# Existence of a common solution of an integral equations system by $(\psi ,\alpha ,\beta )$-weakly contractions

- Parvaneh Lo’lo’
^{1}, - Seiyed Mansour Vaezpour
^{2}, - Reza Saadati
^{3}and - Choonkil Park
^{4}Email author

**2014**:517

https://doi.org/10.1186/1029-242X-2014-517

© Lo’lo’ et al.; licensee Springer. 2014

**Received: **12 August 2014

**Accepted: **9 December 2014

**Published: **23 December 2014

## Abstract

In this paper, we consider a system of integral equations and apply the coincidence and common fixed point theorems for four mappings satisfying a $(\psi ,\alpha ,\beta )$-weakly contractive condition in ordered metric spaces to prove the existence of a common solution to integral equations. Also we furnish suitable examples to demonstrate the validity of the hypotheses of our results.

**MSC:**54H25, 47H10.

### Keywords

coincidence point common fixed point partially weakly increasing mappings compatible pair of mappings weakly compatible pair of mappings semi-compatible pair of mapping $(\psi ,\alpha ,\beta )$-weakly contraction reciprocally continuous mappings*f*-weak reciprocally continuous mappings

## 1 Introduction and preliminary

Fixed point theory has wide and endless applications in many fields of engineering and science. Its core, the Banach contraction principle (see [1]), has attracted many researchers who tried to generalize it in different aspects. In particular, Alber and Guerre-Delabriere [2] introduced the concept of weak contractions in Hilbert spaces. Rhoades [3] showed that the result which Alber *et al.* had proved in Hilbert spaces was also valid in complete metric spaces. Eshaghi Gordji *et al.* [4] proved a new coupled fixed point theorem related to the Pata contraction for mappings having the mixed monotone property in partially ordered metric spaces. Singh *et al.* [5] obtained coincidence and common fixed point theorems for a class of Suzuki hybrid contractions involving two pairs of single-valued and multi-valued maps in a metric space.

**Definition 1.1** ([6])

- (i)
*ψ*is continuous and non-decreasing, - (ii)
$\psi (t)=0$ if and only if $t=0$.

**Definition 1.2** ([3])

where $\phi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ is an altering distance function.

In [3], Rhoades proved that if *X* is complete, then every weak contraction has a unique fixed point.

The weak contraction principle, its generalizations and extensions and other fixed point results for mappings satisfying weak contractive type inequalities have been considered in a number of recent works.

In 2008, Dutta and Choudhury [7] proved the following theorem.

**Theorem 1.3** ([7])

*Let*$(X,d)$

*be a complete metric space and*$f:X\to X$

*be such that*

*where* $\psi ,\phi :[0,+\mathrm{\infty})\to :[0,+\mathrm{\infty})$ *are altering distance functions*. *Then* *f* *has a fixed point in* *X*.

In [8], Eslamian and Abkar introduced the concept of $(\psi ,\alpha ,\beta )$-weak contraction. They stated the following theorem as a generalization of Theorem 1.3.

**Theorem 1.4** ([8])

*Let*$(X,d)$

*be a complete metric space and*$f:X\to X$

*be a mapping satisfying*

*for all*$x,y\in X$,

*where*$\psi ,\alpha ,\beta :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$

*are such that*

*ψ*

*is an altering distance function*,

*α*

*is continuous*,

*β*

*is lower semi*-

*continuous*,

*and*

*and* $\alpha (0)=\beta (0)=0$. *Then* *f* *has a unique fixed point*.

Aydi *et al.* [9] proved that Theorem 1.4 is a consequence of Theorem 1.3. (Define $\phi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ by $\phi (t)=\psi (t)-\alpha (t)+\beta (t)$ for all $t\ge 0$.)

It is also known that common fixed point theorems are generalizations of fixed point theorems. Recently, many researchers have been interested in generalizing fixed point theorems to coincidence point theorems and common fixed point theorems.

