class of transitive relation. To prove our results, we utilized a type of generalized contractive mapping involving two auxiliary functions. We also remarked that various known fixed point results of existing literature can be concluded from our results. Several illustrative examples are also provided to demonstrate the validity of the concept and the degree of applicability of our findings.

In the present work, we used a relatively weaker contractive condition, which is required to hold merely for those elements which are related via underlying relation. Due to such restrictive nature, the fixed point theorems obtained in relational metric space have more scopes of applications in areas of nonlinear matrix equations and periodic boundary value problems. Some applications are available in recent literature, e.g., [15,17,18,21,22,24,25,26]. The results proved herewith and the similar other results in future works can be applied to the area of nonlinear matrix equations, which remains a very important and applicable area by its own.