Common fixed point theorems for generalized contractive mappings with applications

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R E S E A R C H

Open Access

Common ﬁxed point theorems for

generalized contractive mappings with

applications

Nawab Hussain

1

_{, Masood Hussain Shah}

2

_{, Alireza Amini-Harandi}

3

_{and Zahid Akhtar}

4*

*_{Correspondence:}

[email protected]

4_{Department of Mathematics, The}

Islamia University of Bahawalpur, Bahawalpur, 63100, Pakistan Full list of author information is available at the end of the article

Abstract

Certain common ﬁxed point results involving four mappings satisfying generalized contractive conditions on a cone metric type space are obtained. Our results substantially improve and extend a number of known results. An example is given in support of the new results developed here. As an application, we establish the existence of a solution for an implicit integral equation.

MSC: Primary 47H09; 47H10; secondary 49M05

Keywords: cone metric type space; coincidence point; common ﬁxed point; generalized contractive mapping; nonexpansive mapping

1 Introduction

The Banach contraction principle is a remarkable result in the metric fixed point theory. Over the years, it has been generalized in different directions and spaces by several math-ematicians (see [–]). In , Zamfirescu [] definedZ-operators on metric spaces: There exist real numbersa,bandcsatisfying  <a< ,  <b,c< 

 such that for each x,y∈X, one of the following holds:

(Z) d(Tx,Ty)≤ad(x,y);

(Z) d(Tx,Ty)≤b[d(x,Tx) +d(y,Ty)];

(Z) d(Tx,Ty)≤c[d(x,Ty) +d(y,Tx)].

In , Ćirić [] deﬁned quasi-contraction (Cq_{) if for some }_≤_h_{<  and all}_x_,_y_∈_X_,

d(Tx,Ty)≤hmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx). (.) Clearly, the Zamﬁrescu operator is quasi-contractive.

In , Naimpally and Singh [] deﬁned a generalized contractive operator as follows:

d(Tx,Ty)≤hmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx) (.) for some ≤h<  and for allx,y∈X.

A generalized contractive operator in (.) is more general than quasi-contraction. In [, ] the authors investigated quasi-contractions on cone metric spaces and ob-tained ﬁxed point theorems for such mappings under a diﬀerent condition. Recently

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Cvetkovićet al.[] and Hussain and Shah [] introduced the notion of a cone metric type space, which is a generalization of a cone metric space, and proved some common ﬁxed point theorems and KKM-mappings results respectively.

In this paper we consider mappings on a cone metric type space (CMTS) with a solid cone. We prove new common ﬁxed point theorems involving four mappings satisfying a generalized contractive and a generalized nonexpansive conditions. Our results are gen-eralization of theorems proved in [, , , , ] and some other authors. Almost all of our results are proved without the assumption of the continuity of mappings.

The paper is organized as follows. In Section  we review some deﬁnitions and well-known results which will be needed in the sequel. In Section  and  we prove the exis-tence of common ﬁxed points of four mappings. We also present some corollaries which establish the fact that our results are generalization of several recent results in the litera-ture. In Section , we present an application to integral equations to illustrate the usability of the obtained results.

2 Deﬁnitions and notation

LetEbe a real Banach space andPa subset ofE. We use the symbol  to denote the zero element ofEand the symbolintPto denote the interior ofP. The subsetPis called a cone if and only if:

(I) Pis closed, nonempty andP={};

(II) a,b∈R,a,b≥, andx,y∈Psuch thatax+by∈P; (III) P∩(–P) ={}.

For a given coneP, a partial ordering≤with respect toPis introduced as follows:x≤y if and only ify–x∈P;x<ymeansx≤ybutx=y; ify–x∈intP, one writesxy; if

intP=∅, the conePis called a solid cone.

In this paper we always suppose thatEis a real Banach space,P⊆Eis a solid cone and ≤is partial ordering with respect toP.

Deﬁnition . LetX be a nonempty set andE be a real Banach space with a coneP.

A vector-valued functiond:X×X→E is said to be a cone metric type function onX with the constantk≥ if:

(d) ≤d(x,y)for allx,y∈Xandd(x,y) = if and only ifx=y;

(d) d(x,y) =d(y,x)for allx,y∈X;

(d) d(x,y)≤k[d(x,z) +d(z,y)]for allx,y,z∈X.

The pair (X,d) is called a cone metric type space (CMTS).

Remark . Fork= , the above deﬁnition of CMTS reduces to that of a cone metric space introduced in [].

