Fixed point results for cyclic (ψ,ϕ,A,B)-contraction in partial metric spaces

R E S E A R C H

Open Access

Fixed point results for cyclic

(ψ

,

φ,

A

,

B

)

-contraction in partial metric

spaces

Wasfi Shatanawi

_{and Saurabh Manro}

*_{Correspondence:} [email protected] 2_{School of Mathematics and} Computer Applications, Thapar University, Patiala, Punjab, India Full list of author information is available at the end of the article

Abstract

Very recently, Agarwalet al.(Fixed Point Theory Appl. 2012:40, 2012) initiated the study of fixed point theorems for mappings satisfying cyclical generalized contractive conditions in complete partial metric spaces. In the present paper, we study some fixed point theorems for a mapping satisfying a cyclical generalized contractive condition based on a pair of altering distance functions in complete partial metric spaces. Also, we introduce an example and an application to support the usability of our paper.

MSC: Primary 54H25; secondary 47H10

Keywords: partial metric spaces; fixed point; altering distance function; cyclic map

1 Introduction

The existence and uniqueness of fixed and common fixed point theorems of operators has been a subject of great interest since Banach [] proved the Banach contraction principle in . Many authors generalized the Banach contraction principle in various spaces such as quasi-metric spaces, generalized metric spaces, cone metric spaces and fuzzy metric spaces. Matthews [] introduced the notion of partial metric spaces in such a way that each object does not necessarily have to have a zero distance from itself and proved a modified version of the Banach contraction principle. Afterwards, many authors proved many existing fixed point theorems in partial metric spaces (see [–] for examples).

We recall below the definition of partial metric space and some of its properties.

Definition [] A partial metric on a nonempty setXis a functionp:X×X→R+such that for allx,y,z∈X:

(p) x=y⇐⇒p(x,x) =p(x,y) =p(y,y), (p) p(x,x)≤p(x,y),

(p) p(x,y) =p(y,x),

(p) p(x,y)≤p(x,z) +p(z,y) –p(z,z).

A partial metric space is a pair (X,p) such thatXis a nonempty set andpis a partial metric onX. It is clear that, ifp(x,y) = , then from (p) and (p),x=y. But ifx=y,p(x,y)

may not be . The functionp(x,y) =max{x,y} for allx,y∈R+ _{defines a partial metric}

onR+_.

Each partial metric ponXgenerates a Ttopologyτp onX which has as a base the family of openp-balls{Bp(x,ε) :x∈X,ε> }, whereBp(x,ε) ={y∈X:p(x,y) <p(x,x) +ε} for allx∈Xandε> .

Ifpis a partial metric onX, then the functiondp:X×X→R+given by

dp(x,y) = p(x,y) –p(x,x) –p(y,y)

is a metric onX.

Definition  Let (X,p) be a partial metric space. Then

() A sequence{xn}in a partial metric space(X,p)converges to a pointx∈Xif and

only ifp(x,x) =limn→∞p(x,xn).

() A sequence{xn}in a partial metric space(X,p)is called a Cauchy sequence iff

limn,m→∞p(xn,xm)exists (and is finite).

() A partial metric space(X,p)is said to be complete if every Cauchy sequence{xn}in Xconverges, with respect toτp, to a pointx∈Xsuch that

p(x,x) =limn,m→∞p(xn,xm).

() A subsetAof a partial metric space(X,p)is closed if whenever{xn}is a sequence in Asuch that{xn}converges to somex∈X, thenx∈A.

Remark  The limit in a partial metric space is not unique.

Lemma ([, ]) Let(X,p)be a partial metric space.

(a) {xn}is a Cauchy sequence in(X,p)if and only if it is a Cauchy sequence in the metric space(X,dp).

(b) A partial metric space(X,p)is complete if and only if the metric space(X,dp)is complete.Furthermore,lim_n→∞dp(xn,x) = if and only if

p(x,x) = lim

n→∞p(xn,x) =n,mlim→∞p(xn,xm).