**Definition 1.5** ([10])

Let *X* be a non-empty set, *N* be a natural number such that $N\ge 2$ and ${f}_{1},{f}_{2},\dots ,{f}_{N-1},{f}_{N}:X\to X$ be given self-mappings of *X*. If $w={f}_{1}x={f}_{2}x=\cdots ={f}_{N-1}x={f}_{N}x$ for some $x\in X$, then *x* is called a coincidence point of ${f}_{1},{f}_{2},\dots ,{f}_{N-1}$ and ${f}_{N}$, and *w* is called a point of coincidence of ${f}_{1},{f}_{2},\dots ,{f}_{N-1}$ and ${f}_{N}$. If $w=x$, then *x* is called a common fixed point of ${f}_{1},{f}_{2},\dots ,{f}_{N-1}$ and ${f}_{N}$.

On the other hand, compatibility of two mappings introduced by Jungck [11, 12] is an important concept in the context of common fixed point problems in metric spaces.

**Definition 1.6** ([11])

Let $(X,d)$ be a metric space and $f,g:X\to X$ be given self-mappings on *X*. The pair $(f,g)$ is said to be compatible if ${lim}_{n\to \mathrm{\infty}}d(fg{x}_{n},gf{x}_{n})=0$, whenever $\{{x}_{n}\}$ is a sequence in *X* such that ${lim}_{n\to \mathrm{\infty}}f{x}_{n}={lim}_{n\to \mathrm{\infty}}g{x}_{n}=t$, for some $t\in X$.

**Definition 1.7** ([12])

Two mappings $f,g:X\to X$, where $(X,d)$ is a metric space, are weakly compatible if they commute at their coincidence points, that is, if $ft=gt$ for some $t\in X$ implies that $fgt=gft$.

It is clear that if the pair $(f,g)$ is compatible, then $(f,g)$ is weakly compatible.

Recently, fixed point theory has developed rapidly in partially ordered metric spaces (for example, see [13–23] and the references therein). Harjani and Sadarangani in [19, 20] extended Theorem 1.3 in the framework of partially ordered metric spaces in the following way. In 2012, Choudhury and Kundu [24] established the $(\psi ,\alpha ,\beta )$-weak contraction principle to coincidence point and common fixed point results in partially ordered metric spaces and proved the following fixed point theorem as a generalization of Theorem 1.4.

**Theorem 1.8** ([24])

*Let*$(X,d,\u2aaf)$

*be a partially ordered complete metric space*.

*Let*$f,g:X\to X$

*be such that*$fX\subseteq gX$,

*f*

*is*

*g*-

*non*-

*decreasing*,

*gX*

*is closed and*

*where*$\psi ,\alpha ,\beta :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$

*are such that*

*ψ*

*is continuous and monotone non*-

*decreasing*,

*α*

*is continuous*,

*β*

*is lower semi*-

*continuous*,

*and* $\psi (t)=0$ *if and only if* $t=0$ *and* $\alpha (0)=\beta (0)=0$. *Also*, *if any non*-*decreasing sequence* $\{{x}_{n}\}$ *in* *X* *converges to* *z*, *then we assume* ${x}_{n}\u2aafz$ *for all* $n\in \mathbb{N}\cup \{0\}$. *If there exists* ${x}_{0}\in X$ *such that* $g{x}_{0}\u2aaff{x}_{0}$, *then* *f* *and* *g* *have a coincidence point*.

Altun and Simsek [15] introduced the concept of weakly increasing mappings as follows.

**Definition 1.9**Let

*f*,

*g*be two self-maps on a partially ordered set $(X,\u2aaf)$. A pair $(f,g)$ is said to be

Note that a pair $(f,g)$ is weakly increasing if and only if the ordered pairs $(f,g)$ and $(g,f)$ are partially weakly increasing.

Nashine and Samet [25] introduced weakly increasing mappings with respect to another map as follows.

**Definition 1.10** ([25])

*f*and

*g*are weakly increasing with respect to

*h*if and only if for all $x\in X$, we have

where ${h}^{-1}(x):=\{u\in X\mid hu=x\}$ for $x\in X$.

If $f=g$, we say that *f* is weakly increasing with respect to *h*.

If $h:X\to X$ is the identity mapping ($hx=x$ for all $x\in X$), then *f* and *g* being weakly increasing with respect to *h* implies that *f* and *g* are weakly increasing mappings.

