Modified α-ψ-contractive mappings with applications

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R E S E A R C H

Open Access

Modiﬁed

α

-ψ

-contractive mappings with

applications

Peyman Salimi

1

_{, Abdul Latif}

2*

_{and Nawab Hussain}

2

*_{Correspondence: [email protected]} 2_{Department of Mathematics, King}

Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article

Abstract

The aim of this work is to modify the notions of

α

-admissible and

α

-

ψ

-contractive mappings and establish new ﬁxed point theorems for such mappings in complete metric spaces. Presented theorems provide main results of Karapinar and Samet (Abstr. Appl. Anal. 2012:793486, 2012) and Sametet al.(Nonlinear Anal. 75:2154-2165, 2012) as direct corollaries. Moreover, some examples and applications to integral equations are given here to illustrate the usability of the obtained results. MSC: 46N40; 47H10; 54H25; 46T99

1 Introduction and preliminaries

Metric fixed point theory has many applications in functional analysis. The contractive conditions on underlying functions play an important role for finding solutions of metric fixed point problems. The Banach contraction principle is a remarkable result in met-ric fixed point theory. Over the years, it has been generalized in different directions by several mathematicians (see [–]). In , Sametet al.[] introduced the concepts ofα-ψ-contractive andα-admissible mappings and established various fixed point theo-rems for such mappings in complete metric spaces. Afterwards Karapinar and Samet [] generalized these notions to obtain fixed point results. The aim of this paper is to mod-ify further the notions ofα-ψ-contractive andα-admissible mappings and establish fixed point theorems for such mappings in complete metric spaces. Our results are proper gen-eralizations of the recent results in [, ]. Moreover, some examples and applications to integral equations are given here to illustrate the usability of the obtained results.

Denote withthe family of nondecreasing functionsψ: [, +∞)→[, +∞) such that

∞

n=ψn(t) < +∞for allt> , whereψnis thenth iterate ofψ. The following lemma is obvious.

Lemma . Ifψ∈,thenψ(t) <t for all t> .

Deﬁnition .[] LetT be a self-mapping on a metric space (X,d) and letα:X×X→

[, +∞) be a function. We say thatTis anα-admissible mapping if

x,y∈X, α(x,y)≥ ⇒ α(Tx,Ty)≥.

Deﬁnition .[] LetT be a self-mapping on a metric space (X,d). We say thatT is anα-ψ-contractive mapping if there exist two functionsα:X×X→[, +∞) andψ∈

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such that

α(x,y)d(Tx,Ty)≤ψd(x,y)

for allx,y∈X.

For the examples ofα-admissible andα-ψ-contractive mappings, see [, ] and the examples in the next section.

2 Main results

We ﬁrst modify the concept ofα-admissible mapping.

Deﬁnition . LetT be a self-mapping on a metric space (X,d) and letα,η:X×X→

[, +∞) be two functions. We say thatTis anα-admissible mapping with respect toηif

x,y∈X, α(x,y)≥η(x,y) ⇒ α(Tx,Ty)≥η(Tx,Ty).

Note that if we takeη(x,y) = , then this deﬁnition reduces to Deﬁnition .. Also, if we takeα(x,y) = , then we say thatTis anη-subadmissible mapping.

Our ﬁrst result is the following.

Theorem . Let(X,d)be a complete metric space and let T be anα-admissible mapping with respect toη.Assume that

x,y∈X, α(x,y)≥η(x,y) ⇒ d(Tx,Ty)≤ψM(x,y), (.)

whereψ∈and

M(x,y) =max

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

 ,

d(x,Ty) +d(y,Tx) 

.

Also,suppose that the following assertions hold:

(i) there existsx∈Xsuch thatα(x,Tx)≥η(x,Tx);

(ii) eitherTis continuous or for any sequence{xn}inXwithα(xn,xn+)≥η(xn,xn+)for alln∈N∪ {}andxn→xasn→+∞,we haveα(xn,x)≥η(xn,x)for all

n∈N∪ {}. Then T has a ﬁxed point.

