Fixed point theorems for Ćirić type mapping and application to integral equation

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R E S E A R C H

Open Access

Fixed point theorems for ´Ciri´c type

mapping and application to integral equation

Jeong Sheok Ume

*

*_{Correspondence:} [email protected] Department of Mathematics, Changwon National University, Changwon, 641-773, Korea

Abstract

In this paper, we introduce a new class of Ćirić type single-valued mapping with respect tou-distance and prove some fixed point theorems for this mapping. An example is given to show that our results are a proper extension of many well-known results. As an application, we establish the existence of a solution for an integral equation.

1 Introduction

The Banach contraction principle is a remarkable result in metric fixed point theory. Over the years, it has been generalized in different directions and spaces by several mathemati-cians, see [–] and the references therein. In , Ćirić [] proved the following fixed point theorem on a complete metric space, which generalizes the Banach contraction prin-ciple: LetXbe a complete metric space and letT:X→Xbe a quasi-contractive mapping; i.e., there exists a constantq∈[, ) such that, for allx,y∈X,

d(Tx,Ty)≤q·maxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx).

Then:

() Thas a unique ﬁxed pointwinX. () limn→∞Tnx=wfor everyx∈X.

() d(Tn_x,_w)_≤_[qn

–q]d(x,Tx)for everyxinX.

Recently, Ume [] generalized the notion ofτ-distance [] by introducingu-distance as follows.

LetX be metric space with metricd. Then a functionp:X×X→R+is called a

u-distance onXif there exists a functionθ:X×X×R+×R+→R+such that the following

hold forx,y,z∈X:

(u) p(x,z)≤p(x,y) +p(y,z);

(u) θ(x,y, , ) = andθ(x,y,s,t)≥min{s,t}for eachs,t∈R+, and for anyx∈Xand for

everyε> , there existsδ> such that|s–s|<δ,|t–t|<δ,s,s,t,t∈R+andy∈X

imply|θ(x,y,s,t) –θ(x,y,s,t)|<ε;

(u) limn→∞xn = x and limn→∞sup{θ(wn,zn,p(wn,xm),p(zn,xm)) : m ≥ n} =  imply

p(y,x)≤limn→∞infp(y,xn)for ally∈X;

(u) limn→∞sup{p(xn,wm) :m≥n}= ,limn→∞sup{p(yn,zm) :m≥n}= ,limn→∞θ(xn,

wn,sn,tn) = , limn→∞θ(yn,zn,sn,tn) =  imply limn→∞θ(wn,zn,sn,tn) =  or

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limn→∞sup{p(wm,xn) :m≥n}= ,limn→∞sup{p(zm,yn) :m≥n}= ,limn→∞θ(xn,

wn,sn,tn) = ,limn→∞θ(yn,zn,sn,tn) = implylimn→∞θ(wn,zn,sn,tn) = ;

(u) limn→∞θ(wn,zn,p(wn,xn),p(zn,xn)) = ,limn→∞θ(wn,zn,p(wn,yn),p(zn,yn)) = 

im-plylimn→∞d(xn,yn) = orlimn→∞θ(an,bn,p(xn,an),p(xn,bn)) = ,limn→∞θ(an,bn,

p(yn,an),p(yn,bn)) = implylimn→∞d(xn,yn) = .

Remark .([])

(a) Suppose thatθfromX×X×R+×R+intoR+is a mapping satisfying(u)∼(u).

Then there exists a mappingηfromX×X×R+×R+intoR+such thatηis

nondecreasing in its third and fourth variable, satisfying(u)η∼(u)η, where

(u)η∼(u)ηstand for substitutingηforθin(u)∼(u), respectively.

(b) On account of (a), we may assume thatθ is nondecreasing in its third and fourth variables, respectively, for a functionθ fromX×X×R+×R+intoR+satisfying

(u)∼(u).

(c) Eachτ-distancepon a metric space(X,d)is also au-distance onX. We present some example ofu-distance which are notτ-distance. (For details, see [].)

Example . LetX=R+with the usual metric. Deﬁnep:X×X→R+byp(x,y) = (_)x.

Thenpis au-distance on X but not aτ-distance onX.

Example . LetXbe a normed space with norm · . Then a functionp:X×X→R+

deﬁned byp(x,y) =xfor everyx,y∈Xis au-distance onXbut not aτ-distance.

It follows from the above example and Remark .(c) thatu-distance is a proper exten-sion ofτ-distance. Other useful examples onu-distance are given in [].

2 Preliminaries

Throughout this paper we denote byNthe set of all positive integers, byRthe set of all real numbers and byR+the set of all nonnegative real numbers.

Deﬁnition .([]) LetXbe a metric space with a metricdand letpbe au-distance onX. Then a sequence{xn}in X is calledp-Cauchy if there exists a functionθ fromX×

X×R+×R+intoR+satisfying (u)∼(u) and a sequence{zn}ofXsuch that

lim

n→∞sup

θzn,zn,p(zn,xm),p(zn,xm)

:m≥n=  or

lim

n→∞sup

θzn,zn,p(xm,zn),p(xm,zn)

:m≥n= .

Lemma .([]) Let X be a metric space with a metric d and let p a u-distance on X.If {xn}is a p-Cauchy sequence,then{xn}is a Cauchy sequence.