Nashine *et al.* [26] proved some new coincidence point and common fixed point theorems for a pair of weakly increasing mappings with respect to another map.

In [17], Esmaily *et al.* gave the following definition.

**Definition 1.11** ([17])

*h*if and only if for all $x\in X$, we have

**Theorem 1.12** ([17])

*Let*$(X,d,\u2aaf)$

*be a partially ordered complete metric space*.

*Let*$f,g,S,T:X\to X$

*be given mappings satisfying the following*:

- (i)
$fX\subseteq TX$, $gX\subseteq SX$,

- (ii)
*f*,*g*,*S**and**T**are continuous*, - (iii)
*the pairs*$(f,S)$*and*$(g,T)$*are compatible*, - (iv)
$(f,g)$

*is partially weakly increasing with respect to**T**and*$(g,f)$*is partially weakly increasing with respect to**S*.

*Suppose that for every*$x,y\in X$

*such that*

*Sx*

*and*

*Ty*

*are comparable*,

*we have*

*where*

*and* $\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ *is an altering distance function*, *and* $\varphi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ *is a continuous function with* $\varphi (t)=0$ *if only if* $t=0$. *Then the pairs* $(f,S)$ *and* $(g,T)$ *have a coincidence point* $u\in X$; *that is*, $fu=Su$ *and* $gu=Tu$. *Moreover*, *if* *Su* *and* *Tu* *are comparable*, *then* $u\in X$ *is a coincidence point* *f*, *g*, *S* *and* *T*.

**Definition 1.13** ([25])

Let $(X,d,\u2aaf)$ be an ordered metric space. We say that *X* is regular if the following hypothesis holds: if $\{{x}_{n}\}$ is a non-decreasing sequence in *X* with respect to ⪯ such that ${x}_{n}\to x\in X$ as $n\to \mathrm{\infty}$, then ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$.

**Theorem 1.14** ([17])

*Let*$(X,d,\u2aaf)$

*be a partially ordered complete metric space such that*

*X*

*is regular*.

*Let*$f,g,S,T:X\to X$

*be given mappings satisfying the following*:

- (i)
$fX\subseteq TX$, $gX\subseteq SX$,

- (ii)
*SX**and**TX**are closed subsets of*$(X,d)$, - (iii)
*pairs*$(f,S)$*and*$(g,T)$*are weakly compatible*, - (iv)
$(f,g)$

*is partially weakly increasing with respect to**T**and*$(g,f)$*is partially weakly increasing with respect to**S*.

*Suppose that for every* $x,y\in X$ *such that* *Sx* *and* *Ty* *are comparable*, (1) *holds*. *Then the pairs* $(f,S)$ *and* $(g,T)$ *have a coincidence point* $u\in X$.

In this paper, an attempt is made to derive some coincidence and common fixed point theorems for four mappings on complete ordered metric spaces, satisfying a $(\psi ,\alpha ,\beta )$-weak contractive condition, which generalizes the existing results. Our results are supported by some examples.

## 2 Coincidence and common fixed point results

We begin our study with the following result.

**Theorem 2.1**

*Let*$(X,d,\u2aaf)$

*be a partially ordered complete metric space*.

*Let*$f,g,S,T:X\to X$

*be given mappings satisfying*:

- (i)
$fX\subseteq TX$, $gX\subseteq SX$,

- (ii)
*f*,*g*,*S**and**T**are continuous*, - (iii)
*the pairs*$(f,S)$*and*$(g,T)$*are compatible*, - (iv)
$(f,g)$

*is partially weakly increasing with respect to**T**and*$(g,f)$*is partially weakly increasing with respect to**S*.

*Suppose that for every*$x,y\in X$

*such that*

*Sx*

*and*

*Ty*

*are comparable*,

*we have*

*where*

*and*$\psi ,\alpha ,\beta :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$

*are such that*

*ψ*

*is a continuous and monotone non*-

*decreasing function*,

*α*

*is an upper semi*-

*continuous function*,

*β*

*is a lower semi*-

*continuous function and for all*$t>0$,

*Then the pairs* $(f,S)$ *and* $(g,T)$ *have a coincidence point* $u\in X$; *that is*, $fu=Su$ *and* $gu=Tu$. *Moreover*, *if* *Su* *and* *Tu* *are comparable*, *then* $u\in X$ *is a coincidence point of* *f*, *g*, *S* *and* *T*.

