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Existence of a common solution of an integral equations system by $(\psi ,\alpha ,\beta )$weakly contractions
Journal of Inequalities and Applications volume 2014, Article number: 517 (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.
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 GuerreDelabriere [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 singlevalued and multivalued maps in a metric space.
Definition 1.1 ([6])
The function $\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ is called an altering distance function if the following properties are satisfied:

(i)
ψ is continuous and nondecreasing,

(ii)
$\psi (t)=0$ if and only if $t=0$.
Definition 1.2 ([3])
Let $(X,d)$ be a metric space. A mapping $f:X\to X$ is said to be weakly contractive if
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 semicontinuous, 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 nonempty set, N be a natural number such that $N\ge 2$ and ${f}_{1},{f}_{2},\dots ,{f}_{N1},{f}_{N}:X\to X$ be given selfmappings of X. If $w={f}_{1}x={f}_{2}x=\cdots ={f}_{N1}x={f}_{N}x$ for some $x\in X$, then x is called a coincidence point of ${f}_{1},{f}_{2},\dots ,{f}_{N1}$ and ${f}_{N}$, and w is called a point of coincidence of ${f}_{1},{f}_{2},\dots ,{f}_{N1}$ and ${f}_{N}$. If $w=x$, then x is called a common fixed point of ${f}_{1},{f}_{2},\dots ,{f}_{N1}$ 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 selfmappings 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 gnondecreasing, gX is closed and
where $\psi ,\alpha ,\beta :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ are such that ψ is continuous and monotone nondecreasing, α is continuous, β is lower semicontinuous,
and $\psi (t)=0$ if and only if $t=0$ and $\alpha (0)=\beta (0)=0$. Also, if any nondecreasing 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 selfmaps on a partially ordered set $(X,\u2aaf)$. A pair $(f,g)$ is said to be

(i)
weakly increasing if $fx\u2aafg(fx)$ and $gx\u2aaff(gx)$ for all $x\in X$ [15],