Deﬁnition . Let (X,d) be a CMTS and{xn}be a sequence inX:

(I) {xn}converges toxif for everyc∈Ewithcthere existsn∈Nsuch that d(xn,x)cfor alln>n. We writelimn→∞xn=x, orxn→xasn→ ∞. (II) If for everyc∈Ewithcthere existsn∈Nsuch thatd(xn,xm)cfor all

n,m>n, then{xn}is called a Cauchy sequence inX.

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LetE be an ordered real Banach space with a coneP. The following properties hold (a,b,c∈E):

(C) Ifa≤bandbc, thenac;

(C) If≤acfor allc∈intP, thena= ;

(C) Ifa≤λa, wherea∈Pand≤λ< , thena= ;

(C) Letxn→inEandc.

Then there exists a positive integernsuch thatxncfor eachn>n.

Deﬁnition . LetF,G:X→Xbe mappings on a setX:

(i) Ify=Fx=Gxfor somex∈X, thenxis called a coincidence point ofFandG, andy

is called a point of coincidence ofFandG;

(ii) The pair{F,G}is called weakly compatible ifFandGcommute at all of their coincidence points, that is,FGx=GFxfor allx∈C(F,G) ={x∈X:Fx=Gx}. (iii) The pair{F,G}is called occasionally weakly compatible (in brief OWC) [] if

FGx=GFxfor somex∈C(F,G).

3 Generalized contractive operators

First we show the existence of a unique ﬁxed point.

Theorem . Let(X,d)be a cone metric type space with the constant k≥and let P be a solid cone.Let T:X→X be a mapping satisfying the contractive condition

d(Tx,Ty)≤λ

km(x,y), (.)

where

m(x,y)∈kd(x,y),d(x,Tx) +d(y,Ty),d(x,Ty) +d(y,Tx)

for all x,y∈X and for some constantλ∈(,_k₊_k).If TX is complete,then T has a unique

ﬁxed point in X.

Proof Forx∈X, we construct a sequence{xn}inXby

xn+=Txn forn≥.

We show that{xn}is a Cauchy sequence inX. Forn≥, we have

d(xn+,xn) =d(Txn,Txn–)≤ λ

km(xn,xn–),

where

m(xn,xn–)∈k

d(xn,xn–),d(xn,Txn) +d(xn–,Txn–),d(xn,Txn–) +d(xn–,Txn)

=kd(xn,xn–),d(xn,xn+) +d(xn–,xn),d(xn,xn) +d(xn–,xn+)

=kd(xn,xn–),d(xn,xn+) +d(xn–,xn),d(xn–,xn+)

.

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(i) d(xn+,xn)≤λd(xn,xn–);

(ii) d(xn+,xn)≤λ[d(xn,xn+) +d(xn–,xn)]≤(_–λλ )d(xn–,xn);

(iii) d(xn+,xn)≤λd(xn–,xn+)≤λk[d(xn–,xn) +d(xn,xn+)]≤(_–λλk_k)d(xn–,xn).

Letα=max{λ,_–λλ ,_–λλk_k}. Thenα=max{λ,_–λλ ,_–kλ_k_λ}= _–kλ_k_λ and sokα<  if and only if

kλ

–kλ<  if and only ifλ∈(, 

k+k).

Thus

d(xn+,xn)≤αd(xn–,xn)≤ · · · ≤αnd(x,x).

Sincek≥, form>n, we have

d(xn,xm)≤kd(xn,xn+) +kd(xn+,xn+) +· · ·+km–n–d(xm–,xm)

≤kαn+kαn++· · ·+km–n–αm–d(x,x)

≤

k  –kα

αnd(x,x)→ asn→ ∞.

Now, by (C) and (C), it follows that for everyc∈intP, there exists a positive integern such thatd(xn,xm)cfor everym>n>n, so{xn}is a Cauchy sequence. SinceTXis a

complete subspace ofX, there existsp∈TXsuch thatlimn→∞xn=p=limn→∞Txn. We

show thatpis a ﬁxed point ofT. Consider

d(p,Tp)≤kd(p,xn+) +d(xn+,Tp)

=kd(p,xn+) +kd(Txn,Tp)

≤kd(p,xn+) +k λ

km(xn,p),

where

m(xn,p)∈k

d(xn,p),d(xn,xn+) +d(p,Tp),d(xn,Tp) +d(p,xn+)

.