*Proof*Let ${x}_{0}$ be an arbitrary point in

*X*. Since $fX\subseteq TX$, there exists ${x}_{1}\in X$ such that $T{x}_{1}=f{x}_{0}$. Since $gX\subseteq SX$, there exists ${x}_{2}\in X$ such that $S{x}_{2}=g{x}_{1}$. Continuing this process, we can construct sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ in

*X*defined by

*T*, we obtain

*S*, we obtain

We will prove our result in four steps.

Taking the upper limit on both sides of (7) and using (8), (9), the upper semi-continuity of *α*, the lower semi-continuity of *β* and the continuity of *ψ*, we obtain $\psi (r)\le \alpha (r)-\beta (r)$, which by (3) implies that $r=0$. So equation (6) holds and the proof of Step 1 is completed.

*X*. By (6), it suffices to show that the subsequence $\{{y}_{2n}\}$ of $\{{y}_{n}\}$ is a Cauchy sequence in

*X*. If not, then there exists $\u03f5>0$ for which we can find two subsequences $\{{y}_{2m(k)}\}$ and $\{{y}_{2n(k)}\}$ of $\{{y}_{2n}\}$ such that $n(k)$ is the smallest integer and, for all $k>0$,

*ψ*is a non-decreasing function, (17) implies that

*α*, the lower semi-continuity of

*β*and the continuity of

*ψ*, we obtain

By (3), we have $\u03f5=0$, which is a contradiction. Thus $\{{y}_{2n}\}$ is a Cauchy sequence in *X*, and hence $\{{y}_{n}\}$ is a Cauchy sequence.

Step 3. Existence of a coincidence point for $(f,S)$ and $(g,T)$.

*f*,

*g*,

*S*,

*T*and (22), we have

which means that $Su=fu$ and $Tu=gu$.

Step 4. The existence of a coincidence point for *f*, *g*, *S* and *T*.

*Su*and

*Tu*are comparable, we can apply inequality (2)

By (3), we have $d(Su,Tu)=0$; that is, $Su=Tu$. Therefore *u* is a coincidence point of *f*, *g*, *S* and *T*. □

Now, we relax the conditions of Theorem 2.1, the continuity of *f*, *g*, *S* and *T* and the compatibility of the pairs $(f,S)$ and $(g,T)$, and we replace them by other conditions in order to find the same result. This will be the purpose of the next theorems.

**Theorem 2.2**

*Let*$(X,d,\u2aaf)$

*be a partially ordered complete metric space such that*

*X*

*is regular*.

*Let*$f,g,S,T:X\to X$

*be given mappings satisfying*:

- (i)
$fX\subseteq TX$, $gX\subseteq SX$,

- (ii)
*SX**and**TX**are closed subsets of*$(X,d)$, - (iii)
*pairs*$(f,S)$*and*$(g,T)$*are weakly compatible*, - (iv)
*is partially weakly increasing with respect to**T**and*$(g,f)$*is partially weakly increasing with respect to**S*.

*Suppose that for every* $x,y\in X$ *such that* *Sx* *and* *Ty* *are comparable*, (2) *holds*.

*Then the pairs* $(f,S)$ *and* $(g,T)$ *have a coincidence point* $u\in X$; *that is*, $fu=Su$ *and* $gu=Tu$. *Moreover*, *if* *Su* *and* *Tu* *are comparable*, *then* $u\in X$ *is a coincidence point of* *f*, *g*, *S* *and T*.

*Proof*We take the same sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ as in the proof of Theorem 2.1. In particular, $\{{y}_{n}\}$ is a Cauchy sequence in $(X,d)$. Hence, there exists $v\in X$ such that

*SX*and

*TX*are closed subsets of $(X,d)$, there exist ${u}_{1},{u}_{2}\in X$ such that

Therefore $v=T{u}_{1}=S{u}_{2}$.