Now, we define the cyclic map.

Definition  LetAandBbe nonempty subsets of a metric space (X,d) andT:A∪B→ A∪B. ThenTis called a cyclic map ifT(A)⊆BandT(B)⊆A.

In , Kirket al.[] gave the following fixed point theorem for a cyclic map.

Theorem [] Let A and B be nonempty closed subsets of a complete metric space(X,d). Suppose that T:A∪B→A∪B is a cyclic map such that

d(Tx,Ty)≤kd(x,y) ∀x∈A,∀y∈B.

If k∈[, ),then T has a unique fixed point in A∩B.

Definition  [] Let A andB be nonempty closed subsets of a metric space (X,d). A cyclic mapT:A∪B→A∪Bis said to be a Kannan type cyclic contraction if there existsk∈(,_) such that

d(Tx,Ty)≤kd(Tx,x) +d(Ty,y) ∀x∈A,∀y∈B.

Definition  [] Let A and B be nonempty closed subsets of a metric space (X,d). A cyclic mapT:A∪B→A∪Bis said to be a Reich type cyclic contraction if there exists k∈(,_) such that

d(Tx,Ty)≤kd(x,y) +d(Tx,x) +d(Ty,y) ∀x∈A,∀y∈B.

Definition  [] Let A andB be nonempty closed subsets of a metric space (X,d). A cyclic mapT:A∪B→A∪Bis said to be a Ćirić type cyclic contraction if there exists k∈(,_) such that

d(Tx,Ty)≤kmaxd(x,y),d(Tx,x),d(Ty,y) ∀x∈A,∀y∈B.

Moreover, Karapınar and Erhan [] obtained the following results:

Theorem [] Let A and B be nonempty closed subsets of a complete metric space(X,d), and let T:A∪B→A∪B be a Kannan type cyclic contraction.Then T has a unique fixed point in A∩B.

Theorem [] Let A and B be nonempty closed subsets of a complete metric space(X,d), and let T :A∪B→A∪B be a Reich type cyclic contraction.Then T has a unique fixed point in A∩B.

Theorem [] Let A and B be nonempty closed subsets of a complete metric space(X,d), and let T:A∪B→A∪B be a Ćirić type cyclic contraction.Then T has a unique fixed point in A∩B.

For more results on cyclic contraction mappings, see [, ].

Very recently, Agarwalet al.[] initiated the study of fixed point theorems for mappings satisfying cyclical generalized contractive conditions in complete partial metric spaces.

Khanet al.[] introduced the notion of altering distance function as follows.

Definition  (Altering distance function []) The function φ: [, +∞)→[, +∞) is called an altering distance function if the following properties are satisfied:

() φis continuous and nondecreasing. () φ(t) = if and only ift= .

For some work on altering distance function, we refer the reader to [–].

2 Main result

We start with the following definition.

Definition  Let (X,p) be a partial metric space andA,Bbe nonempty closed subsets ofX. A mappingT:X→Xis called a cyclic (ψ,φ,A,B)-contraction if

() ψandφare altering distance functions;

() A∪Bhas a cyclic representation w.r.t.T; that is,T(A)⊆BandT(B)⊆A; and ()

ψp(Tx,Ty)≤ψ

max

p(x,y),p(x,Tx),p(y,Ty), 

p(x,Ty) +p(Tx,y)

–φmaxp(x,y),p(y,Ty) (.)

for allx∈Aandy∈B.

From now on, byψandφwe mean altering distance functions unless otherwise stated.

In the rest of this paper,Nstands for the set of nonnegative integer numbers.

Theorem  Let A and B be nonempty closed subsets of a complete partial metric space (X,p).If T:X→X is a cyclic (ψ,φ,A,B)-contraction,then T has a unique fixed point u∈A∩B.