Let c. Then at least one of the following three cases holds: .

d(p,Tp)≤kd(p,xn+) +λkd(xn,p)k

c k+λk

c λk =c,

whenevern≥K. .

d(p,Tp)≤kd(p,xn+) +λkd(xn,xn+) +λkd(p,Tp)

implies

d(p,Tp)≤ k

 –λkd(p,xn+) + λk

 –λkd(xn,xn+)

k

 –λk  –λk

k c+ λk  –λk

 –λk λk c = c,

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.

d(p,Tp)≤kd(p,xn+) +λkd(xn,Tp) +λkd(p,xn+)

=kd(p,xn+) +λkd(p,xn+) +λkd(xn,Tp)

≤kd(p,xn+) +λkd(p,xn+) +λkd(xn,p) +λkd(p,Tp)

or

d(p,Tp)≤ k

 –λkd(p,xn+) + λk

 –λkd(p,xn+) + λk

 –λkd(xn,p)

k+λk  –λk

 –λk (k+λk)c+

λk  –λk

 –λk λk c = c,

whenevern≥K.

Hence, forn≥max{K,K,K},d(p,Tp)≤c.

ThusTp=p. Suppose that there existsq∈Xsuch thatTq=q; then from (.), we have

d(p,q) =d(Tp,Tq)≤λ km(p,q),

where

m(p,q)∈kd(p,q),d(p,Tp) +d(q,Tq),d(p,Tq) +d(q,Tp)=kd(p,q), d(p,q).

Thus we have the following two cases:

(i) d(p,q) =d(Tp,Tq)≤λd(p,q), and (ii) d(Tp,Tq)≤λd(p,q).

Sinceλ∈(,_k₊_k), from both the cases we getp=q.

Remark . In Theorem . we have generalized and uniﬁed the ﬁxed point theorems of Huang and Zhang [] and corresponding results in [].

Now we present some new common ﬁxed point theorems involving four mappings de-ﬁned on a cone metric type space.

Theorem . Let(X,d)be a cone metric type space with the constant k≥and let P be a solid cone.Suppose that the self-mappings F,G,S,T:X→X are such that SX⊂GX and TX⊂FX. Let x∈X,let the sequences {xn}and{zn}be deﬁned by zn=Gxn+=Sxn

and zn+=Fxn+=Txn+for n≥and letO(x) ={zn:n≥}.Assume that there exists

x∈X such that for some constantλ∈(,_k₊_k),

d(Sx,Ty)≤λ

km(x,y), (.)

where

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for all x,y∈F–_{_O₍_x₎_{} ∪}_G–_{_O₍_x₎_}_._{If one of SX}_,_TX_,_FX_,_{or GX is a complete subspace} of X,then{S,F}and{T,G}have a point of coincidence in X.

Proof We consider two cases of an odd integernand of an evenn. Forn= l+ ,l∈N, we haved(zl+,zl+) =d(Sxl+,Txl+), and from (.) there exists some

m(xl+,xl+)

∈kd(Fxl+,Gxl+),d(Fxl+,Sxl+) +d(Gxl+,Txl+),

d(Fxl+,Txl+) +d(Sxl+,Gxl+)

=kd(zl+,zl),d(zl+,zl+) +d(zl,zl+),d(zl+,zl)

such that

d(zl+,zl+) =d(Sxl+,Txl+)≤

λ k

m(xl+,xl+).

Thus we have the following three cases:

(i) d(zl+,zl+)≤λd(zl+,zl);

(ii) d(zl+,zl+)≤λd(zl+,zl+) +d(zl,zl+)≤(_–λλ )d(zl+,zl);

(iii) d(zl+,zl+)≤λd(zl+,zl)≤λk[d(zl+,zl+) +d(zl+,zl)]≤(_–λλk_k)d(zl+,zl).

Letα=max{λ, λ –λ,

λk

–λk}. It is easy to see thatα,kα∈(, ), such that

d(zn+,zn)≤αd(zn,zn–), n≥. (.)

Forn= l,l∈N, we have

d(zl,zl+) =d(Sxl,Txl+)≤ λ

km(xl,xl+), (.)

where

m(xl,xl+)

∈kd(Fxl,Gxl+),d(Fxl,Sxl) +d(Gxl+,Txl+),

d(Fxl,Txl+) +d(Sxl,Gxl+)

=kd(zl–,zl),d(zl–,zl) +d(zl,zl+),d(zl–,zl+)

.

Thus we have the following three cases:

(i) d(zl,zl+)≤λd(zl–,zl);

(ii) d(zl,zl+)≤λ[d(zl–,zl) +d(zl,zl+)]≤(_–λλ )d(zl–,zl);

(iii) d(zl,zl+)≤λd(zl–,zl+)≤λk[d(zl–,zl) +d(zl,zl+)]≤(_–λλk_k)d(zl–,zl).