(ii)
partially weakly increasing if $fx\u2aafg(fx)$ for all $x\in X$ [13].
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])
Let $(X,\u2aaf)$ be a partially ordered set and $f,g,h:X\to X$ be given mappings such that $fX\subseteq hX$ and $gX\subseteq hX$. We say that f and g are weakly increasing with respect to h if and only if for all $x\in X$, we have
and
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])
Let $(X,\u2aaf)$ be a partially ordered set and $f,g,h:X\to X$ be given mappings such that $fX\subseteq hX$. We say that $(f,g)$ is partially weakly increasing with respect to 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 nondecreasing 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 nondecreasing function, α is an upper semicontinuous function, β is a lower semicontinuous 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
By construction we have ${x}_{2n+1}\in {T}^{1}(f{x}_{2n})$. Then, using the fact that $(f,g)$ is partially weakly increasing with respect to T, we obtain
On the other hand, we have ${x}_{2n+2}\in {S}^{1}(g{x}_{2n+1})$. Then, using the fact that $(g,f)$ is partially weakly increasing with respect to S, we obtain
Therefore, we can then write
or
We will prove our result in four steps.
Step 1.
Since $S{x}_{2n}$ and $T{x}_{2n+1}$ are comparable, by applying inequality (2), we have
where
Since $\frac{1}{2}d({y}_{2n1},{y}_{2n+1})\le \frac{1}{2}[d({y}_{2n1},{y}_{2n})+d({y}_{2n},{y}_{2n+1})]$, it follows that
and
If $d({y}_{2n1},{y}_{2n})<d({y}_{2n},{y}_{2n+1})$, then it follows from (8) and (9) that
Therefore, (7) implies that
By (3), we have $d({y}_{2n},{y}_{2n+1})=0$; that is, ${y}_{2n}={y}_{2n+1}$, and consequently we obtain
Now, by applying inequality (2), we have
and (3) implies that $d({y}_{2n+1},{y}_{2n+2})=0$; that is, ${y}_{2n+1}={y}_{2n+2}$. Repeating the above process inductively, one obtains ${y}_{k}={y}_{2n}$ for all $k\ge 2n$, which implies that (6) holds. On the other hand, if
by a similar calculation we obtain
Thus by (11) and (12) we obtain
which implies that the sequence $\{d({y}_{n},{y}_{n+1})\}$ is monotonically nonincreasing. Hence, there exists $r\ge 0$ such that
Taking the upper limit on both sides of (7) and using (8), (9), the upper semicontinuity of α, the lower semicontinuity 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.
Step 2. We claim that $\{{y}_{n}\}$ is a Cauchy sequence in 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 $k>0$,
This means that
Therefore we use (13), (14) and the triangular inequality to get
Letting $k\to \mathrm{\infty}$ in the above inequality and using (6), we obtain
Again, using the triangular inequality, we have
Letting again $k\to \mathrm{\infty}$ in the above inequality and using (6), (15), we get
On the other hand we have
Thanks to (6), (15), letting $k\to \mathrm{\infty}$, we have from the above inequality that
Also, by the triangular inequality, we have
Letting again $k\to \mathrm{\infty}$ in the above inequality and using (6) and (15), we obtain
Similarly, we can show that ${lim}_{k\to \mathrm{\infty}}d({y}_{2m(k)1},{y}_{2n(k)+1})\le \u03f5$, so
From (2) we have
where
and
Since ψ is a nondecreasing function, (17) implies that
Taking the upper limit on both sides of (19) and using (6), (15), (16), (18), (20) and the upper semicontinuity of α, the lower semicontinuity 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 $(g,T)$.
From the completeness of $(X,d)$, there is $u\in X$ such that
From (4) and (21), we obtain
Since the pairs $(f,S)$ and $(g,T)$ are compatible,
Using the continuity of f, g, S, T and (22), we have
The triangular inequality and (4) yield
Taking $n\to \mathrm{\infty}$ and using (23) and (24), we obtain
which means that $Su=fu$ and $Tu=gu$.
Step 4. The existence of a coincidence point for f, g, S and T.
Since Su and Tu are comparable, we can apply inequality (2)
where
and
Therefore we have
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)
$(f,g)$ 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, (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
Since SX and TX are closed subsets of $(X,d)$, there exist ${u}_{1},{u}_{2}\in X$ such that
Therefore $v=T{u}_{1}=S{u}_{2}$.
Since $\{{y}_{n}\}$ is a nondecreasing sequence and X is regular, it follows from (25) that ${y}_{n}\u2aafv$ for all $n\in \mathbb{N}\cup \{0\}$. Hence,
Applying inequality (2), we have
where
and
Letting $n\to \mathrm{\infty}$ in (26) and using (25), we obtain
or
By (3), we have $d(v,f{u}_{2})=0$, and hence $v=f{u}_{2}$. Similarly, we have
Therefore we can apply inequality (2) to obtain
where
and
Letting $n\to \mathrm{\infty}$ in (27) and using (25), we obtain
or
By (3), we have $d(v,g{u}_{1})=0$ and hence $v=g{u}_{1}$.
Therefore we have obtained
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])
Let $(X,d)$ be a metric space and $f,g:X\to X$ be given selfmappings on X. The pair $(f,g)$ is said to be semicompatible 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 semicompatibility by condition (ii) only. It is clear that if the pair $(f,g)$ is semicompatible, 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 selfmappings 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 selfmappings on X. The pair $(f,g)$ is said to be fweak 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 semicompatibility and fweakly 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 fweak reciprocally continuous and semicompatible,