Proof Letx∈X be such thatα(x,Tx)≥η(x,Tx). Deﬁne a sequence{xn} inX by

xn=Tnx=Txn–for alln∈N. Ifxn+=xnfor somen∈N, thenx=xnis a ﬁxed point forT and the result is proved. Hence, we suppose thatxn+=xnfor alln∈N. SinceT is a generalizedα-admissible mapping with respect toηandα(x,Tx)≥η(x,Tx), we de-duce thatα(x,x) =α(Tx,Tx)≥η(Tx,Tx) =η(x,x). Continuing this process, we getα(xn,xn+)≥η(xn,xn+) for alln∈N∪ {}. Now, by (.) withx=xn–,y=xn, we get

d(Txn–,Txn)≤ψ

M(xn–,xn)

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On the other hand,

M(xn–,xn) =max

d(xn–,xn),

d(xn–,Txn–) +d(xn,Txn)

 ,

d(xn–,Txn) +d(xn,Txn–) 

=max

d(xn–,xn),

d(xn–,xn) +d(xn,xn+)

 ,

d(xn–,xn+) 

≤max

d(xn–,xn),

d(xn–,xn) +d(xn,xn+) 

≤maxd(xn–,xn),d(xn,xn+)

,

which implies

d(xn,xn+)≤ψ

maxd(xn–,xn),d(xn,xn+)

.

Now, ifmax{d(xn–,xn),d(xn,xn+)}=d(xn,xn+) for somen∈N, then

d(xn,xn+)≤ψ

maxd(xn–,xn),d(xn,xn+)

=ψd(xn,xn+)

<d(xn,xn+),

which is a contradiction. Hence, for alln∈N, we have

d(xn,xn+)≤ψ

d(xn–,xn)

.

By induction, we have

d(xn,xn+)≤ψn

d(x,x)

.

Fix> , there existsN∈Nsuch that

n≥N

ψnd(xn,xn+)

< for alln∈N.

Letm,n∈Nwithm>n≥N. Then, by the triangular inequality, we get

d(xn,xm)≤ m–

k=n

d(xk,xk+)≤ n≥N

ψnd(xn,xn+)

<.

Consequently,limm,n,→+∞d(xn,xm) = . Hence{xn}is a Cauchy sequence. SinceXis com-plete, there isz∈Xsuch thatxn→zasn→ ∞. Now, if we suppose thatTis continuous, then we have

Tz= lim

n→∞Txn=nlim→∞xn+=z.

So,zis a ﬁxed point ofT. On the other hand, since

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for alln∈N∪ {}andxn→zasn→ ∞, we get

α(xn,z)≥η(xn,z)

for alln∈N∪ {}. Then from (.) we have

d(xn+,Tz)≤ψ

M(xn,z)

,

where

M(xn,z) =max

d(xn,z),

d(xn,xn+) +d(z,Tz)

 ,

d(xn,Tz) +d(z,xn+) 

.

SinceM(xn,z) > , then

d(xn+,Tz)≤ψ

M(xn,z)

<M(xn,z).

By taking limit asn→ ∞in the above inequality, we have

d(z,Tz) = lim

n→∞d(xn+,Tz)≤nlim→∞M(xn,z) =

d(z,Tz)  ,

which impliesd(z,Tz) = ,i.e.,z=Tz.

By takingη(x,y) =  in Theorem ., we have the following result.

Corollary . Let(X,d)be a complete metric space and let T be anα-admissible mapping.

Assume that forψ∈,

x,y∈X, α(x,y)≥ ⇒ d(Tx,Ty)≤ψM(x,y).

(i) there existsx∈Xsuch thatα(x,Tx)≥;

(ii) eitherTis continuous or for any sequence{xn}inXwithα(xn,xn+)≥for all

n∈N∪ {}andxn→xasn→+∞,we haveα(xn,x)≥for alln∈N∪ {}.

Then T has a ﬁxed point.

By takingα(x,y) =  in Theorem ., we have the following corollary.

Corollary . Let(X,d)be a complete metric space and let T be anη-subadmissible map-ping.Assume that forψ∈,

x,y∈X, η(x,y)≤ ⇒ d(Tx,Ty)≤ψM(x,y).

(i) there existsx∈Xsuch thatη(x,Tx)≤;

(ii) eitherTis continuous or for any sequence{xn}inXwithη(xn,xn+)≤for all

n∈N∪ {}andxn→xasn→+∞,we haveη(xn,x)≤for alln∈N∪ {}.

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Clearly, Corollary . implies the following results.