So, inequality (.) is satisﬁed in this case, too. Therefore, (.) is satisﬁed for alln∈N, and by iterating we get

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Sincek≥, form>nwe have

d(zn,zm)≤kd(zn,zn+) +kd(zn+,zn+) +· · ·+Km–n–d(zm–,zm)

≤kαn+kαn++· · ·+Km–n–αm–d(z,z)

≤

k  –kα

αnd(z,z)→ asn→ ∞.

Now, by (C) and (C), it follows that for everyc∈intPthere exists a positive integern such thatd(zn,zm)cfor everym>n>n, so{zn}is a Cauchy sequence.

Let us suppose thatSXis a complete subspace ofX, then there existsz∈SXsuch that

limn→∞zn=limn→∞Sxn=z. Then we have

lim

n→∞Gxn+=nlim→∞Sxn=nlim→∞Fxn=nlim→∞Txn+=z,

that is, for any c, for suﬃcient largen, we haved(zn,z)c. Sincez∈SX⊂GX, then

there existsy∈Xsuch thatz=Gy. Let us prove thatz=Ty. From (d) and (.), we get

d(Ty,z)≤kd(Ty,Sxn) +d(Sxn,z)

≤λm(xn,y) +kd(zn,z),

where

m(xn,y)∈k

d(Fxn,Gy),d(Fxn,Sxn) +d(Gy,Ty),d(Fxn,Ty) +d(Sxn,Gy)

=kd(zn–,z),d(zn–,zn) +d(z,Ty),d(zn–,Ty) +d(zn,z).

Therefore we have three cases:

(i) d(Ty,z)≤kλd(zn–,z) +kd(zn,z)kλ__kcλ+k

c

k=casn→ ∞;

(ii) d(Ty,z)≤kλ[d(zn–,zn) +d(z,Ty)] +kd(zn,z)_–λk_k[λ–λ__k_λkc+–λ__kkc] =cas

n→ ∞(since≤k≤andλ∈(,_k₊_k), we haveλ<_k, and therefore –λk> ); (iii)

d(Ty,z)≤λkd(zn–,Ty) +d(zn,z) +kd(zn,z)

≤λkkd(zn–,z) +d(z,Ty)

+d(zn,z) +kd(zn,z)

≤ k

 –λk

λkd(zn–,z) +λd(zn,z) +d(zn,z)

k

 –λk

λk –λk 

λk c+λ  –λk

λk c+  –λk

k

=c asn→ ∞

(sincek≥andλ∈(,_k₊_k), we haveλ<_k₊_k <_k, and therefore –λk> ).

Therefore,d(Ty,z)cfor eachc∈intP. So, by (C) we haved(Ty,z) = , that is,

Ty=Gy=z,

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SinceTX⊂FX, there existsv∈Xsuch thatz=Fv. From (d) and (.), we have

d(Sv,z)≤kd(Sv,Txn+) +d(Txn+,z) ≤λm(v,xn+) +kd(zn+,z),

where

m(v,xn+)∈k

d(Fv,Gxn+),d(Fv,Sv) +d(Gxn+,Txn+),

d(Fv,Txn+) +d(Sv,Gxn+)

=kd(z,zn),d(z,Sv) +d(zn,zn+),d(z,zn+) +d(Sv,zn)

.

Therefore we have the following three cases:

(i) d(Sv,z)≤λkd(z,zn) +kd(zn+,z);

(ii) d(Sv,z)≤λk[d(z,Sv) +d(zn,zn+)] +kd(zn+,z);

(iii) d(Sv,z)≤λk[d(z,zn+) +d(Sv,zn)] +kd(zn+,z).

By the same argument as above, we getd(Sv,z) = , that is,Sv=Fv=z. So,zis also a point

of coincidence ofSandF.

Theorem . In addition to the hypothesis of Theorem.,suppose that{S,F}and{T,G} are weakly compatible pairs,then F,G,S,and T have a common ﬁxed point.

Proof From Theorem ., we get that the mappings{S,F}and{T,G}have a point of coin-cidence inz∈X. Now we prove thatzis a unique point of coincidence of pairs{S,F}and {T,G}. Suppose that there existsz∗∈Xsuch thatFv∗=Gy∗=Sv∗=Ty∗=z∗, from (.),

dz,z∗=dSv,Ty∗≤λ km

v,y∗, (.)

where

mv,y∗∈kdFv,Gy∗,d(Fv,Sv) +dGy∗,Ty∗,dFv,Ty∗+dSv,Gy∗

=kdz,z∗,dz,z∗+dz,z∗=kdz,z∗, dz,z∗.