*X*is regular, it follows from (25) that ${y}_{n}\u2aafv$ for all $n\in \mathbb{N}\cup \{0\}$. Hence,

By (3), we have $d(v,g{u}_{1})=0$ and hence $v=g{u}_{1}$.

Now, if $(f,S)$ and $(g,T)$ are weakly compatible, then $fv=fS{u}_{2}=Sf{u}_{2}=Sv$ and $gv=gT{u}_{1}=Tg{u}_{1}=Tv$, and *v* is a coincidence point of $(f,S)$ and $(g,T)$.

The rest of the conclusion follows as in the proof of Theorem 2.1. □

**Definition 2.3** ([27])

*X*. The pair $(f,g)$ is said to be semi-compatible if the two conditions hold:

- (i)
$ft=gt$ implies $fgt=gft$,

- (ii)
${lim}_{n\to \mathrm{\infty}}f{x}_{n}={lim}_{n\to \mathrm{\infty}}g{x}_{n}=t$ for some $t\in X$, implies ${lim}_{n\to \mathrm{\infty}}fg{x}_{n}=gt$.

Singh and Jain [28] observe that (ii) implies (i). Hence, they defined the semi-compatibility by condition (ii) only. It is clear that if the pair $(f,g)$ is semi-compatible, then $(f,g)$ is weakly compatible.

**Definition 2.4** ([29])

Let $(X,d)$ be a metric space and $f,g:X\to X$ be given self-mappings on *X*. The pair $(f,g)$ is said to be reciprocally continuous if ${lim}_{n\to \mathrm{\infty}}fg{x}_{n}=ft$ and ${lim}_{n\to \mathrm{\infty}}gf{x}_{n}=gt$ whenever $\{{x}_{n}\}$ is a sequence such that ${lim}_{n\to \mathrm{\infty}}f{x}_{n}={lim}_{n\to \mathrm{\infty}}g{x}_{n}=t$ for some $t\in X$.

**Definition 2.5** ([30])

Let $(X,d)$ be a metric space and $f,g:X\to X$ be given self-mappings on *X*. The pair $(f,g)$ is said to be *f*-weak reciprocally continuous if ${lim}_{n\to \mathrm{\infty}}fg{x}_{n}=ft$ whenever $\{{x}_{n}\}$ is a sequence such that ${lim}_{n\to \mathrm{\infty}}f{x}_{n}={lim}_{n\to \mathrm{\infty}}g{x}_{n}=t$ for some $t\in X$.

In the next theorem, the concepts of semi-compatibility and *f*-weakly reciprocal continuity are used.

**Theorem 2.6**

*Let*$(X,d,\u2aaf)$

*be a partially ordered complete metric space*.

*Let*$f,g,S,T:X\to X$

*be given mappings satisfying*:

- (i)
$fX\subseteq TX$, $gX\subseteq SX$,

- (ii)
*the pair*$(f,S)$*is**f*-*weak reciprocally continuous and semi*-*compatible*, - (iii)
*the pair*$(g,T)$*is**g*-*weak reciprocally continuous and semi*-*compatible*, - (iv)
*is partially weakly increasing with respect to**T**and*$(g,f)$*is partially weakly increasing with respect to**S*.

*Suppose that for every* $x,y\in X$ *such that* *Sx* *and* *Ty* *are comparable*, (2) *holds*.

*Then the pairs* $(f,S)$ *and* $(g,T)$ *have a coincidence point* $u\in X$; *that is*, $fu=Su$ *and* $gu=Tu$. *Moreover*, *if* *Su* *and* *Tu* *are comparable*, *then* $u\in X$ *is a coincidence point* *f*, *g*, *S* *and* *T*.

*Proof*We take the same sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ as in the proof of Theorem 2.1. In particular, $\{{y}_{n}\}$ is a Cauchy sequence in $(X,d)$. Hence, there exists $u\in X$ such that

which implies that $gu=Tu$. Therefore, we have proved that *u* is a coincidence point of $(f,S)$ and $(g,T)$.

The rest of the conclusion follows as in the proof of Theorem 2.1. □

Now, we shall prove the existence and uniqueness theorem of a common fixed point.