Corollary .(Theorem . and Theorem . of []) Let(X,d)be a complete metric space and let T be anα-admissible mapping.Assume that forψ∈,

α(x,y)d(Tx,Ty)≤ψd(x,y)

holds for all x,y∈X.Also,suppose that the following assertions hold:

Corollary . (Theorem . and Theorem . of []) Let(X,d)be a complete metric space and let T be anα-admissible mapping.Assume that forψ∈,

α(x,y)d(Tx,Ty)≤ψM(x,y) ∀x,y∈X,

where

M(x,y) =max

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

 ,

d(x,Ty) +d(y,Tx) 

.

Example . LetX= [,∞) be endowed with the usual metricd(x,y) =|x–y|for allx,y∈ Xand letT:X→Xbe deﬁned by

Tx=

⎧ ⎨ ⎩

x

(x+) ifx∈[, ],

lnx+|sinx| ifx∈(,∞).

Deﬁne alsoα:X×X→[, +∞) andψ: [,∞)→[,∞) by

α(x,y) =

⎧ ⎨ ⎩

 ifx,y∈[, ],

 otherwise

and ψ(t) =  t.

We prove that Corollary . can be applied toT. But Theorem . of [] and Theo-rem . of [] cannot be applied toT.

Clearly, (X,d) is a complete metric space. We show thatTis anα-admissible mapping. Letx,y∈X, ifα(x,y)≥, thenx,y∈[, ]. On the other hand, for allx∈[, ] we have

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Now, if{xn}is a sequence inXsuch thatα(xn,xn+)≥ for alln∈N∪ {}andxn→xas

n→+∞, then{xn} ⊂[, ] and hencex∈[, ]. This implies thatα(xn,x)≥ for alln∈N. Letα(x,y)≥. Thenx,y∈[, ]. We get

d(Tx,Ty) =Ty–Tx= y (y+ )–

x

(x+ )

= y–x ( +x)( +y)

≤y–x  =



d(x,y)≤ 

M(x,y) =ψ

M(x,y).

That is,

α(x,y)≥ ⇒ d(Tx,Ty)≤ψM(x,y).

All of the conditions of Corollary . hold. Hence,T has a ﬁxed point. Letx=  and

y= , then

α(, )d(T,T) =  > / =ψd(, ).

That is, Theorem . of [] cannot be applied toT.

Also, by a similar method, we can show that Theorem . of [] cannot be applied toT. By the following simple example, we show that our results improve the results of Samet

et al.[] and the results of Karapinar and Samet [].

Example . LetX= [,∞) be endowed with the usual metricd(x,y) =|x–y| for all

x,y∈Xand letT:X→Xbe deﬁned byTx=_x. Also, deﬁneα:X_→_[,_∞_{) by}_α₍_x_,_y_{) = } andψ: [,∞)→[,∞) byψ(t) =_t.

Clearly,Tis anα-admissible mapping. Also,α(x,y) = ≥ for allx,y∈X. Hence,

d(Tx,Ty) = 

|x–y| ≤ 

|x–y|=ψ

d(x,y)≤ψM(x,y).

Then the conditions of Corollary . hold andT has a ﬁxed point. But if we choosex=  andy= , then

α(, )d(T,T) =  >  =ψd(, ).

That is, Theorem . of [] cannot be applied toT. Similarly, we can show that Theo-rem . of [] cannot be applied toT. Further notice that the Banach contraction prin-ciple holds for this example.

Example . LetX= [,∞) be endowed with the usual metricd(x,y) =|x–y| for all

x,y∈Xand letT:X→Xbe deﬁned by

Tx=

⎧ ⎨ ⎩

 x

 _if_x_∈_{[, ],}

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Deﬁne alsoα,η:X×X→[, +∞) andψ: [,∞)→[,∞) by

η(x,y) =

⎧ ⎨ ⎩

 ifx,y∈[, ],

 otherwise and

ψ(t) =  t.

We prove that Corollary . can be applied toT. But the Banach contraction principle cannot be applied toT.

Clearly, (X,d) is a complete metric space. We show thatT is anη-subadmissible map-ping. Letx,y∈X, ifη(x,y)≤, thenx,y∈[, ]. On the other hand, for allx∈[, ], we haveTx≤. It follows thatη(Tx,Ty)≤. Also,η(,T)≤.