Using (C), we getd(z,z∗) = , that is,z=z∗. Therefore,zis a unique point of coincidence of the pairs{S,F}and{T,G}. That is,Sv=Fv=Ty=Gy=z.

The weak compatibility of the pair{S,F}impliesSFv=FSv=SSv=FFv=Sz=Fz. Now we show thatzis a common ﬁxed point ofSandF.

Consider

d(Sz,z) =d(Sz,Ty)≤λ km(z,y),

where

m(z,y)∈kd(Fz,Gy),d(Fz,Sz) +d(Gy,Ty),d(Fz,Ty) +d(Sz,Gy)

=kd(Sz,z),d(Sz,z) +d(Sz,z)

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Using (C), we getSz=z=Fz. Similarly, the weak compatibility of the pair{T,G}implies

Tz=z=Gz. HenceSz=Fz=Tz=Gz=z.

Remark . Theorem . is a generalization of Theorem . of [].

If we chooseT=SandG=Fin Theorems . and ., we get the following results for two mappings on a cone metric type space.

Corollary . Let(X,d)be a cone metric type space with the constant k≥and let P be a solid cone.Suppose that the self-mappings F,S:X→X are such that SX⊂FX.Let x∈X, let the sequences{xn}and{zn}be deﬁned by zn=Fxn+=Sxnand zn+=Fxn+=Sxn+ for n≥,and letO(x) ={zn:n≥}.Assume that there exists x∈X such that for some

constantλ∈(,_k₊_k),

d(Sx,Sy)≤λ km(x,y),

where

m(x,y)∈kd(Fx,Fy),d(Fx,Sx) +d(Fy,Sy),d(Fx,Sy) +d(Sx,Fy)

for all x,y∈F–_{_O₍_x₎_{} ∪}_S–_{_O₍_x₎_}_._{If one of SX or FX is a complete subspace of X}_,_then S and F have a unique point of coincidence in X.Moreover,if{S,F}is a weakly compatible pair,then S and F have a common ﬁxed point.

The following example shows that Theorem . is diﬀerent from Theorem . of Cvetkovićet al.[].

Example . LetX={, , }, and letd(x,y) =|x–y|for eachx,y∈X. LetT:X→Xbe given byT = ,T =  andT = . Let S=T and letF=G=I. We ﬁrst show that we cannot invoke Theorem . of [] to show the existence of a ﬁxed point forT. On the contrary, assume that there existsλ∈(, ) such that

|Tx–Ty| ≤λmax

|x–y|,|x–Tx|,|y–Ty|,|x–Ty|+|y–Tx| 

for eachx,y∈X. Letx=  andy= . Then, from the above, we get ≤λ, a contradiction. Now letx= . ThenO(x) ={, }and soF–_{_O₍_x₎_{} ∪}_G–_{_O₍_x₎_}₌_{_{, }_}_{. Since for} eachx,y∈ {, },Tx=Ty, then the assumptions of Theorem . are satisﬁed by anyλ∈ (,_). ThusThas a ﬁxed point.

4 Generalized nonexpansive maps

The aim of this section is to present coincidence points results for four mappings without satisfying the condition of continuity and commutation on cone metric type spaces with a non-normal cone. Common ﬁxed point theorems are obtained under the weak compatible condition. Our results generalize and unify main results in [, ] and many others.

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Theorem . Let(X,d)be a cone metric type space with the constant k≥and let P be a

solid cone.Suppose that the mappings F,G,S,T:X→X are such that SX⊂GX,TX⊂FX satisfying

d(Sx,Ty)≤ad(Fx,Gy) +bd(Fx,Sx) +d(Gy,Ty)+cd(Fx,Ty) +d(Gy,Sx) (.)

for all x,y∈X,where a,b and c(= )are nonnegative constants such that b+ (a+b+ c)k+ck_{< .}_{If one of SX}_,_TX_,_FX_,_{or GX is a complete subspace of X}_, _then_{S_,_F}_and {T,G}have a unique point of coincidence in X.Moreover,if{S,F}and{T,G}are weakly compatible pairs,then F,G,S,and T have a unique common ﬁxed point.

Proof We deﬁne sequences{xn}and{zn}as in the proof of Theorem .. We prove that

{zn}is a Cauchy sequence by considering the cases whennis odd and even, respectively.