Proof Letx∈A. SinceTA⊆B, we choosex∈Bsuch thatTx=x. Also, sinceTB⊆A,

we choosex∈Asuch thatTx=x. Continuing this process, we can construct sequences {xn}inXsuch thatxn∈A,xn+∈B,xn+=Txnandxn+=Txn+. Ifxn+=xn+for

somen∈N, thenxn+=Txn+. Thus,xn+is a fixed point ofTinA∩B. Thus, we may

assume thatxn+=xn+for alln∈N.

Givenn∈N. Ifnis even, thenn= tfor somet∈N. By (.), we have

ψp(xn+,xn+)

=ψp(xt+,xt+)

=ψp(Txt,Txt+)

≤ψ

max

p(xt,xt+),p(Txt,xt),p(Txt+,xt+),

 

p(xt,Txt+) +p(Txt,xt+)

–φmaxp(xt,xt+),p(Txt+,xt+)

=ψ

max

p(xt,xt+),p(xt+,xt+),

 

p(xt,xt+) +p(xt+,xt+)

–φmaxp(xt,xt+),p(xt+,xt+)

By (p), we have

ψp(xn+,xn+)

=ψp(xt+,xt+)

≤ψ

max

p(xt,xt+),p(xt+,xt+),

 

p(xt,xt+) +p(xt+,xt+)

–φmaxp(xt,xt+),p(xt+,xt+)

≤ψmaxp(xt,xt+),p(xt+,xt+)

–φmaxp(xt,xt+),p(xt+,xt+)

≤ψmaxp(xt,xt+),p(xt+,xt+)

.

If

maxp(xt,xt+),p(xt+,xt+)

=p(xt+,xt+),

then

ψp(xt+,xt+)

≤ψp(xt+,xt+)

–φp(xt+,xt+)

.

Therefore, φ(p(xt+,xt+)) = , and hencep(xt+,xt+) = . By (p) and (p), we have

xt+=xt+, which is a contradiction. Therefore,

maxp(xt,xt+),p(xt+,xt+)

=p(xt,xt+).

Hence,

p(xn+,xn+) =p(xt+,xt+)≤p(xt,xt+) =p(xn,xn+) (.)

and

ψp(xn+,xn+)

≤ψp(xn,xn+)

–φp(xn,xn+)

. (.)

Ifnis odd, thenn= t+  for somet∈N. By (.), we have

ψp(xn+,xn+)

=ψp(xt+,xt+)

=ψp(xt+,xt+)

=ψp(Txt+,Txt+)

≤ψ

max

p(xt+,xt+),p(Txt+,xt+),p(Txt+,xt+),

 

p(xt+,Txt+) +p(Txt+,xt+)

=ψ

max

p(xt+,xt+),p(xt+,xt+),

 

p(xt+,xt+) +p(xt+,xt+)

–φmaxp(xt+,xt+),p(xt+,xt+)

.

By (p), we have

ψp(xn+,xn+)

=ψp(xt+,xt+)

≤ψ

max

p(xt+,xt+),p(xt+,xt+),

 

p(xt+,xt+) +p(xt+,xt+)

–φmaxp(xt+,xt+),p(Txt+,xt+)

≤ψmaxp(xt+,xt+),p(xt+,xt+)

–φmaxp(xt+,xt+),p(Txt+,xt+)

≤ψmaxp(xt+,xt+),p(xt+,xt+)

.

If

maxp(xt+,xt+),p(xt+,xt+)

=p(xt+,xt+),

then

φp(xt+,xt+)

≤ψp(xt+,xt+)

–φp(xt+,xt+)

.

Therefore, φ(p(xt+,xt+)) = , and hencep(xt+,xt+) = . By (p) and (p), we have

xt+=xt+, which is a contradiction. Therefore,

maxp(xt+,xt+),p(xt+,xt+)

=p(xt+,xt+).

Hence,

p(xn+,xn+) =p(xt+,xt+)≤p(xt+,xt+) =p(xn+,xn), (.)

ψp(xn+,xn+)

≤ψp(xn,xn+)

–φp(xn,xn+)

. (.)