Now, if{xn}is a sequence inXsuch thatη(xn,xn+)≤ for alln∈N∪ {}andxn→xas

n→+∞, then{xn} ⊂[, ] and hencex∈[, ]. This implies thatη(xn,x)≤ for alln∈N. Letη(x,y)≤. Thenx,y∈[, ]. We get

d(Tx,Ty) = 

|x–y||x+y| ≤ 

|x–y| ≤ 

M(x,y) =ψ

M(x,y).

That is,

η(x,y)≤ ⇒ d(Tx,Ty)≤ψM(x,y).

Then the conditions of Corollary . hold. Hence,Thas a ﬁxed point. Letx= ,y=  and

r∈[, ). Then

d(T,T) =  >  >r=rd(, ).

That is, the Banach contraction principle cannot be applied toT. From our results, we can deduce the following corollaries.

Corollary . Let(X,d)be a complete metric space and let T be anα-admissible mapping.

Assume that

α(x,y) +d(Tx,Ty)≤( +)ψ(d(x,y)) (.)

holds for all x,y∈X,whereψ∈ and> .Also,suppose that the following assertions hold:

Proof Letα(x,y)≥. Then by (.) we have

( +)d(Tx,Ty)≤α(x,y) +d(Tx,Ty)≤( +)ψ(d(x,y)).

Thend(Tx,Ty)≤ψ(d(x,y)). Hence, the conditions of Corollary . hold andf has a ﬁxed

point.

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Corollary . Let(X,d)be a complete metric space and let T be anα-admissible mapping.

Assume that

d(Tx,Ty) +α(x,y)≤ψd(x,y)+ (.)

hold for all x,y∈X,whereψ∈ and> .Also,suppose that the following assertions hold:

Notice that the main theorem of Dutta and Choudhury [] remains true ifφ is lower semi-continuous instead of continuous (see,e.g., [, ]).

We assume that

=

ψ: [,∞)→[,∞) such thatψis non-decreasing and continuous

and

=ϕ: [,∞)→[,∞) such thatϕis lower semicontinuous,

whereψ(t) =ϕ(t) =  if and only ift= .

Theorem . Let(X,d)be a complete metric space and let T be anα-admissible mapping

with respect toη.Assume that forψ∈andϕ∈ ,

x,y∈X, α(x,Tx)α(y,Ty)≥η(x,Tx)η(y,Ty)

⇒ ψd(Tx,Ty)≤ψd(x,y)–ϕd(x,y). (.)

(i) there existsx∈Xsuch thatα(x,Tx)≥η(x,Tx);

(ii) eitherTis continuous or for any sequence{xn}inXwithα(xn,xn+)≥η(xn,xn+)for alln∈N∪ {}andxn→xasn→+∞,we haveα(x,Tx)≥η(x,Tx)for all

n∈N∪ {}. Then T has a ﬁxed point.

Proof Let x ∈X such that α(x,Tx)≥η(x,Tx). Deﬁne a sequence {xn} in X by

xn =Tnx=Txn– for all n∈ N. If xn+ =xn for some n∈N, then x =xn is a ﬁxed point for T and the result is proved. We suppose thatxn+ =xnfor alln∈N. SinceT is an α-admissible mapping with respect to η and α(x,Tx)≥η(x,Tx), we deduce thatα(x,x) =α(Tx,Tx)≥η(Tx,Tx) =η(x,x). By continuing this process, we get

α(xn,xn+)≥η(xn,xn+) for alln∈N∪ {}. Clearly,

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Now, by (.) withx=xn–,y=xn, we have

ψd(Txn–,Txn)

≤ψd(xn–,xn)

–ϕd(xn–,xn)

,

which implies

ψd(xn,xn+)

≤ψd(xn–,xn)

–ϕd(xn–,xn)

≤ψd(xn–,xn)

. (.)

Sinceψis increasing, we get

d(xn,xn+)≤d(xn–,xn)

for alln∈N. That is,{dn:=d(xn,xn+)}is a non-increasing sequence of positive real num-bers. Then there existsr≥ such thatlim_n_→∞dn=r. We shall show thatr= . By taking the limit inﬁmum asn→ ∞in (.), we have

ψ(r)≤ψ(r) –ϕ(r).

Henceφ(r) = . That is,r= . Then

lim

n→∞d(xn,xn+) = . (.)