**Theorem 2.7** *If*, *in addition to the hypotheses of Theorems* 2.1, 2.2 *and* 2.6, *we suppose that* *Tu* *with* ${T}^{2}u$ *and* *Su* *with* ${S}^{2}u$ *are comparable*, *where* *u* *is a coincidence point of* *f*, *g*, *S* *and* *T*, *then* *f*, *g*, *S* *and* *T* *have a common fixed point in* *X*. *Moreover*, *if a set of fixed points of one of the mappings* *f*, *g*, *S* *and* *T* *is totally ordered*, *then* *f*, *g*, *S* *and* *T* *have a unique common fixed point*.

*Proof*We set

*Tu*and $TTu$ are comparable, it follows that

*Su*and

*Tw*are comparable. Applying inequality (2) and using (29) and (30), we obtain

Hence, by (32) and (33), we deduce that $w=fw=gw=Sw=Tw$. Therefore *w* is a common fixed point of *f*, *g*, *S* and *T*.

*f*is totally ordered. Assume on the contrary that $fp=gp=Sp=Tp=p$ and $fq=gq=Sq=Tq$ but $p\ne q$. Since

*p*and

*q*contain a set of fixed points of

*f*, we obtain $p=Sp$ and $q=Tq$ are comparable, by inequality (2), we have

by (3), $d(p,q)=0$, a contradiction. Therefore *f*, *g*, *S* and *T* have a unique common fixed point. Similarly, the result follows when the set of fixed points of *g*, *S* or *T* is totally ordered. This completes the proof of Theorem 2.7. □

## 3 Some examples

In this section we present some examples which illustrate our results.

Now, we present an example to illustrate the obtained result given by the previous theorems.

**Example 3.1**Let $X=[0,+\mathrm{\infty})$. We define an order ⪯ on

*X*as $x\u2aafy$ if and only if $x\ge y$ for all $x,y\in X$. We take the usual metric $d(x,y)=|x-y|$ for $x,y\in X$. It is easy to see that $(X,d,\u2aaf)$ is a partially ordered complete metric space. Let $f,g,S,T:X\to X$ be defined by

Then *f*, *g*, *S*, *T*, *ψ*, *α* and *β* satisfy all the hypotheses of Theorems 2.1 and 2.7.

*Proof*The proof of (i) and (ii) is clear. To prove (iii), let $\{{x}_{n}\}$ be any sequence in

*X*such that

*f*and

*S*are continuous, we have $f{x}_{n}\to f0=0$ and $S{x}_{n}\to S0=0$ as $n\to \mathrm{\infty}$. Therefore,

Thus, the pair $(f,S)$ is compatible. Similarly, one can show that the pair $(g,T)$ is compatible.

*T*, let $x,y\in X$ be such that $y\in {T}^{-1}(fx)$. Then $Ty=fx$. By the definition of

*f*and

*T*, we have $ln(1+x)={e}^{y}-1$. So, we have $y=ln(1+ln(1+x))$. Now, since ${e}^{3x}-1\ge 3x\ge x\ge ln(1+x)$, we have

Therefore, $fx\u2aafgy$. Thus, we have proved that $(f,g)$ is partially weakly increasing with respect to *T*. Similarly, one can show that $(g,f)$ is partially weakly increasing with respect to *S*.

Now, we prove that *ψ*, *α* and *β* do satisfy the inequality of (3). If $t>1$, then $\psi (t)-\alpha (t)+\beta (t)=t-\frac{1}{3}t-\frac{1}{2}+\frac{1}{2}=\frac{2}{3}t>0$; if $t=1$, then $\psi (1)-\alpha (1)+\beta (1)=1-\frac{1}{3}-\frac{1}{2}=\frac{1}{6}>0$. And if $0\le t<1$, then $\psi (t)-\alpha (t)+\beta (t)=t-\frac{1}{3}t=\frac{2}{3}t>0$.

*f*,

*g*,

*S*,

*T*,

*ψ*,

*α*and

*β*do satisfy the contractive condition (2) in Theorem 2.1, using a mean value theorem, we have, for $x,y\in X$,

Then the following cases are possible.

Case I. $M(x,y)\ge 1$.