Forn= l+ ,l∈N, we have

d(zl+,zl+) =d(Sxl+,Txl+)

≤ad(Fxl+,Gxl+) +b

d(Fxl+,Sxl+) +d(Gxl+,Txl+)

+cd(Fxl+,Txl+) +d(Gxl+,Sxl+)

=ad(zl+,zn) +b

d(zl+,zl+) +d(zl,zl+)

+cd(zl+,zl+) +d(zl,zl+)

= (a+b)d(zl+,zn) +cd(zl,zl+) +bd(zl+,zl+)

≤(a+b)d(zl+,zn) +ck

d(zl,zl+) +d(zl+,zl+) +bd(zl+,zl+)

= (a+b+kc)d(zl+,zn) + (b+ck)d(zl+,zl+).

Therefore

d(zl+,zl+)≤

a+b+ck  –b–ck

d(zl+,zn). (.)

Similarly, forn= l,l∈N, we have

d(zl,zl+) =d(Sxl,Txl+)

≤ad(Fxl,Gxl+) +b

d(Fxl,Sxl) +d(Gxl+,Txl+)

+cd(Fxl,Txl+) +d(Gxl+,Sxl)

=ad(zl–,zn) +b

d(zl–,zl) +d(zl,zl+)

+cd(zl–,zl+) +d(zl,zl)

= (a+b)d(zl–,zn) +bd(zl,zl+) +cd(zl–,zl+)

≤(a+b)d(zl–,zn) +bd(zl,zl+)

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Thus,

d(zl,zl+)≤

a+b+ck  –b–ck

d(zl–,zn).

Let α = a_–+b_b+_–ck_ck. Clearly α< . Hence, inequality (.) holds for both the cases. By the same arguments as in Theorem ., we conclude that {zn} is a Cauchy sequence. Let

us suppose that SX is a complete subspace of X, then there exists z∈SX such that

lim_n_→∞zn=limn→∞Sxn=z. Then we have

lim

n→∞Gxn+=nlim→∞Sxn=nlim→∞Fxn=nlim→∞Txn+=z, (.)

that is, for any c, for suﬃcient largen, we haved(zn,z)c. Sincez∈SX⊂GX, then

there existsy∈Xsuch thatz=Gy. We prove thatz=Ty. From (d) and (.), we have

d(Ty,z)≤kd(Sxn,Ty) +d(Sxn,z)

≤kad(Fxn,Gy) +b

d(Fxn,Sxn) +d(Gy,Ty)

+cd(Fxn,Ty) +d(Gy,Sxn) +kd(Sxn,z)

=kad(zn–,z) +b

d(zn–,zn) +d(z,Ty)

+cd(zn–,Ty) +d(z,zn) +kd(zn,z)

≤kad(zn–,z) +b

d(zn–,zn) +d(z,Ty)

+ckd(zn–,z) +d(z,Ty) +d(z,zn) +kd(zn,z)

=k(a+kc)d(zn–,z) + (c+ )d(z,zn) + (b+ck)d(z,Ty) +bd(zn–,zn).

Since{zn}converges toz, so for eacht∈intP, there existsn∈Nsuch that for everyn>n,

d(Ty,z)≤

k  –bk–ck

(a+kc)d(zn–,z) + (c+ )d(z,zn) +bd(zn–,zn)

k  –bk–ck

(a+kc)t 

( –bk–ck) k(a+kc)

+(c+ )t 

( –bk–ck₎ k(c+ ) +

bt 

( –bk–ck₎ bk

= t.

Now, by (C) it follows thatd(z,Ty) = , that is,Ty=z. So, we haveTy=Gy=z, that is,z is a point of coincidence ofTandG.

SinceTX⊂FX, there existsu∈Xsuch thatz=Fu. From (d) and (.), we have

d(Su,z)≤kd(Su,Txn+) +d(Txn+,z)

≤kad(Fu,Gxn+) +b

d(Fu,Su) +d(Gxn+,Txn+)

+cd(Fu,Txn+) +d(Gxn+,Su)

+d(Txn+,z)

=kad(z,zn) +b

d(z,Su) +d(zn,zn+)

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+cd(z,zn+) +d(zn,Su)

+d(zn+,z)

≤kad(z,zn) +b

d(z,Su) +d(zn,zn+)

+cd(z,zn+) +k

d(zn,z) +d(z,Su) +d(zn+,z),

so by the same arguments as above, we haved(Su,z) = , that is,Su=Fu=Ty=Gy=z. Suppose thatz∗ is another point of coincidence of these four mappings, that is,Fu∗= Su∗=Gy∗=Ty∗=z∗. From (.) we get

dz,z∗=dSu,Ty∗

≤adFu,Gy∗+bd(Fu,Su) +dGy∗,Ty∗+cdFu,Ty∗+dGy∗,Su ≤(a+ ck)dz,z∗,

because of (C), we getz=z∗. Therefore,zis the unique point of coincidence ofS,F,G andT. The weak compatibility of the pair{S,F}impliesSFu=FSu=SSu=FFu=Sz=Fz, and the weak compatibility of the pair{T,G}givesGTy=TGy=GGy=TTy=Gz=Tz. Consider

d(z,Sz) =d(Ty,Sz)

≤ad(Fz,Gy) +bd(Fz,Sz) +d(Gy,Ty)+cd(Fz,Ty) +d(Gy,Sz)

= (a+ c)d(Sz,z).