Suppose, to the contrary, that{xn}is not a Cauchy sequence. Then there isε>  and se-quences{m(k)}and{n(k)}such that for all positive integersk,

n(k) >m(k) >k, d(xn(k),xm(k))≥ε and d(xn(k),xm(k)–) <ε.

Now, for allk∈N, we have

ε≤d(xn(k),xm(k))≤d(xn(k),xm(k)–) +d(xm(k)–,xm(k))

< ε+d(xm(k)–,xm(k)).

Taking limit ask→+∞in the above inequality and using (.), we get

lim

k→+∞d(xn(k),xm(k)) =ε. (.)

Since

d(xn(k),xm(k))≤d(xm(k),xm(k)+) +d(xm(k)+,xn(k)+) +d(xn(k)+,xn(k))

and

d(xn(k)+,xm(k)+)≤d(xm(k),xm(k)+) +d(xm(k),xn(k)) +d(xn(k)+,xn(k)),

then by taking the limit ask→+∞in the above inequality, and by using (.) and (.), we deduce that

lim

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On the other hand,

α(xn(k),Txn(k))α(xm(k),Txm(k))≥η(xn(k),Txn(k))η(xm(k),Txm(k)).

Then, by (.) withx=xn(k)andy=xm(k), we get

ψd(xn(k)+,xm(k)+)

≤ψd(xn(k),xm(k))

–ϕd(xn(k),xm(k))

.

By taking limit ask→ ∞in the above inequality and applying (.) and (.), we obtain

ψ(ε)≤ψ(ε) –ϕ(ε).

That is,ε= , which is a contradiction. Hence{xn}is a Cauchy sequence. SinceXis com-plete, then there isz∈Xsuch thatxn→z. First we assume thatTis continuous. Then we deduce

Tz= lim

α(xn,xn+)≥η(xn,xn+)

for alln∈N∪ andxn→zasn→ ∞, so

α(z,Tz)≥η(z,Tz),

which implies

α(xn,xn+)α(z,Tz)≥η(xn,xn+)η(z,Tz).

Now, by (.) we get

ψd(xn+,Tz)

=ψd(Txn,Tz)

≤ψd(xn,z)

–ϕd(xn,z)

.

Passing limit inf asn→ ∞in the above inequality, we have

ψd(z,Tz)= lim

n→∞ψ

d(xn+,Tz)

= .

That is,z=Tz.

By takingη(x,y) =  in Theorem ., we deduce the following corollary.

Corollary . Let(X,d)be a complete metric space and let T be anα-admissible mapping.

Assume that forψ∈andϕ∈ ,

x,y∈X, α(x,Tx)α(y,Ty)≥ ⇒ ψd(Tx,Ty)≤ψd(x,y)–ϕd(x,y).

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n∈N∪ {}andxn→xasn→+∞,we haveα(x,Tx)≥.

By takingα(x,y) =  in Theorem ., we deduce the following corollary.

Corollary . Let(X,d)be a complete metric space and let T be anη-subadmissible map-ping.Assume that forψ∈andϕ∈ ,

x,y∈X, η(x,Tx)η(y,Ty)≤ ⇒ ψd(Tx,Ty)≤ψd(x,y)–ϕd(x,y).

(i) there existsx∈Xsuch thatη(x,Tx)≤;

(ii) eitherTis continuous or for any sequence{xn}inXwithη(xn,xn+)≤for all

n∈N∪ {}andxn→xasn→+∞,we haveη(x,Tx)≤.

Example . LetX= [,∞) be endowed with the usual metric

d(x,y) =

⎧ ⎨ ⎩

max{x,y} ifx=y,

 ifx=y

for allx,y∈X, and letT:X→Xbe deﬁned by

Tx=

⎧ ⎨ ⎩

x–x

 ifx∈[, ], x ifx∈(,∞).

Deﬁne alsoα:X×X→[, +∞) andψ,ϕ: [,∞)→[,∞) by

α(x,y) =

⎧ ⎨ ⎩

 ifx,y∈[, ], 

 otherwise,

ψ(t) =t and ϕ(t) =  t.

We prove that Corollary . can be applied toT, but the main theorem in [] cannot be applied toT.