Therefore in this case (2) is satisfied.

Case II. $M(x,y)<1$.

Therefore in this case (2) is satisfied.

Thus, *f*, *g*, *S*, *T*, *ψ* and *φ* satisfy all the hypotheses of Theorems 2.1. Therefore, *f*, *g*, *S* and *T* have a coincidence point. Moreover, since *f*, *g*, *S* and *T* satisfy all the hypotheses of Theorem 2.7, we obtain that *f*, *g*, *S* and *T* have a unique common fixed point. In fact, 0 is the unique common fixed point of *f*, *g*, *S* and *T*. □

Clearly, the above example satisfies all the hypotheses of Theorem 2.6.

**Example 3.2**Let $X=\{1,2,3,4\}$. Let $d:X\times X\to \mathbb{R}$ be given as

and $\u2aaf:=\{(1,1),(2,2),(3,3),(4,4),(1,4),(2,4),(3,4)\}$ on *X*. Clearly, $(X,d,\u2aaf)$ is a partially ordered complete metric space.

Let $\{{x}_{n}\}$ be a non-decreasing sequence in *X* with respect to ⪯ such that ${x}_{n}\to x$. By the definition of metric *d*, there exists $k\in \mathbb{N}$ such that ${x}_{n}=x$ for all $n\ge k$. So $(X,d,\u2aaf)$ is regular.

*f*,

*g*,

*S*and

*T*on

*X*be given by

It is easy to see that *f*, *g*, *S* and *T* satisfy all the conditions given in Theorem 2.2. Thus 1, 2 and 3 are coincidence points of the pairs $(f,S)$ and $(g,T)$. Since $S1=2$ and $T1=2$ are comparable, 1 is a coincidence point *f*, *g*, *S* and *T*. Moreover, since $S1=2$ and $SS1=1$ are not comparable, so Theorem 2.7 is not applicable for this example. It is observed that 1 is not a common fixed point *f*, *g*, *S* and *T*.

## 4 Application: existence of a common solution to integral equations

where $b>a\ge 0$. The purpose of this section is to give an existence theorem for a solution of (35) using Theorem 2.1 or 2.2.

**Theorem 4.1**

*Consider the integral equations*(35).

- (i)
${K}_{1},{K}_{2}:[a,b]\times [a,b]\times \mathbb{R}\to \mathbb{R}$

*are continuous*; - (ii)
*for all*$t,s\in [a,b]$,$\begin{array}{c}{K}_{1}(t,s,x(s))\le {K}_{2}(t,s,{\int}_{a}^{b}{K}_{1}(s,\tau ,x(\tau ))\phantom{\rule{0.2em}{0ex}}d\tau ),\hfill \\ {K}_{2}(t,s,x(s))\le {K}_{1}(t,s,{\int}_{a}^{b}{K}_{2}(s,\tau ,x(\tau ))\phantom{\rule{0.2em}{0ex}}d\tau ),\hfill \end{array}$ - (iii)
*for all*$s,t\in [a,b]$*and comparable*$u,v\in \mathbb{R}$,$|{K}_{1}(t,s,u)-{K}_{2}(t,s,v){|}^{2}\le p(t,s)log(1+{|u-v|}^{2}),$

*where*$p:[a,b]\times [a,b]\to [0,+\mathrm{\infty})$

*is a continuous function satisfying*

*Then integral equations* (35) *have a solution* $x\in C[a,b]$.

*Proof*Let $X:=C[a,b]$ (the set of continuous functions defined on $C[a,b]$ and taking value in ℝ) with the usual supremum norm, that is, $\parallel x\parallel ={sup}_{a\le t\le b}|x(t)|$, for $x\in C[a,b]$. Consider on

*X*the partial order defined by

and $\psi (t)-\alpha (t)+\beta (t)>0$ for each $t>0$. By taking $S=T={I}_{X}$ (the identity mapping on *X*), all the required hypotheses of Theorem 2.1 (or Theorem 2.2) are satisfied. Then there exists $x\in X$, a common fixed point of *f* and *g*, that is, *x* is a solution to (35). □

## Declarations

## Authors’ Affiliations

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