Sinceb+ (a+b+c)k+ck<  and using (C), we getSz=Fz=z; similarly we haveTz= Gz=z. This shows thatzis a common ﬁxed point ofG,T,FandS. The proof for the cases in whichFXorGXorTXis complete are similar. Hence the theorem follows.

Choosingk=  in Theorem ., we obtain the following generalized form of the results in [] and [].

Corollary . Let(X,d)be a cone metric space and P be a solid cone.Suppose that the mappings F,G,S,T:X→X are such that SX⊂GX,TX⊂FX and that there exist non-negative constants a,b andc satisfying

a+ b+ c< 

such that for all x,y∈X,

d(Sx,Ty)≤ad(Fx,Gy) +bd(Fx,Sx) +d(Gy,Ty)+cd(Fx,Ty) +d(Gy,Sx).

If one of SX,TX,FX,or GX is a complete subspace of X,then{S,F}and{T,G}have a unique point of coincidence in X.Moreover,if{S,F}and{T,G}are weakly compatible pairs,then F,G,S,and T have a unique common ﬁxed point.

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Corollary . Let(X,d)be a cone metric type space with the constant k≥and let P be a solid cone.Suppose that the mappings G,T:X→X are such that TX⊂GX and that there exist nonnegative constants a,b andc,satisfying b+ (a+b+c)k+ck_{< ,}_{such that for all} x,y∈X,

d(Sx,Ty)≤ad(Gx,Gy) +bd(Gx,Tx) +d(Gy,Ty)+cd(Gx,Ty) +d(Gy,Tx).

If one of TX or GX is a complete subspace of X,then G and T have a unique point of coincidence in X.Moreover,if{T,G}is a weakly compatible pair,then G and T have a unique common ﬁxed point.

Now, we use the following lemma, that is a consequence of the axiom of choice, to obtain coincidence point results for two self-mappings deﬁned on a subset of a partially ordered cone metric type space (see [, ]).

Lemma . Let X be a nonempty set and let g:X→X be a mapping.Then there exists a subset E⊆X such that g(E) =g(X)and g:E→X is one-to-one.

The following result contains properly the recent main result (Theorem .) of Shi and Xu []. Here we shall establish this result in the set up of a partially ordered cone metric type space relative to a solid coneP.

Theorem .(Theorem . of []) Let(X,d)be a cone metric type space with the con-stant k≥and let P be a cone having a nonempty interior.Suppose that the mappings f,g:X→X are such that fX⊆gX and fX or gX is a complete subspace of X and that there exist nonnegative constants a,a,a,a,a∈[, )withka+ (k+ )(a+a) + (k+k)(a+ a) < such that,for each x,y∈X,

d(fx,fy)≤ad(gx,gy) +ad(gx,fx) +ad(gy,fy) +ad(gx,fy) +ad(gy,fx).

Then f and g have a unique point of coincidence in X.Moreover,if f and g are OWC,f and g have a unique common ﬁxed point.

Recall that if (X,) is a partially ordered set andf,g:X→Xis such that forx,y∈X, xyimpliesfxfy, then the mappingf is said to be nondecreasing. Further, forx,y∈X, gxgyimpliesfxfy, then the mappingf is said to beg-nondecreasing [].

The following result extends Theorem . and corresponding results in Altunet al.[, ] and many others (see also Lemma . []).

Theorem . Let(X,d,)be a partially ordered cone metric type space relative to a solid cone P with the constant k≥.Suppose that the mappings f,g:X→X are such that f is g-nondecreasing mapping w.r.t.,fX⊆gX and gX is a complete subspace of X.Suppose that the following assertions hold:

(a) f is continuous orXhas the following property:if a nondecreasing sequence {gxn} →gx,thengxngxfor alln≥;

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(c) for eachx,y∈Xwithgygx,we have

d(fx,fy)≤ad(gx,gy) +ad(gx,fx) +ad(gy,fy)

+ad(gx,fy) +ad(gy,fx). (.)

If there exists x∈X such that g(x)f(x),then f and g have a coincidence point in X.