(iii)
the pair $(g,T)$ is gweak reciprocally continuous and semicompatible,

(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, (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
From (4) and (28), we obtain
Hence by (ii) we deduce that
which implies that $fu=Su$. Similarly, we can apply (4) and (28) to obtain
Hence by (iii) we deduce 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
Since the pair $(g,T)$ is compatible in Theorem 2.1, the pair $(g,T)$ is weakly compatible in Theorem 2.2 and the pair $(g,T)$ is semicompatible in Theorem 2.6, we have
Since Tu and $TTu$ are comparable, it follows that Su and Tw are comparable. Applying inequality (2) and using (29) and (30), we obtain
where
and
Therefore, (31) implies that
By (3), we have $d(w,gw)=0$, that is, $w=gw$. Then, by (30), we have
Similarly, we can show that
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.
Now, suppose that the set of fixed points of 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
where
and
Therefore, (34) implies that
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 $d(x,y)=xy$ 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
Define $\psi ,\alpha ,\beta :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ by $\psi (t)=t$,
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
for some $t\in X$. Since $f{x}_{n}=ln(1+{x}_{n})$ and $S{x}_{n}={e}^{3{x}_{n}}1$, we have ${x}_{n}\to {e}^{t}1$ and ${x}_{n}\to \frac{1}{3}ln(1+t)$. By the uniqueness of limit, we get that ${e}^{t}1=\frac{1}{3}ln(1+t)$ and hence $t=0$. Thus, ${x}_{n}\to 0$ as $n\to \mathrm{\infty}$. Since 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.
To prove that $(f,g)$ is partially weakly increasing with respect to 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
or
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 $\psi (t)\alpha (t)+\beta (t)=t\frac{1}{3}t=\frac{2}{3}t>0$.
In order to show that 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. $M(x,y)\ge 1$.
In this case, we have $N(x,y)>1$ or $N(x,y)\le 1$. If $N(x,y)>1$, then $\alpha (M(x,y))=\frac{1}{3}M(x,y)+\frac{1}{2}$ and $\beta (N(x,y))=\frac{1}{2}$. Therefore, we have
If $N(x,y)\le 1$, then $\alpha (M(x,y))=\frac{1}{3}M(x,y)+\frac{1}{2}$ and $\beta (N(x,y))=0$. Therefore, we have
Therefore in this case (2) is satisfied.
Case II. $M(x,y)<1$.
In this case, since $N(x,y)\le M(x,y)$, we obtain $N(x,y)<1$. Therefore, we have $\alpha (M(x,y))=\frac{1}{3}M(x,y)$ and $\beta (N(x,y))=0$. So, we obtain
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 nondecreasing 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.
Let $\psi ,\alpha ,\beta :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ be defined by $\psi (t)=t$,
and selfmaps 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
Consider the 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 $u,v\in \mathbb{R}$,
$${K}_{1}(t,s,u){K}_{2}(t,s,v){}^{2}\le p(t,s)log(1+{uv}^{2}),$$
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
Then $(X,\u2aaf)$ is a partially ordered set and regular. Also $(X,\parallel \cdot \parallel )$ is a complete metric space. Define $f,g:X\to X$ by
and
Now, let $x,y\in X$ such that $x\u2aafy$. From condition (iii), for all $t\in [a,b]$, we can write
Since $M(x,y)\ge N(x,y)$ and $\varphi (t)={t}^{2}log(1+{t}^{2})$ is a nondecreasing function in $[0,\mathrm{\infty})$, we have
Put $\psi (t)={t}^{2}$, $\alpha (t)=\frac{4}{3}{t}^{2}$ and $\beta (t)={t}^{2}log(1+{t}^{2})$, we get
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). □
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Lo’lo’, P., Vaezpour, S.M., Saadati, R. et al. Existence of a common solution of an integral equations system by $(\psi ,\alpha ,\beta )$weakly contractions. J Inequal Appl 2014, 517 (2014). https://doi.org/10.1186/1029242X2014517
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Keywords
 coincidence point
 common fixed point
 partially weakly increasing mappings
 compatible pair of mappings
 weakly compatible pair of mappings
 semicompatible pair of mapping
 $(\psi ,\alpha ,\beta )$weakly contraction
 reciprocally continuous mappings
 fweak reciprocally continuous mappings