Existence of a common solution of an integral equations system by (ψ,α,β) weakly contractions

(1)

R E S E A R C H

Open Access

Existence of a common solution of an integral

equations system by

(

ψ

,

α

,

β

)

-weakly

contractions

Parvaneh Lo’lo’

1

_{, Seiyed Mansour Vaezpour}

2

_{, Reza Saadati}

3

_{and Choonkil Park}

4*

*_{Correspondence:}

[email protected]

4_{Department of Mathematics,}

Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Republic of Korea Full list of author information is available at the end of the article

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 (ψ,

α,

β)-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; (ψ,

α,

β)-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 sci-ence. Its core, the Banach contraction principle (see []), has attracted many researchers who tried to generalize it in different aspects. In particular, Alber and Guerre-Delabriere [] introduced the concept of weak contractions in Hilbert spaces. Rhoades [] showed that the result which Alberet al.had proved in Hilbert spaces was also valid in complete metric spaces. Eshaghi Gordjiet al.[] proved a new coupled fixed point theorem related to the Pata contraction for mappings having the mixed monotone property in partially ordered metric spaces. Singhet al.[] obtained coincidence and common fixed point the-orems for a class of Suzuki hybrid contractions involving two pairs of single-valued and multi-valued maps in a metric space.

Deﬁnition .([]) The functionψ : [, +∞)→[, +∞) is called an altering distance function if the following properties are satisﬁed:

(i) ψ is continuous and non-decreasing, (ii) ψ(t) = if and only ift= .

Deﬁnition .([]) Let (X,d) be a metric space. A mappingf :X→Xis said to be weakly contractive if

d(fx,fy)≤d(x,y) –ϕd(x,y) for eachx,y∈X,

whereϕ: [, +∞)→[, +∞) is an altering distance function.

(2)

In [], Rhoades proved that ifXis 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 , Dutta and Choudhury [] proved the following theorem.

Theorem .([]) Let(X,d)be a complete metric space and f :X→X be such that

ψd(fx,fy)≤ψd(x,y)–ϕd(x,y) for each x,y∈X,

whereψ,ϕ: [, +∞)→: [, +∞)are altering distance functions.Then f has a ﬁxed point in X.

In [], Eslamian and Abkar introduced the concept of (ψ,α,β)-weak contraction. They stated the following theorem as a generalization of Theorem ..

Theorem .([]) Let(X,d)be a complete metric space and f :X→X be a mapping satisfying

ψd(fx,fy)≤αd(x,y)–βd(x,y)

for all x,y∈X,whereψ,α,β: [, +∞)→[, +∞)are such thatψ is an altering distance function,αis continuous,βis lower semi-continuous,and

ψ(t) –α(t) +β(t) >  for each t> ,

andα() =β() = .Then f has a unique ﬁxed point.

Aydiet al.[] proved that Theorem . is a consequence of Theorem .. (Deﬁne ϕ: [, +∞)→[, +∞) byϕ(t) =ψ(t) –α(t) +β(t) for allt≥.)

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 the-orems to coincidence point thethe-orems and common fixed point thethe-orems.

Deﬁnition .([]) LetXbe a non-empty set,Nbe a natural number such thatN≥ andf,f, . . . ,fN–,fN:X→Xbe given self-mappings ofX. Ifw=fx=fx=· · ·=fN–x=

fNxfor somex∈X, thenxis called a coincidence point off,f, . . . ,fN–andfN, andwis called a point of coincidence off,f, . . . ,fN–andfN. Ifw=x, thenxis called a common ﬁxed point off,f, . . . ,fN–andfN.

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

(3)

Deﬁnition . ([]) Two mappings f,g :X→X, where (X,d) is a metric space, are weakly compatible if they commute at their coincidence points, that is, ifft=gtfor some

t∈Ximplies thatfgt=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 [–] and the references therein). Harjani and Sadarangani in [, ] extended Theorem . in the framework of partially ordered metric spaces in the follow-ing way. In , Choudhury and Kundu [] established the (ψ,α,β)-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 ..

Theorem .([]) Let(X,d,)be a partially ordered complete metric space.Let f,g:

X→X be such that fX⊆gX,f is g-non-decreasing,gX is closed and

ψd(fx,fy)≤αd(gx,gy)–βd(gx,gy) for all x,y∈X such that gxgy,

where ψ,α,β : [, +∞)→[, +∞) are such that ψ is continuous and monotone non-decreasing,αis continuous,βis lower semi-continuous,

ψ(t) –α(t) +β(t) >  for all t> ,

andψ(t) = if and only if t= andα() =β() = .Also,if any non-decreasing sequence {xn}in X converges to z,then we assume xnz for all n∈N∪ {}.If there exists x∈X

such that gxfx,then f and g have a coincidence point.

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

Deﬁnition . Letf,gbe two self-maps on a partially ordered set (X,). A pair (f,g) is said to be

(i) weakly increasing iffxg(fx)andgxf(gx)for allx∈X[], (ii) partially weakly increasing iffxg(fx)for allx∈X[].

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 [] introduced weakly increasing mappings with respect to another map as follows.