From (.) and (.), we have{p(xn+,xn) :n∈N}is a nonincreasing sequence and hence there existsr≥ such that

lim

n→+∞p(xn,xn+) =r.

Also, from (.) and (.), we have

ψp(xn+,xn+)

≤ψp(xn,xn+)

–φp(xn,xn+)

∀n∈N. (.)

Lettingn→+∞in (.) and using the fact thatψandφare continuous, we get that

Therefore,φ(r) =  and hencer= . Thus

lim

n→+∞p(xn,xn+) = . (.)

By (p), we get that

lim

n→+∞p(xn,xn) = . (.)

Sincedp(x,y)≤p(x,y) for allx,y∈X, we get that

lim

n→+∞dp(xn,xn+) = . (.)

Next, we show that{xn}is a Cauchy sequence in the metric space (A∪B,dp). It is suf-ficient to show that{xn}is a Cauchy sequence in (A∪B,dp). Suppose the contrary; that is,{xn}is not a Cauchy sequence in (A∪B,dp). Then there exists>  for which we can find two subsequences{xm(i)}and{xn(i)}of{xn}such thatn(i) is the smallest index for which

n(i) >m(i) >i, dp(xm(i),xn(i))≥. (.)

This means that

dp(xm(i),xn(i)–) <. (.)

From (.), (.) and the triangular inequality, we get that

≤dp(xm(i),xn(i))

≤dp(xm(i),xn(i)–) +dp(xn(i)–,xn(i)–)

+dp(xn(i)–,xn(i))

<+dp(xn(i)–,xn(i)–) +dp(xn(i)–,xn(i)).

On lettingi→+∞in the above inequalities and using (.), we have

lim

i→+∞dp(xm(i),xn(i)) =. (.)

Again, from (.) and the triangular inequality, we get that

≤dp(xm(i),xn(i))

≤dp(xn(i),xn(i)–) +dp(xn(i)–,xm(i))

Lettingi→+∞in the above inequalities and using (.) and (.), we get that

lim

i→+∞dp(xm(i),xn(i)) =i→lim+∞dp(xm(i)+,xn(i)–)

= lim

i→+∞dp(xm(i)+,xn(i))

= lim

i→+∞dp(xm(i),xn(i)–)

=.

Since

dp(x,y) = p(x,y) –p(x,x) –p(y,y)

for allx,y∈X, then

lim

i→+∞p(xm(i),xn(i)) =i→lim+∞p(xm(i)+,xn(i)–)

= lim

i→+∞p(xm(i)+,xn(i))

= lim

i→+∞p(xm(i),xn(i)–)

= .

By (.), we have

ψp(xm(i)+,xn(i))

=ψp(Txm(i),Txn(i)–)

≤ψ

max

p(xm(i),xn(i)–),p(xm(i),Txm(i)),p(xn(i)–,Txn(i)–),

 

p(xm(i),Txn(i)–) +p(xn(i)–,Txm(i))

–φmaxp(xm(i),xn(i)–),p(xn(i)–,Txn(i)–)

=ψ

max

p(xm(i),xn(i)–),p(xm(i),xm(i)+),p(xn(i)–,xn(i)),

 

p(xm(i),xn(i)) +p(xn(i)–,xm(i)+)

–φmaxp(xm(i),xn(i)–),p(xn(i)–,xn(i))

.

Lettingi→+∞and using the continuity ofφandψ, we get that

ψ

 ≤ψ

 –φ

 .

Therefore, we get thatφ(

) = . Hence,=  is a contradiction. Thus{xn}is a Cauchy

sequence in (A∪B,dp). Since (X,p) is complete andA∪Bis a closed subspace of (X,p), then we have (A∪B,p) is complete. From Lemma , the sequence{xn}converges in the metric space (A∪B,dp), saylimn→∞dp(xn,u) = . Again from Lemma , we have

p(u,u) = lim

Moreover, since{xn}is a Cauchy sequence in the metric space (A∪B,dp), we have

lim

n,m→∞dp(xn,xm) = . (.)