By a similar proof to that of Example ., we show thatTis anα-admissible mapping. As-sume thatα(x,Tx)α(y,Ty)≥. Now, ifx∈/[, ], thenα(x,Tx) =_and soα(x,Tx)α(y,Ty) < , which is contradiction. Ify∈/[, ]. Similarly,α(x,Tx)α(y,Ty) < , which is contradiction. Hence,α(x,Tx)α(y,Ty)≥ impliesx,y∈[, ]. Therefore, we get

ψd(Tx,Ty)=max

x–x  ,

y–y 

≤ 

max{x,y}=ψ

d(x,y)–ϕd(x,y).

That is,

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The conditions of Corollary . are satisﬁed. Hence,T has a ﬁxed point. Letx=  and

y= , then

ψd(T,T)=  > / =ψd(, )–ϕd(, ).

That is, the main theorem in [] cannot be applied toT.

Example . LetX= [,∞) be endowed with the usual metricd(x,y) =|x–y| for all

x,y∈X, and letT:X→Xbe deﬁned by

Tx=

⎧ ⎨ ⎩

 ( –x

₎ _if_x_∈_{[, ],} |sin(π_x)| ifx∈(,∞).

Deﬁne alsoη:X×X→[, +∞) andψ,ϕ: [,∞)→[,∞) by

η(x,y) =

⎧ ⎨ ⎩

 ifx,y∈[, ],

 otherwise,

ψ(t) =t and ϕ(t) =  t.

We prove that Corollary . can be applied toT, but the main theorem in [] cannot be applied toT.

By a similar proof to that of Example ., we can show that T is an η-subadmissible mapping.

Assume thatη(x,Tx)η(y,Ty)≤. Now, ifx∈/[, ], thenη(x,Tx)η(y,Ty) > , which is a contradiction. Similarly,y∈/ [, ] is a contradiction. Hence,η(x,Tx)η(y,Ty)≤ implies

x,y∈[, ]. We get

ψd(Tx,Ty)= |x–y|x

₊_xy₊_y_≤

|x–y|=ψ

d(x,y)–ϕd(x,y).

That is,

η(x,Tx)η(y,Ty)≤ ⇒ d(Tx,Ty)≤ψd(x,y)–ϕd(x,y).

Then the conditions of Corollary . hold andT has a ﬁxed point. Letx= ,y= . Then

T =  andT = , which implies

ψd(T,T)=  > =ψ

d(, )–ϕd(, ).

That is, the main theorem in [] cannot be applied toT. In  Khanet al.[] proved the following theorem.

Theorem . Let(X,d)be a complete metric space and let T be a self-mapping on X.

Assume that

ψd(Tx,Ty)≤cψd(x,y) ∀x,y∈X,

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Theorem . Let(X,d)be a complete metric space and let T be a generalizedα-admissible mapping with respect toη.Assume that

x,y∈X, α(x,x)α(y,y)≥η(x,x)η(y,y) ⇒ ψd(Tx,Ty)≤cψd(x,y), (.)

whereψ∈and <c< .Also,suppose that the following assertions hold: (i) there existsx∈Xsuch thatα(x,x)≥η(x,x);

(ii) eitherTis continuous or for any sequence{xn}inXwithα(xn,xn)≥η(xn,xn)for all

n∈N∪ {}andxn→xasn→+∞,we haveα(x,x)≥η(x,x).

Proof Let x∈X such that α(x,x)≥η(x,x). Deﬁne a sequence {xn} in X by xn=

Tn_x

=Txn–for alln∈N. Ifxn+ =xn for somen∈N, thenx=xnis a ﬁxed point for

T and the result is proved. Hence, we suppose that xn+=xn for alln∈N. SinceT is a generalizedα-admissible mapping with respect to ηandα(x,x)≥η(x,x), we de-duce thatα(x,x) =α(Tx,Tx)≥η(Tx,Tx) =η(x,x). By continuing this process, we getα(xn,xn)≥η(xn,xn) for alln∈N∪ {}. Clearly,

α(xn–,xn–)α(xn,xn)≥η(xn–,xn–)η(xn,xn).

Now, by (.) withx=xn–,y=xn, we have

ψd(xn,xn+)

=ψd(Txn–,Txn)

≤cψd(xn–,xn)

<ψd(xn–,xn)

. (.)