Deﬁnition .([]) Let (X,) be a partially ordered set andf,g,h:X→Xbe given mappings such thatfX⊆hXandgX⊆hX. We say thatf andgare weakly increasing with respect tohif and only if for allx∈X, we have

fxgy, ∀y∈h–(fx)

and

gxfy, ∀y∈h–(gx),

(4)

Iff =g, we say thatf is weakly increasing with respect toh.

Ifh:X→Xis the identity mapping (hx=xfor allx∈X), thenf andg being weakly increasing with respect tohimplies thatf andgare weakly increasing mappings.

Nashineet al.[] proved some new coincidence point and common ﬁxed point theo-rems for a pair of weakly increasing mappings with respect to another map.

In [], Esmailyet al.gave the following deﬁnition.

Deﬁnition .([]) Let (X,) be a partially ordered set andf,g,h:X→Xbe given mappings such thatfX⊆hX. We say that (f,g) is partially weakly increasing with respect tohif and only if for allx∈X, we have

fxgy, ∀y∈h–(fx).

Theorem . ([]) Let (X,d,) be a partially ordered complete metric space. Let f,g,S,T:X→X be given mappings satisfying the following:

(i) fX⊆TX,gX⊆SX,

(ii) f,g,SandTare continuous,

(iii) the pairs(f,S)and(g,T)are compatible,

(iv) (f,g)is partially weakly increasing with respect toTand(g,f)is partially weakly increasing with respect toS.

Suppose that for every x,y∈X such that Sx and Ty are comparable,we have

ψd(fx,gy)≤ψM(x,y)–φN(x,y), ()

where

M(x,y) =max

d(Sx,Ty),d(Sx,fx),d(Ty,gy), 

d(Sx,gy) +d(fx,Ty),

N(x,y) =maxd(Sx,Ty),d(Sx,gy),d(Ty,fx) ,

andψ: [, +∞)→[, +∞)is an altering distance function,andφ: [, +∞)→[, +∞)is a continuous function withφ(t) = if only if t= .Then the pairs(f,S)and(g,T)have a co-incidence point u∈X;that is,fu=Su and gu=Tu.Moreover,if Su and Tu are comparable,

then u∈X is a coincidence point f,g,S and T.

Deﬁnition .([]) Let (X,d,) be an ordered metric space. We say thatXis regular if the following hypothesis holds: if{xn}is a non-decreasing sequence inXwith respect to

such thatxn→x∈Xasn→ ∞, thenxnxfor alln∈N.

Theorem .([]) Let(X,d,)be a partially ordered complete metric space such that X is regular.Let f,g,S,T:X→X be given mappings satisfying the following:

(ii) SXandTXare closed subsets of(X,d), (iii) pairs(f,S)and(g,T)are weakly compatible,

(5)

Suppose that for every x,y∈X such that Sx and Ty are comparable, ()holds.Then the pairs(f,S)and(g,T)have a coincidence point u∈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 (ψ,α,β )-weak contractive condition, which generalizes the existing results. Our results are sup-ported by some examples.

2 Coincidence and common ﬁxed point results We begin our study with the following result.

Theorem . Let(X,d,)be a partially ordered complete metric space.Let f,g,S,T:X→ X be given mappings satisfying:

(ii) f,g,SandT are continuous,

(iii) the pairs(f,S)and(g,T)are compatible,

Suppose that for every x,y∈X such that Sx and Ty are comparable,we have

ψd(fx,gy)≤αM(x,y)–βN(x,y), ()

where

M(x,y) =max

d(Sx,Ty),d(Sx,fx),d(Ty,gy), 

d(Sx,gy) +d(fx,Ty),

N(x,y) =maxd(Sx,Ty),d(Sx,fx),d(Ty,gy) ,

and ψ,α,β : [, +∞)→[, +∞) are such that ψ is a continuous and monotone non-decreasing function,αis an upper semi-continuous function,βis a lower semi-continuous function and for all t> ,

ψ(t) –α(t) +β(t) > . ()

Then the pairs(f,S)and(g,T)have a coincidence point u∈X;that is,fu=Su and gu=Tu.

Moreover,if Su and Tu are comparable,then u∈X is a coincidence point of f,g,S and T.

Proof Letxbe an arbitrary point inX. SincefX⊆TX, there existsx∈Xsuch thatTx=

fx. SincegX⊆SX, there existsx∈Xsuch thatSx=gx. Continuing this process, we

can construct sequences{xn}and{yn}inXdeﬁned by

yn=fxn=Txn+, yn+=gxn+=Sxn+, ∀n∈N∪ {}. ()

By construction we havexn+∈T–(fxn). Then, using the fact that (f,g) is partially weakly increasing with respect toT, we obtain

(6)

On the other hand, we havexn+∈S–(gxn+). Then, using the fact that (g,f) is partially

weakly increasing with respect toS, we obtain

Sxn+=gxn+fxn+=Txn+, ∀n∈N∪ {}.