From the definition ofdpwe have

dp(xn,xm) = p(xn,xm) –p(xn,xn) –p(xm,xm).

Lettingn,m→+∞in the above equality and using (.) and (.), we get

lim

n,m→∞p(xn,xm) = .

Thus by (.), we have

lim

n→+∞p(xn,u) =p(u,u) = . (.)

Sincep(xn,u)→ =p(u,u),{xn}is a sequence inA, andAis closed in (X,p), we have u∈A. Similarly, we haveu∈B, that isu∈A∩B. Again, from the definition ofp, we have

p(xn,Tu)≤p(xn,u) +p(u,Tu) –p(u,u)

≤p(xn,u) +p(u,xn) +p(xn,Tu) –p(xn,xn) –p(u,u).

Lettingn→+∞in the above inequalities and using (.) and (.), we get that

lim

n→+∞p(xn,Tu) =p(u,Tu).

Now, we claim thatTu=u.

Sincexn∈Aandu∈B, by (.) we have

ψp(xn+,Tu)

=ψp(Txn,Tu)

≤ψ

max

p(xn,u),p(Txn,xn),p(Tu,u),

 

p(xn,Tu) +p(u,Txn)

–φmaxp(xn,u),p(Tu,u)

=ψ

max

p(xn,u),p(xn,xn+),p(Tu,u),

 

p(xn,Tu) +p(u,xn+)

–φmaxp(xn,u),p(u,Tu)

.

Lettingn→+∞, we get that

ψp(u,Tu)≤ψp(u,Tu)–φp(u,Tu).

Therefore,uis a fixed point ofT. To prove the uniqueness of the fixed point, we letvbe any other fixed point ofTinA∩B. It is an easy matter to prove thatp(v,v) = . Now, we prove thatu=v. Sinceu∈A∩B⊆Aandv∈A∩B⊆B, we have

ψp(u,v)=ψp(Tu,Tv)

≤ψmaxp(u,v),p(u,u),p(v,v)–φmaxp(u,v),p(v,v)

=ψp(u,v)–φp(u,v).

Thusφ(p(u,v)) =  and hencep(u,v) = . Therefore,u=v.

Takingψ=I[,+∞)(the identity function) in Theorem , we have the following result.

Corollary  Let A and B be nonempty closed subsets of a complete partial metric space (X,p).Let T:X→X be a mapping such that A∪B has a cyclic representation w.r.t.T. Suppose there exists an altering distance functionφsuch that

p(Tx,Ty)≤max

p(x,y),p(x,Tx),p(y,Ty), 

p(x,Ty) +p(Tx,y)

–φmaxp(x,y),p(y,Ty)

for all x∈A and y∈B.Then T has a unique fixed point u∈A∩B.

Corollary  Let A and B be nonempty closed subsets of a complete partial metric space (X,p).Let T:X→X be a mapping such that A∪B has a cyclic representation w.r.t.T. Suppose there exists an altering distance functionφsuch that

p(Tx,Ty)≤maxp(x,y),p(x,Tx),p(y,Ty)–φmaxp(x,y),p(x,Tx),p(y,Ty)

for all x∈A and y∈B.Then T has a unique fixed point u∈A∩B.

Now, we introduce an example to support the usability of our results.

Example  LetX= [, ]. Define the partial metricponXby

p(x,y) =

⎧ ⎨ ⎩

, ifx=y;

max{x,y}, ifx=y.

Also, define the mapping T:X→X byT(x) = _+x_x and the functionsψ,φ: [, +∞)→ [, +∞) byψ(t) = tandφ(t) = t

+t. TakeA= [, 

] andB= [, ]. Then () (X,p)is a complete partial metric space.