Proof Using Lemma ., there existsE⊆Xsuch thatg(E) =g(X) andg:E→Xis one-to-one. We deﬁne a mappingG:g(E)→g(E) by

G(gx) =f(x) (.)

for allgx∈g(E). Asgis one-to-one ong(E) =g(X) andf(X)⊆g(X), soGis well deﬁned. Thus, it follows from (.) and (.) that

d(Ggx,Ggy) =d(fx,fy)≤ad(gx,gy) +ad(gx,fx) +ad(gy,fy) +ad(gx,fy) +ad(gy,fx)

for allgx,gy∈g(X) for whichg(y)g(x). Sincef is ag-nondecreasing mapping, for all gygx∈g(X), it impliesfyfx, which givesGgyGgx. ThusGis a nondecreasing map-ping ong(X). Also, there existsx∈Xsuch thatg(x)f(x) =G(gx). Suppose that the assumption (a) holds. Sincef is continuous,Gis also continuous. Using Theorem . [] to the mappingG, it follows thatGhas a ﬁxed pointu∈g(X).

Suppose that the assumption (b) holds. We conclude similarly from Theorem . [] that the mappingGhas a ﬁxed pointu∈g(X). Finally, we prove thatf andghave a coincidence point. Sinceuis a ﬁxed point ofG, we getu=Gu. Sinceu∈g(X), there exists a point u∈Xsuch thatu=gu. It follows thatgu=u=Gu=Ggu=fu. Thus,uis a required coincidence point off andg. This completes the proof.

5 Application to the existence of solutions of integral equations

Fixed point theorems for operators in ordered Banach spaces are widely investigated and have found various applications in differential and integral equations (see [, ] and refer-ences therein). LetX=C([;T];R) be the set of real continuous functions defined on [;T] and letd:X×X→[; +∞) be defined byd(x,y) =sup_t_∈_[,_T_]|x(t) –y(t)|_{. Then (}_X_,_d_{) is a} complete metric type space with the constantk= .

Consider the integral equation

Fu(t) =p(t) +

T



G(t,s)fs,u(s)ds, (.)

and letS:X→Xbe deﬁned by

(Su)(t) =p(t) +

T



G(t,s)fs,u(s)ds for eacht∈[,T]. (.)

We assume that

(i) p: [,T]→Ris continuous;

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(iii) |f(s,u(s)) –f(s,v(s))| ≤μmax{|Fu(s) –Fv(s)|,|Fu(s) –Su(s)|+|Fv(s) –Sv(s)|, |Fu(s) –Sv(s)|+|Su(s) –Fv(s)|}for eachu,v∈Xands∈[,T], whereμis a constant such thatTμ_sup

t∈[,T]

T

 |G(t,s)|ds<  .

Theorem . Under the assumptions(i)-(iii),the integral equation(.)has a solution in X=C([;T];R).

Proof Consider the mappingS:X→Xdefined by (.). Notice first that the existence of a solution for the integral equation (.) is equivalent to the existence of a common fixed point for the mappingsFandS. For eachu,v∈X, we have

d(Su,Sv) = sup

t∈[,T]

Su(t) –Sv(t)

≤ sup

t∈[,T]

T



G(t,s)fs,u(s)–fs,v(s)ds



≤ sup

t∈[,T]

T



G(t,s)ds

T



fs,u(s)–fs,v(s)ds

≤μ sup

t∈[,T]

T



G(t,s)ds

×

T



maxFu(s) –Fv(s),Fu(s) –Su(s)+Fv(s) –Sv(s),

Fu(s) –Sv(s)+Su(s) –Fv(s)ds

≤Tμ sup

t∈[,T]

T



G(t,s)ds

×maxd(Fu,Fv),d(Fu,Su) +d(Fv,Sv),d(Fu,Sv) +d(Su,Fv).

Hence all the hypotheses of Corollary . are satisﬁed, and so the mappingsFandShave a common ﬁxed point that is a solution inX=C([;T];R) of the integral equation (.).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, King Abdul Aziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.}2_{Department of}

Mathematics, Lahore University of Management Sciences, Lahore, 54792, Pakistan.3_{Department of Mathematics,}

University of Shahrekord, Shahrekord, 88186-34141, Iran.4_{Department of Mathematics, The Islamia University of}

Bahawalpur, Bahawalpur, 63100, Pakistan.

Acknowledgements

The authors would like to express their thanks to the referees for their helpful comments and suggestions. The ﬁrst author gratefully acknowledges the support from the Deanship of Scientiﬁc Research (DSR) at King Abdulaziz University (KAU) during this research. The third author was partially supported by the Center of Excellence for Mathematics, University of Shahrekord, Iran.

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