Therefore, we can then write

TxSxTx · · · Txn+Sxn+Txn+ · · ·

or

yyy · · · ynyn+yn+ · · ·. ()

We will prove our result in four steps. Step .

lim

n→∞d(yn,yn+) = . ()

SinceSxnandTxn+are comparable, by applying inequality (), we have

ψd(yn,yn+)

=ψd(fxn,gxn+)

≤αM(xn,xn+)

–βN(xn,xn+)

, ()

where

M(xn,xn+) =max

d(Sxn,Txn+),d(Sxn,fxn),d(Txn+,gxn+),

 

d(Sxn,gxn+) +d(fxn,Txn+)

=max

d(yn–,yn),d(yn–,yn),d(yn,yn+),

 

d(yn–,yn+) +d(yn,yn)

=max

d(yn–,yn),d(yn,yn+),



d(yn–,yn+)

.

Since_d(yn–,yn+)≤_[d(yn–,yn) +d(yn,yn+)], it follows that

M(xn,xn+) =max

d(yn–,yn),d(yn,yn+) ()

and

N(xn,xn+) =max

d(Sxn,Txn+),d(Sxn,fxn),d(Txn+,gxn+)

=maxd(yn–,yn),d(yn–,yn),d(yn,yn+)

(7)

Ifd(yn–,yn) <d(yn,yn+), then it follows from () and () that

M(xn,xn+) =N(xn,xn+) =d(yn,yn+).

Therefore, () implies that

ψd(yn,yn+)

≤αd(yn,yn+)

–βd(yn,yn+)

. ()

By (), we haved(yn,yn+) = ; that is,yn=yn+, and consequently we obtain

M(xn+,xn+) =N(xn+,xn+) =d(yn+,yn+).

Now, by applying inequality (), we have

ψd(yn+,yn+)

=ψd(fxn+,gxn+)

≤αM(xn+,xn+)

–βN(xn+,xn+)

=αd(yn+,yn+)

–βd(yn+,yn+)

,

and () implies thatd(yn+,yn+) = ; that is,yn+=yn+. Repeating the above process

inductively, one obtainsyk=ynfor allk≥n, which implies that () holds. On the other hand, if

d(yn,yn+)≤d(yn–,yn), ()

by a similar calculation we obtain

d(yn+,yn+)≤d(yn,yn+). ()

Thus by () and () we obtain

d(yn+,yn+)≤d(yn,yn+),

which implies that the sequence {d(yn,yn+)} is monotonically non-increasing. Hence,

there existsr≥ such that

lim

n→∞d(yn,yn+) =r.

Taking the upper limit on both sides of () and using (), (), the upper semi-continuity ofα, the lower semi-continuity ofβand the continuity ofψ, we obtainψ(r)≤α(r) –β(r), which by () implies thatr= . So equation () holds and the proof of Step  is completed. Step . We claim that{yn}is a Cauchy sequence in X. By (), it suﬃces to show that the subsequence{yn}of{yn}is a Cauchy sequence inX. If not, then there exists>  for which we can ﬁnd two subsequences{ym(k)}and{yn(k)}of{yn}such thatn(k) is the smallest integer and, for allk> ,

(8)

This means that

d(ym(k),yn(k)–) <. ()

Therefore we use (), () and the triangular inequality to get

≤d(ym(k),yn(k))≤d(ym(k),yn(k)–) +d(yn(k)–,yn(k)–) +d(yn(k)–,yn(k))

<+d(yn(k)–,yn(k)–) +d(yn(k)–,yn(k)).

Lettingk→ ∞in the above inequality and using (), we obtain

lim

k→∞d(ym(k),yn(k)) =. ()

Again, using the triangular inequality, we have

d(ym(k)–,yn(k)) –d(ym(k),yn(k))≤d(ym(k)–,ym(k)).

Letting againk→ ∞in the above inequality and using (), (), we get

lim

k→∞d(ym(k)–,yn(k)) =. ()

On the other hand we have

d(ym(k),yn(k))≤d(ym(k),yn(k)+) +d(yn(k)+,yn(k)).

Thanks to (), (), lettingk→ ∞, we have from the above inequality that

≤ lim

n→∞d(ym(k),yn(k)+). ()

Also, by the triangular inequality, we have

d(ym(k),yn(k))≤d(ym(k),ym(k)–) +d(ym(k)–,yn(k)+) +d(yn(k)+,yn(k)).

Letting againk→ ∞in the above inequality and using () and (), we obtain

≤ lim

k→∞d(ym(k)–,yn(k)+).

Similarly, we can show thatlim_k→∞d(ym(k)–,yn(k)+)≤, so

lim

n→∞d(ym(k)–,yn(k)+) =. ()

From () we have

ψd(ym(k),yn(k)+)

=ψd(fxm(k),gxn(k)+)

≤αM(xm(k),xn(k)+)

–βN(xm(k),xn(k)+)

(9)

where

M(xm(k),xn(k)+) =max

d(Sxm(k),Txn(k)+),d(Sxm(k),fxm(k)),d(Txn(k)+,gxn(k)+),

 

d(Sxm(k),gxn(k)+) +d(fxm(k),Txn(k)+)

=max

d(ym(k)–,yn(k)),d(ym(k)–,ym(k)),d(yn(k),yn(k)+),

 

d(ym(k)–,yn(k)+) +d(ym(k),yn(k))

and

N(xm(k),xn(k)+) =max

d(Sxm(k),Txn(k)+),d(Sxm(k),fxm(k)),d(Txn(k)+,gxn(k)+)

=maxd(ym(k)–,yn(k)),d(ym(k)–,ym(k)),d(yn(k),yn(k)+) .