Since,ψis increasing, we get

d(xn,xn+)≤d(xn–,xn)

for alln∈N. That is,{dn:=d(xn,xn+)}is a non-increasing sequence of positive real num-bers. Then there existsr≥ such thatlimn→∞dn=r. We shall show thatr= . By taking the limit asn→ ∞in (.), we have

ψ(r)≤cψ(r),

which impliesψ(r) = ,i.e.,r= . Then

lim

n→∞d(xn,xn+) = . (.)

Suppose, to the contrary, that{xn}is not a Cauchy sequence. Proceeding as in the proof of Theorem ., there exists>  such that for allk∈Nthere existn(k),m(k)∈Nwith

m(k) >n(k)≥ksuch that

lim

k→+∞d(xn(k),xm(k)) =ε (.)

and

lim

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Clearly,

α(xn(k),xn(k))α(xm(k),xm(k))≥η(xn(k),xn(k))η(xm(k),xm(k)).

Then, by (.) withx=xn(k)andy=xm(k), we get

ψd(xn(k)+,xm(k)+)

=ψd(Txn(k),Txm(k))

≤cψd(xn(k),xm(k))

.

Taking limit ask→ ∞in the above inequality and applying (.) and (.), we get

ψ()≤cψ(),

and so= , which is a contradiction. Hence{xn}is a Cauchy sequence. SinceXis com-plete, then there isz∈Xsuch thatxn→z. First, we assume thatTis continuous. Then, we deduce

Tz= lim

α(xn,xn)≥η(xn,xn)

for alln∈N∪ {}andxn→zasn→ ∞, we get

α(z,z)≥η(z,z),

which implies

α(z,z)α(xn,xn)≥η(z,z)η(xn,xn).

Then by (.) we deduce

ψd(xn+,Tz)

=ψd(Txn,Tz)

≤cψd(xn,z)

.

Taking limit asn→ ∞in the above inequality, we have

ψd(z,Tz)≤ψ() = 

and thenz=Tz.

Corollary . Let(X,d)be a complete metric space and let T be anα-admissible mapping.

Assume that

x,y∈X, α(x,x)α(y,y)≥ ⇒ ψd(Tx,Ty)≤cψd(x,y),

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(i) there existsx∈Xsuch thatα(x,x)≥;

(ii) eitherTis continuous or for any sequence{xn}inXwithα(xn,xn)≥for all

n∈N∪ {}andxn→xasn→+∞,we haveα(x,x)≥.

Corollary . Let (X,d) be a complete metric space and let T be a generalized α -admissible mapping with respect toη.Assume that

x,y∈X, η(x,x)η(y,y)≤ ⇒ ψd(Tx,Ty)≤cψd(x,y),

whereψ∈and <c< .Also,suppose that the following assertions hold: (i) there existsx∈Xsuch thatη(x,x)≤;

(ii) eitherTis continuous or for any sequence{xn}inXwithη(xn,xn)≤for all

n∈N∪andxn→xasn→+∞,we haveη(x,x)≤.

Example . LetX= [,∞) be endowed with the usual metric

d(x,y) =

⎧ ⎨ ⎩

max{x,y} ifx=y,

 ifx=y

for allx,y∈X, and letT:X→Xbe deﬁned by

Tx=

⎧ ⎨ ⎩

x–x

 ifx∈[, ],

x₊_|₍_x_{– )(}_x_{– )}_| _if_x_∈_(,_∞_).

Deﬁne alsoα:X×X→[, +∞) andψ,ϕ: [,∞)→[,∞) by

α(x,y) =

⎧ ⎨ ⎩

 ifx,y∈[, ], 

 otherwise

and ψ(t) =t.

We prove that Corollary . can be applied toT. But Theorem . cannot be applied toT.

By a similar proof to that of Example ., we show thatT is anα-admissible mapping. Assume thatα(x,x)α(y,y)≥. Now, ifx∈/ [, ], then α(x,x) = _ and soα(x,x)α(y,y) < , which is contradiction. Ify∈/[, ]. Similarly,α(x,x)α(y,y) < , which is contradiction. Hence,α(x,x)α(y,y)≥ impliesx,y∈[, ]. Therefore, we get

ψd(Tx,Ty)=

max

x_–_x

 ,

y_–_y 

 ≤ 



max{x,y}=  ψ

d(x,y).