() A∪Bhas a cyclic representation w.r.t.T. () For allx∈Aandy∈B, we have

ψp(Tx,Ty)≤ψ

max

p(x,y),p(x,Tx),p(y,Ty), 

p(x,Ty) +p(Tx,y)

Proof Note thatTA= [,_]⊆BandTB= [,_]⊆A. ThusA∪Bhas a cyclic representa-tion ofT. To prove (), givenx∈Aandy∈B, without loss of generality, we may assume thatx≤y. So,

ψp(Tx,Ty)=ψ

p

x

 +x, y

 +y

=ψ

y

 +y = y

 +y,

ψ

max

p(x,y),p(x,Tx),p(y,Ty), 

P(x,Ty) +p(Tx,y)

=ψ

max

y,p

x, x



 +x ,p

y, y



 +y ,  

p

x, y



 +y +p

x  +x,y

≤ψ(y) = y,

and

φmaxp(x,y),p(y,Ty)=φ

max

y,p

y, y



 +y

=φ(y) = y  + y.

Since

y

 +y≤y– y  + y,

we have

ψp(Tx,Ty)≤ψ

max

p(x,y),p(x,Tx),p(y,Ty), 

p(x,Ty) +p(Tx,y)

–φmaxp(x,y),p(y,Ty).

Note that Example  satisfies all the hypotheses of Theorem .

3 Application

Denote bythe set of functionsμ: [, +∞)→[, +∞) satisfying the following hypothe-ses:

(h) μis a Lebesgue-integrable mapping on each compact of[, +∞). (h) For every> , we have



μ(t)dt> .

Theorem  Let A and B be nonempty closed subsets of a complete partial metric space (X,p).Let T:X→X be a mapping such that A∪B has a cyclic representation w.r.t.T. Suppose that for x∈A and y∈B,we have

p(Tx,Ty)



μ(t)dt≤

max{p(x,y),p(x,Tx),p(y,Ty),_(p(x,Ty)+p(Tx,y))}



μ(t)dt

–

max{p(x,y),p(y,Ty)}



μ(t)dt,

Proof Follows from Theorem  by definingψ,φ: [, +∞)→[, +∞)viaψ(t) =_tμ(s)ds

andφ(t) =_tμ(s)dsand noting thatψ,φare altering distance functions.

Remark  Theorem . of [] is a special case of Corollary .

Remark  Theorem . of [] is a special case of Corollary .

Remark  Theorem . of [] is a special case of Corollary .

Remark  Theorem . of [] is a special case of Corollary .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the manuscript.

Author details

1_{Department of Mathematics, Hashemite University, P.O. Box 150459, Zarqa, 13115, Jordan.}2_{School of Mathematics and} Computer Applications, Thapar University, Patiala, Punjab, India.

Acknowledgements

The authors thank the Editor and the referees for their useful comments and suggestions.

Received: 2 July 2012 Accepted: 31 August 2012 Published: 28 September 2012 References

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

2. Matthews, SG: Partial metric topology. Proc. 8th Summer Conference on General Topology and Applications. Ann. N.Y. Acad. Sci.728, 183-197 (1994)

3. Abdeljawad, T, Karapinar, E, Ta¸s, K: Existence and uniqueness of a common fixed point on partial metric spaces. Appl. Math. Lett.24, 1900-1904 (2011)

4. Abdeljawad, T, Karapinar, E, Ta¸s, K: A generalized contraction principle with control functions on partial metric spaces. Comput. Math. Appl.6, 716-719 (2012)

5. Abdeljawad, T: Fixed points for generalized weakly contractive mappings in partial metric spaces. Math. Comput. Model.54, 2923-2927 (2011)

6. Altun, I, Erduran, A: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl. 2011, Article ID 508730 (2011)

7. Altun, I, Sola, F, Simsek, H: Generalized contractions on partial metric spaces. Topol. Appl.157, 2778-2785 (2010) 8. Aydi, H: Some fixed point results in ordered partial metric spaces. J. Nonlinear Sci. Appl4, 1-12 (2011)

9. Aydi, H: Some coupled fixed point results on partial metric spaces. Int. J. Math. Math. Sci.2011, Article ID 647091 (2011)