Sinceψis a non-decreasing function, () implies that

ψ()≤ψ

lim

k→∞d(ym(k),yn(k)+)

. ()

Taking the upper limit on both sides of () and using (), (), (), (), () and the upper semi-continuity ofα, the lower semi-continuity ofβ and the continuity ofψ, we obtain

ψ()≤α

max

, , , (+)

–βmax{, , }.

By (), we have= , which is a contradiction. Thus{yn}is a Cauchy sequence inX, and hence{yn}is a Cauchy sequence.

Step . Existence of a coincidence point for (f,S) and (g,T). From the completeness of (X,d), there isu∈Xsuch that

lim

n→∞yn=u. ()

From () and (), we obtain

d(Sxn,u)→, d(fxn,u)→,

d(Sxn+,u)→, d(gxn+,u)→, d(Txn+,u)→.

()

Since the pairs (f,S) and (g,T) are compatible,

dS(fxn),f(Sxn)

→, dT(gxn+),g(Txn+)

→. ()

Using the continuity off,g,S,Tand (), we have

df(Sxn),fu

→, dg(Txn+),gu

→,

dS(Txn+),Su

→, dT(Sxn+),Tu

→.

(10)

The triangular inequality and () yield

d(Su,fu)≤dSu,S(Txn+)

+dS(fxn),f(Sxn)

+df(Sxn),fu

,

d(Tu,gu)≤dTu,T(Sxn+)

+dT(gxn+),g(Txn+)

+dg(Txn+),gu

.

Takingn→ ∞and using () and (), we obtain

d(Su,fu)≤, d(Tu,gu)≤,

which means thatSu=fuandTu=gu.

Step . The existence of a coincidence point forf,g,SandT. SinceSuandTuare comparable, we can apply inequality ()

ψd(fu,gu)≤αM(u,u)–βN(u,u),

where

M(u,u) =max

d(Su,Tu),d(Su,fu),d(Tu,gu), 

d(Su,gu) +d(fu,Tu)

=max

d(Su,Tu), , , 

d(Su,Tu) +d(Su,Tu)=d(Su,Tu)

and

N(u,u) =maxd(Su,Tu),d(Su,fu),d(Tu,gu)

=maxd(Su,Tu), ,  =d(Su,Tu).

Therefore we have

ψd(Su,Tu)≤αd(Su,Tu)–βd(Su,Tu).

By (), we haved(Su,Tu) = ; that is,Su=Tu. Thereforeuis a coincidence point off,g,S

andT.

Now, we relax the conditions of Theorem ., the continuity off,g,SandT 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 . Let(X,d,)be a partially ordered complete metric space such that X is regular.Let f,g,S,T:X→X be given mappings satisfying:

(ii) SXandTXare closed subsets of(X,d), (iii) pairs(f,S)and(g,T)are weakly compatible,

(11)

Then the pairs(f,S)and(g,T)have a coincidence point u∈X;that is,fu=Su and gu=

Tu.Moreover,if Su and Tu are comparable,then u∈X is a coincidence point of f,g,S and T.

Proof We take the same sequences{xn}and{yn}as in the proof of Theorem .. In partic-ular,{yn}is a Cauchy sequence in (X,d). Hence, there existsv∈Xsuch that

lim

n→∞yn=v. ()

SinceSXandTXare closed subsets of (X,d), there existu,u∈Xsuch that

yn=Txn+→Tu, yn+=Sxn+→Su.

Thereforev=Tu=Su.

Since{yn}is a non-decreasing sequence andXis regular, it follows from () thatynv for alln∈N∪ {}. Hence,

Txn+=ynv=Su.

Applying inequality (), we have

ψd(fu,yn+)

=ψd(fu,gxn+)

≤αM(u,xn+)

–βN(u,xn+)

, ()

where

M(u,xn+) =max

d(Su,Txn+),d(Su,fu),d(Txn+,gxn+),

 

d(Su,gxn+) +d(fu,Txn+)

=max

d(v,yn),d(v,fu),d(yn,yn+),

 

d(v,yn+) +d(fu,yn)

and

N(u,xn+) =max

d(Su,Txn+),d(Su,fu),d(Txn+,gxn+)

=maxd(v,yn),d(v,fu),d(yn,yn+) .

Lettingn→ ∞in () and using (), we obtain

ψd(fu,v)≤α

max

,d(v,fu), , 

 +d(fu,v)–βmax,d(v,fu), 

or

(12)

By (), we haved(v,fu) = , and hencev=fu. Similarly, we have

Sxn=yn–v=Tu.