That is,

α(x,x)α(y,y)≥ ⇒ ψd(Tx,Ty)≤  ψ

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Then the conditions of Corollary . hold. Hence,T has a ﬁxed point. Letx=  and

y= , thenT =  andT = , and hence

ψd(T,T)=  >  =

 ψ

d(, ).

That is, Theorem . cannot be applied toT.

3 Application to the existence of solutions of integral equations

Integral equations like (.) were studied in many papers (see [, ] and references therein). In this section, we look for a nonnegative solution to (.) inX=C([,T],R). Let X=C([,T],R) be the set of real continuous functions deﬁned on [,T] and let

d:X×X→R+be deﬁned by

d(x,y) =x–y∞

for allx,y∈X. Then (X,d) is a complete metric space. Consider the integral equation

x(t) =p(t) +

T



S(t,s)fs,x(s)ds, (.)

and letF:X→Xdeﬁned by

F(x)(t) =p(t) +

T



S(t,s)fs,x(s)ds. (.)

We assume that

(A) f: [,T]×R→Ris continuous; (B) p: [,T]→Ris continuous;

(C) S: [,T]×R→[, +∞)is continuous;

(D) there existψ∈andθ:X×X→Rsuch that ifθ(x,y)≥forx,y∈X, then for everys∈[,T]we have

≤fs,x(s)–fs,y(s)

≤ψ

maxx(s) –y(s),

x(s) –F

x(s)+y(s) –Fy(s),



x(s) –F

y(s)+y(s) –Fx(s)

;

(F) there existsx∈Xsuch thatθ(x,F(x))≥; (G) ifθ(x,y)≥,x,y∈X, thenθ(Fx,Fy)≥;

(H) if{xn}is a sequence inXsuch thatθ(xn,xn+)≥for alln∈N∪ {}andxn→xas

n→+∞, thenθ(xn,x)≥for alln∈N∪ {};

(J) _TS(t,s)ds≤for allt∈[,T]ands∈R.

Theorem . Under assumptions(A)-(J),the integral equation(.)has a solution in X=

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Proof Consider the mappingF:X→Xdeﬁned by (.). By the condition (D), we deduce

F(x)(t) –F(y)(t)

=

T



S(t,s)fs,x(s)–fs,y(s)ds

≤

T



S(t,s)fs,x(s)–fs,y(s)ds

≤

T



S(t,s)

ψ

maxx(s) –y(s),

x(s) –F

x(s)+y(s) –Fy(s),



x(s) –F

y(s)+y(s) –Fx(s)

ds

≤

T



S(t,s)

ψ

maxx(s) –y(s),

x(s) –F

x(s)+y(s) –Fy(s),



x(s) –F

y(s)+y(s) –Fx(s)

ds

=

T



S(t,s)ds

ψ

maxx(s) –y(s),

x(s) –F

x(s)+y(s) –Fy(s),



x(s) –F

y(s)+y(s) –Fx(s)

≤ψ

maxx(s) –y(s),

x(s) –F

x(s)+y(s) –Fy(s),



x(s) –F

y(s)+y(s) –Fx(s)

.

Then

Fx–Fy∞≤ψ

maxx(s) –y(s),

x(s) –F

x(s)+y(s) –Fy(s),



x(s) –F

y(s)+y(s) –Fx(s)

.

Now, deﬁneα:X×X→[, +∞) by

α(x,y) =

⎧ ⎨ ⎩

 ifθ(x,y)≥,

 otherwise.

That is,α(x,y)≥ implies

Fx–Fy∞≤ψ

maxx(s) –y(s),

x(s) –F

x(s)+y(s) –Fy(s),



x(s) –F

y(s)+y(s) –Fx(s)

,

Fx–Fy∞≤ψ

max

x–y∞,

x–F(x)∞+y–F(y)∞

,



x–F(y)∞+y–F(x)∞

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All of the hypotheses of Corollary . are satisﬁed, and hence the mappingFhas a 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, Astara Branch, Islamic Azad University, Astara, Iran.}2_{Department of Mathematics, King}

Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.

Acknowledgements

The authors thank the referees for their valuable comments and suggestions. This research was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the second and third authors acknowledge with thanks DSR, KAU for financial support. The first author is thankful for support of Astara Branch, Islamic Azad University, during this research.

Received: 14 February 2013 Accepted: 21 May 2013 Published: 10 June 2013

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doi:10.1186/1687-1812-2013-151