10. Aydi, H: Fixed point theorems for generalized weakly contractive condition in ordered partial metric spaces. J. Nonlinear Anal. Optim.2, 33-48 (2011)

11. Aydi, H, Karapinar, E, Shatanawi, W: Coupled fixed point results for (ψ,ϕ)-weakly contractive condition in ordered partial metric spaces. Comput. Math. Appl.62, 4449-4460 (2011)

12. Golubovi´c, Z, Kadelburg, Z, Radenovi´c, S: Coupled coincidence points of mappings in ordered partial metric spaces. Abstr. Appl. Anal.2012, Article ID 192581 (2012)

13. Heckmann, R: Approximation of metric spaces by partial metric spaces. Appl. Categ. Struct.7, 71-83 (1999) 14. Karapinar, E: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl. (2011, in

press)

15. Karapinar, E, Erhan, IM: Fixed point theorems for operators on partial metric spaces. Appl. Math. Lett. (2011). doi:10.1016/j.aml.2011.05.013

16. Nashine, HK, Kadelburg, Z, Radenovi´c, S: Common fixed point theorems for weakly isotone increasing mappings in ordered partial metric spaces. Math. Comput. Model. (2012, in press)

17. Oltra, S, Valero, O: Banach’s fixed point theorem for partial metric spaces. Rend. Ist. Mat. Univ. Trieste36, 17-26 (2004) 18. Romaguera, S: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl.2010,

Article ID 493298 (2010)

19. Samet, B, Rajovi´c, M, Lazovi´c, R, Stoiljkovi´c, R: Common fixed point results for nonlinear contractions in ordered partial metric spaces. Fixed Point Theory Appl.2011, 71 (2011)

20. Shatanawi, W, Nashine, HK: A generalization of Banach’s contraction principle for nonlinear contraction in a partial metric space. J. Nonlinear Sci. Appl5, 37-43 (2012)

22. Kirk, WA, Srinavasan, PS, Veeramani, P: Fixed points for mapping satisfying cyclical contractive conditions. Fixed Point Theory4, 79-89 (2003)

23. Karapinar, E, Erhan, IM: Best proximity point on different type contractions. Appl. Math. Inf. Sci.5, 342-353 (2011) 24. Karapinar, E, Erhan, IM, Ulus, AY: Fixed point theorem for cyclic maps on partial metric spaces. Appl. Math. Inf. Sci.6,

239-244 (2012)

25. Karapinar, E, Erhan, IM: Cyclic contractions and fixed point theorems. Filomat26, 777-782 (2012)

26. Agarwal, RP, Alghamdi, MA, Shahzad, N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl.2012, 40 (2012)

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

28. Cho, YJ, Rhoades, BE, Saadati, R, Samet, B, Shatanawi, W: Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type. Fixed Point Theory Appl.2012, 8 (2012). doi:10.1186/1687-1812-2012-8 29. Aydi, H, Postolache, M, Shatanawi, W: Coupled fixed point results for (ψ,φ)-weakly contractive mappings in ordered

G-metric spaces. Comput. Math. Appl.63, 298-309 (2012)

30. Lakzian, H, Samet, B: Fixed points for (ψ,φ)-weakly contractive mappings in generalized metric spaces. Appl. Math. Lett.25, 902-906 (2012)

31. Shatanawi, W, Al-Rawashdeh, A: Common fixed points of almost generalized (ψ,φ)-contractive mappings in ordered metric spaces. Fixed Point Theory Appl. (accepted)

32. Shatanawi, W, Mustafa, Z, Tahat, N: Some coincidence point theorems for nonlinear contraction in ordered metric spaces. Fixed Point Theory Appl.2011, 68 (2011)

33. Shatanawi, W, Samet, B: On (ψ,φ)-weakly contractive condition in partially ordered metric spaces. Comput. Math. Appl.62, 3204-3214 (2011)

doi:10.1186/1687-1812-2012-165