Therefore we can apply inequality () to obtain

ψd(yn,gu)

=ψd(fxn,gu)

≤αM(xn,u)

–βN(xn,u)

, ()

where

M(xn,u) =max

d(Sxn,Tu),d(Sxn,fxn),d(Tu,gu),

 

d(Sxn,gu) +d(fxn,Tu)

=max

d(yn–,v),d(yn–,yn),d(v,gu),  

d(yn–,gu) +d(yn,v)

and

N(xn,u) =max

d(Sxn,Tu),d(Sxn,fxn),d(Tu,gu)

=maxd(yn–,v),d(yn–,yn),d(v,gu) .

Lettingn→ ∞in () and using (), we obtain

ψd(v,gu)≤α

max

, ,d(v,gu), 

d(v,gu) + –βmax, ,d(v,gu)

or

ψd(v,gu)≤αd(v,gu)–βd(v,gu).

By (), we haved(v,gu) =  and hencev=gu. Therefore we have obtained

v=Su=fu, v=Tu=gu.

Now, if (f,S) and (g,T) are weakly compatible, thenfv=fSu=Sfu=Svandgv=gTu=

Tgu=Tv, andvis a coincidence point of (f,S) and (g,T).

The rest of the conclusion follows as in the proof of Theorem ..

Deﬁnition .([]) Let (X,d) be a metric space andf,g:X→Xbe given self-mappings onX. The pair (f,g) is said to be semi-compatible if the two conditions hold:

(i) ft=gtimpliesfgt=gft,

(ii) limn→∞fxn=limn→∞gxn=tfor somet∈X, implieslimn→∞fgxn=gt.

(13)

Deﬁnition . ([]) Let (X,d) be a metric space and f,g : X → X be given self-mappings on X. The pair (f,g) is said to be reciprocally continuous iflim_n→∞fgxn=ft andlim_n→∞gfxn=gtwhenever{xn}is a sequence such thatlimn→∞fxn=limn→∞gxn=t for somet∈X.

Deﬁnition .([]) Let (X,d) be a metric space andf,g:X→Xbe given self-mappings onX. The pair (f,g) is said to bef-weak reciprocally continuous iflim_n→∞fgxn=ft when-ever{xn}is a sequence such thatlimn→∞fxn=limn→∞gxn=tfor somet∈X.

In the next theorem, the concepts of semi-compatibility andf-weakly reciprocal conti-nuity are used.

Theorem . Let(X,d,)be a partially ordered complete metric space.Let f,g,S,T:X→ X be given mappings satisfying:

(ii) the pair(f,S)isf-weak reciprocally continuous and semi-compatible, (iii) the pair(g,T)isg-weak reciprocally continuous and semi-compatible,

Suppose that for every x,y∈X such that Sx and Ty are comparable, ()holds.

Then the pairs(f,S)and(g,T)have a coincidence point u∈X;that is,fu=Su and gu=

Tu.Moreover,if Su and Tu are comparable,then u∈X is a coincidence point f,g,S and T.

Proof We take the same sequences{xn}and{yn}as in the proof of Theorem .. In partic-ular,{yn}is a Cauchy sequence in (X,d). Hence, there existsu∈Xsuch that

lim

n→∞yn=u. ()

From () and (), we obtain

d(Sxn,u)→ and d(fxn,u)→.

Hence by (ii) we deduce that

lim

n→∞fSxn=fu and n→∞limfSxn=Su,

which implies thatfu=Su. Similarly, we can apply () and () to obtain

d(gxn+,u)→ and d(Txn+,u)→.

Hence by (iii) we deduce that

lim

n→∞gTxn+=gu and n→∞limgTxn+=Tu,

which implies thatgu=Tu. Therefore, we have proved thatuis a coincidence point of (f,S) and (g,T).

(14)

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

Theorem . If,in addition to the hypotheses of Theorems., .and.,we suppose that Tu with T_{u and Su with S}_{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

w:=Su=fu=Tu=gu. ()

Since the pair (g,T) is compatible in Theorem ., the pair (g,T) is weakly compatible in Theorem . and the pair (g,T) is semi-compatible in Theorem ., we have

gw=gTu=Tgu=Tw. ()

SinceTuandTTuare comparable, it follows thatSuandTware comparable. Applying inequality () and using () and (), we obtain

ψd(w,gw)=ψd(fu,gw)≤αM(u,w)–βN(u,w), ()

where

M(u,w) =max

d(Su,Tw),d(Su,fu),d(Tw,gw), 

d(Su,gw) +d(fu,Tw)

=max

d(w,gw),d(w,w),d(gw,gw), 

d(w,gw) +d(w,gw)

=maxd(w,gw), , ,d(w,gw) =d(w,gw)

and

N(u,w) =maxd(Su,Tw),d(Su,fu),d(Tw,gw)

=maxd(w,gw),d(w,w),d(gw,gw) =d(w,gw).

Therefore, () implies that

ψd(w,gw)≤αd(w,gw)–βd(w,gw).

By (), we haved(w,gw) = , that is,w=gw. Then, by (), we have

w=gw=Tw. ()

Similarly, we can show that

(15)

Hence, by () and (), we deduce thatw=fw=gw=Sw=Tw. Thereforewis a common ﬁxed point off,g,SandT.

Now, suppose that the set of ﬁxed points offis totally ordered. Assume on the contrary thatfp=gp=Sp=Tp=pandfq=gq=Sq=Tqbutp=q. Sincepandqcontain a set of ﬁxed points off, we obtainp=Spandq=Tqare comparable, by inequality (), we have

ψd(p,q)=ψd(fp,gq)≤αM(p,q)–βN(p,q), ()

where

M(p,q) =max

d(Sp,Tq),d(Sp,fp),d(Tq,gq), 

d(Sp,gq) +d(fp,Tq)

=max

d(p,q),d(p,p),d(q,q), 

d(p,q) +d(p,q)=d(p,q)

and

N(p,q) =maxd(Sp,Tq),d(Sp,fp),d(Tq,gq)

=maxd(p,q),d(p,p),d(q,q) =d(p,q).

Therefore, () implies that

ψd(p,q)≤αd(p,q)–βd(p,q),

by (),d(p,q) = , a contradiction. Thereforef,g,SandT have a unique common ﬁxed point. Similarly, the result follows when the set of ﬁxed points ofg,SorTis totally ordered.

This completes the proof of Theorem ..

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 the-orems.

Example . LetX= [, +∞). We deﬁne an orderonXasxyif and only if x≥y

for allx,y∈X. We take the usual metricd(x,y) =|x–y|forx,y∈X. It is easy to see that (X,d,) is a partially ordered complete metric space. Letf,g,S,T:X→Xbe deﬁned by

fx=ln( +x), gx=ln

 +x 

, Sx=ex– , Tx=ex– .

Deﬁneψ,α,β: [, +∞)→[, +∞) byψ(t) =t,

α(t) =



t if ≤t< , 

t+ 

 ift≥

and β(t) =

 if ≤t≤,



 ift> .

(16)

Proof The proof of (i) and (ii) is clear. To prove (iii), let{xn}be any sequence inXsuch that

lim

n→∞fxn=n→∞limSxn=t

for somet∈X. Sincefxn=ln( +xn) andSxn=exn– , we havexn→et–  andxn→



ln( +t). By the uniqueness of limit, we get thatet–  = 

ln( +t) and hencet= . Thus, xn→ asn→ ∞. Sincef andSare continuous, we havefxn→f =  andSxn→S =  asn→ ∞. Therefore,

lim n→∞d

S(fxn),f(Sxn)

=d(S,f) =d(, ) = .

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 toT, letx,y∈Xbe such thaty∈T–(fx). ThenTy=fx. By the deﬁnition off andT, we haveln( +x) =ey– . So, we havey=ln( +ln( +x)). Now, sinceex– ≥x≥x≥ln( +x), we have

 +x≥ + ln

 +ln( +x)

or

fx=ln( +x)≥ln

 + ln

 +ln( +x)=ln

 +y 

=gy.

Therefore, fxgy. Thus, we have proved that (f,g) is partially weakly increasing with respect toT. Similarly, one can show that (g,f) is partially weakly increasing with respect toS.

Now, we prove thatψ,αandβdo satisfy the inequality of (). Ift> , thenψ(t) –α(t) + β(t) =t–_t–_+_ =_t> ; ift= , thenψ() –α() +β() =  –_ –_=_ > . And if ≤t< , thenψ(t) –α(t) +β(t) =t–_t=_t> .

In order to show thatf,g,S,T,ψ,αandβ do satisfy the contractive condition () in Theorem ., using a mean value theorem, we have, forx,y∈X,

|fx–gy|=ln( +x) –ln

 +y 

≤

|x–y| ≤  e

x_–_ey₌

|Sx–Ty| ≤  M(x,y).

Then the following cases are possible. Case I.M(x,y)≥.

In this case, we haveN(x,y) >  orN(x,y)≤. IfN(x,y) > , thenα(M(x,y)) =_M(x,y) +_ andβ(N(x,y)) =_. Therefore, we have

ψd(fx,gy)=|fx–gy| ≤ 

M(x,y) = 

M(x,y) +  –

 =α

M(x,y)–βN(x,y).

IfN(x,y)≤, thenα(M(x,y)) =_M(x,y) +_ andβ(N(x,y)) = . Therefore, we have

ψd(fx,gy)=|fx–gy| ≤ 

M(x,y) < 

M(x,y) +   –  =α

M(x,y)–βN(x,y).

(17)

Case II.M(x,y) < .

In this case, sinceN(x,y)≤M(x,y), we obtainN(x,y) < . Therefore, we haveα(M(x,y)) =



M(x,y) andβ(N(x,y)) = . So, we obtain

ψd(fx,gy)=|fx–gy| ≤ 

M(x,y) –  =α

M(x,y)–βN(x,y).

Therefore in this case () is satisﬁed.

Thus,f,g,S,T,ψ andϕ satisfy all the hypotheses of Theorems .. Therefore,f,g,S

andThave a coincidence point. Moreover, sincef,g,SandTsatisfy all the hypotheses of Theorem ., we obtain thatf,g,SandT have a unique common ﬁxed point. In fact,  is the unique common ﬁxed point off,g,SandT.

Clearly, the above example satisﬁes all the hypotheses of Theorem ..

Example . LetX={, , , }. Letd:X×X→Rbe given as

d(x,y) =

, x=y,

x+y, x=y

and:={(, ), (, ), (, ), (, ), (, ), (, ), (, )}onX. Clearly, (X,d,) is a partially or-dered complete metric space.

Let{xn}be a non-decreasing sequence inXwith respect tosuch thatxn→x. By the deﬁnition of metricd, there existsk∈Nsuch thatxn=xfor alln≥k. So (X,d,) is regular.

Letψ,α,β: [,∞)→[,∞) be deﬁned byψ(t) =t,

α(t) =

 t+



t, t≥,

, t<  and β(t) =



t, t> ,  +t_, _t_≤_

and self-mapsf,g,SandTonXbe given by

f =

       

, g=

       

,

S=

       

, T=

       

.

It is easy to see thatf,g,SandTsatisfy all the conditions given in Theorem .. Thus ,  and  are coincidence points of the pairs (f,S) and (g,T). SinceS =  andT =  are comparable,  is a coincidence pointf,g,SandT. Moreover, sinceS =  andSS =  are not comparable, so Theorem . is not applicable for this example. It is observed that  is not a common ﬁxed pointf,g,SandT.

4 Application: existence of a common solution to integral equations Consider the integral equations:

x(t) =

b

a

Kt,s,x(s)ds,

x(t) =

b

a

Kt,s,x(s)ds,

(18)

whereb>a≥. The purpose of this section is to give an existence theorem for a solution of () using Theorem . or ..

Theorem . Consider the integral equations().

(i) K,K: [a,b]×[a,b]×R→Rare continuous; (ii) for allt,s∈[a,b],

K

t,s,x(s)≤K

t,s,

b

a

K

s,τ,x(τ)dτ

,

Kt,s,x(s)≤K

t,s,

b

a

Ks,τ,x(τ)dτ

,

(iii) for alls,t∈[a,b]and comparableu,v∈R,

K(t,s,u) –K(t,s,v)≤p(t,s)log +|u–v|,

wherep: [a,b]×[a,b]→[, +∞)is a continuous function satisfying

sup a≤t≤b

b

a

p(s,t)ds< 

b–a.

Then integral equations()have a solution x∈C[a,b].

Proof LetX:=C[a,b] (the set of continuous functions deﬁned onC[a,b] and taking value inR) with the usual supremum norm, that is,x=sup_a≤t≤b|x(t)|, forx∈C[a,b]. Consider onXthe partial order deﬁned by

x,y∈X, xy ⇐⇒ x(t)≤y(t), ∀t∈[a,b].

Then (X,) is a partially ordered set and regular. Also (X, · ) is a complete metric space. Deﬁnef,g:X→Xby

fx(t) =

b

a

Kt,s,x(s)ds, ∀t∈[a,b]

and

gx(t) =

b

a

Kt,s,x(s)ds, ∀t∈[a,b].

Now, letx,y∈Xsuch thatxy. From condition (iii), for allt∈[a,b], we can write

fx(t) –gy(t)≤

b

a

Kt,s,x(s)–Kt,s,y(s)ds 

≤ b

a ds

b

a

Kt,s,x(s)–Kt,s,y(s)ds

≤(b–a)

b

a

(19)

≤(b–a)

b

a

p(t,s)log +d(x,y)ds

= (b–a)

b

a

p(t,s)ds

log +d(x,y)

< log +d(x,y)≤log +M(x,y)

=  M(x,y)

_–_M₍_x_,_y₎_–_log_{ +}_M₍_x_,_y₎_.

SinceM(x,y)≥N(x,y) andφ(t) =t_–_log_{( +}_t_{) is a non-decreasing function in [,}_∞_),

we have

sup a≤t≤b

fx(t) –gy(t)≤ M(x,y)

_–_N₍_x_,_y₎_–_log_{ +}_N₍_x_,_y₎_.

Putψ(t) =t_,_α₍_t_{) =}

tandβ(t) =t–log( +t), we get

ψd(fx,gy)≤αM(x,y)–βN(x,y),

andψ(t) –α(t) +β(t) >  for eacht> . By takingS=T =IX (the identity mapping on

X), all the required hypotheses of Theorem . (or Theorem .) are satisﬁed. Then there existsx∈X, a common ﬁxed point off andg, that is,xis a solution to ().

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU), Tehran, Iran.}2_{Department of}

Mathematics and Computer Science, Amirkabir University of Technology, Hafez Ave., P.O. Box 15914, Tehran, Iran.

3_{Department of Mathematics, Iran University of Science and Technology, Tehran, Iran.}4_{Department of Mathematics,}

Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Republic of Korea.

Received: 12 August 2014 Accepted: 9 December 2014 Published:23 Dec 2014

References

1. Banach, S: Sur les opérations dans les ensembles et leur application aux équation sitegrales. Fundam. Math.3, 133-181 (1922)

2. Alber, T, Guerre-Delabriere, S: Principles of weakly contractive maps in Hilbert spaces. Oper. Theory, Adv. Appl.98, 7-22 (1997)

3. Rhoades, BE: Some theorems on weakly contractive maps. Nonlinear Anal.47, 2683-2693 (2001)

4. Eshaghi Gordji, M, Mohseni, S, Rostamian Delavar, M, De La Sen, M, Kim, G, Arian, A: Pata contractions and coupled type ﬁxed points. Fixed Point Theory Appl.2014, 130 (2014)

5. Singh, SL, Kamal, R, De La Sen, M: Coincidence and common ﬁxed point theorems for Suzuki type hybrid contraction and applications. Fixed Point Theory Appl.2014, 147 (2014)

6. Khan, MS, Swaleh, M, Sessa, S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 30, 1-9 (1984)

7. Dutta, PN, Choudhury, BS: A generalization of contraction principle in metric spaces. Fixed Point Theory Appl.2008, Article ID 406368 (2008)

8. Eslamian, M, Abkar, A: A ﬁxed point theorem for generalized weakly contractive mappings in complete metric space. Ital. J. Pure Appl. Math. (in press)

9. Aydi, H, Karapinar, E, Samet, B: Remarks on some recent ﬁxed point theorems. Fixed Point Theory Appl.2012, 76 (2012)

10. Aydi, H: Coincidence and common ﬁxed point result for contraction type maps in partially ordered metric spaces. Int. J. Math. Anal.5(13), 631-642 (2011)

(20)

12. Jungck, G, Rhoades, BE: Fixed point for set valued functions without continuity. Indian J. Pure Appl. Math.29, 227-238 (1998)

13. Abbas, M, Nazir, T, Radenovi´c, S: Common ﬁxed points of four maps in partially ordered metric spaces. Appl. Math. Lett.24, 1520-1526 (2011)

14. Agarwal, RP, El-Gebeily, MA, O’Regan, D: Generalized contractions in partially ordered metric spaces. Appl. Anal.87, 109-116 (2008)

15. Altun, I, Simsek, H: Some ﬁxed point theorems on ordered metric spaces and application. Fixed Point Theory Appl. 2010, Article ID 621469 (2010)

16. Erduran, A, Kadelburg, Z, Nashine, HK, Vetro, C: A ﬁxed point theorem for (φ,L)-weak contraction mappings on a partial metric space. J. Nonlinear Sci. Appl.7, 196-204 (2014)

17. Esmaily, J, Vaezpour, SM, Rhoades, BE: Coincidence point theorem for generalized weakly contractions in ordered metric spaces. Appl. Math. Comput.219, 1536-1548 (2012)

18. Esmaily, J, Vaezpour, SM, Rhoades, BE: Coincidence and common ﬁxed point theorems for a sequence of mappings in ordered metric spaces. Appl. Math. Comput.219, 5684-5692 (2013)

19. Harjani, J, Sadarangani, K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal.71, 3403-3410 (2009)

20. Harjani, J, Sadarangani, K: Generalized contractions in partially ordered metric spaces and applications to ordinary diﬀerential equations. Nonlinear Anal.72, 1188-1197 (2010)

21. Kumam, P, Hamidreza, R, Ghasem, SR: The existence of ﬁxed and periodic point theorems in cone metric type spaces. J. Nonlinear Sci. Appl.7, 255-263 (2014)

22. Ran, ACM, Reurings, MCB: A ﬁxed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc.132, 1435-1443 (2004)

23. Nieto, JJ, Rodriguez-Lopez, R: Contractive mapping theorems in partially ordered sets and applications to ordinary diﬀerential equations. Order22, 223-239 (2005)

24. Choudhury, BS, Kundu, A: (ψ,α,β)-Weak contractions in partially ordered metric spaces. Appl. Math. Lett.25, 6-10 (2012)

25. Nashine, HK, Samet, B: Fixed point results for mappings satisfying (ψ,ϕ)-weakly contractive condition in partially ordered metric spaces. Nonlinear Anal.74, 2201-2209 (2011)

26. Nashine, HK, Samet, B, Kim, JK: Fixed point results for contractions involving generalized altering distances in ordered metric spaces. Fixed Point Theory Appl.2011, 5 (2011)

27. Cho, YJ, Sharma, BK, Sahu, DR: Semi-compatibility and ﬁxed points. Math. Jpn.42, 91-98 (1995)

28. Singh, B, Jain, S: Semi-compatibility, compatibility and ﬁxed point theorems in fuzzy metric spaces. J. Chungcheong Math. Soc.18, 1-22 (2005)

29. Pant, RP, Bisht, RK, Arora, D: Weak reciprocal continuity and ﬁxed point theorems. Ann. Univ. Ferrara57, 181-190 (2011)

30. Srinivas, V, Reddy, BVB, Rao, RU: A common ﬁxed point theorem usingA-compatible andS-compatible mappings. Int. J. Theor. Appl. Sci.5, 154-161 (2013)

10.1186/1029-242X-2014-517