Lemma .([]) Let X be a metric space with a metric d and let p be a u-distance on X. () If sequences{xn}and{yn}ofXsatisfylimn→∞p(z,xn) = andlimn→∞p(z,yn) = for

somez∈X,thenlimn→∞d(xn,yn) = .

() Ifp(z,x) = andp(z,y) = ,thenx=y.

() Suppose that sequences{xn}and{yn}ofXsatisfylimn→∞p(xn,z) = and

limn→∞p(yn,z) = for somez∈X,thenlimn→∞d(xn,yn) = .

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Lemma .([]) Let X be a metric space with a metric d and let p be a u-distance on X. Suppose that a sequence{xn}of X satisﬁes

lim

n→∞sup

p(xn,xm) :m≥n

=  or

lim

n→∞sup

p(xm,xn) :m≥n

= .

Then:

(i) {xn}is ap-Cauchy sequence.

(ii) If{xn}is ap-Cauchy sequence,then{xn}is a Cauchy sequence.

3 Fixed point theorems

The following lemma plays an important role in proving our theorems.

Lemma . Let(X,d)be a metric space with a u-distance p on X and{an}and{bn}be

sequences of X such that

lim

n→∞sup

p(an,am) :m>n

=  and

lim

n→∞sup

p(an,bm) :m>n

= .

Then there exist a subsequence{akn}of {an} and a subsequence{bkn} of{bn}such that limn→∞d(akn,bkn) = .

Proof Sincepis au-distance onX,

there exists a mappingθ:X×X×R+×R+→R+

such thatθis nondecreasing in its third and (.)

fourth variable respectively, satisfying (u)∼(u).

For eachn∈N, let

αn=sup

p(an,am) :m>n

and βn=sup

p(an,bm) :m>n

. (.)

By the hypotheses and (.), we have

lim

n→∞(αn+βn) = . (.)

Letk∈Nbe an arbitrary and ﬁxed element. Then, by (u), for thisak∈Xandε= , there existsδ>  such that

|s|=s<δ, |t|=t<δ, y∈X imply θ(ak,y,s,t) < . (.)

By virtue of (.) and (.), for thisδ> , there existsM∈Nsuch that

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Letk∈Nbe such that

k≥max{ +k,M}. (.)

Due to (.), we have

k<k and k≥M. (.)

From (.), (.), (.), and (.) we get

θ(ak,ak,αk+βk,αk+βk) < . (.) In terms of (u) and (.), for thisak∈Xandε=



, there existsδ>  such that|s|=s<δ,

|t|=t<δ,y∈Ximply

θ(ak,y,s,t) < 

. (.)

In view of (.) and (.), for thisδ> , there existsM∈Nsuch that

n≥M implies αn+βn<δ. (.)

Letk∈Nbe such that

k≥max{ +k,M}. (.)

On account of (.), (.), (.), we obtain

k<k and θ(ak,ak,αk+βk,αk+βk) < 

. (.)

Continuing this process, there exist a subsequence{akn}of{an}, and a subsequence{bkn}

of{bn}such that, for alln∈N,

θ(akn,akn+,αkn++βkn+,αkn++βkn+) < 

n. (.)

Using (.), (.), and (.), we know that

lim

n→∞

supp(akn,akm+) :m≥n

≤ lim

n→∞

supp(akn,al) :l>kn

= lim

n→∞αkn=  and lim

n→∞θ(akn,akn+,αkn++βkn+,αkn++βkn+) = .

(.)

Using (.), (.), (.) and puttingxn=yn=akn,wm=zm=akm+andsn=tn=αkn++βkn+ in (u) we deduce

lim

n→∞θ

akn+,akn+,p(akn+,akn+),p(akn+,akn+)

=  and

lim

n→∞θ

akn+,akn+,p(akn+,bkn+),p(akn+,bkn+)

= .

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Using (.) and puttingwn=zn=akn+,xn=akn+, andyn=bkn+in (u), we have

lim

n→∞d(akn+,bkn+) = . (.)

Due to (.) and (.), there exist a subsequence{akn}of{an}and a subsequence{bkn}

of{bn}such that

lim

n→∞d(akn,bkn) = . (.)

Deﬁnition . Let (X,d) be a metric space with au-distanceponXand letTbe a self-mapping onX. ForA⊆X, letδ(A) =sup{p(x,y) :x,y∈A} and for eachx,y∈X,n∈N, let

O(x,y,n) =Tix,Tjy: ≤i,j≤n,i,j∈N∪ {},

whereT_x₌_x_and_Ti_{is the}_i_{times repeated composition of}_T _{with itself. Let}

O(x,y,∞) =Tix,Tjy:i,j∈N∪ {} for eachx,y∈X.

Lemma . Let(X,d)be a metric space with a u-distance p on X.Let T :X→X and ϕ:R+→R+be mappings that satisfy the following conditions:

(i) p(Tx,Ty)≤ϕmaxp(x,y),p(x,Tx),p(y,Ty),p(x,Ty),p(y,Tx),

p(y,x),p(Tx,x),p(Ty,y),p(Ty,x),p(Tx,y) (.)

for all x,y∈X;

(ii) ϕis nondecreasing andϕ(t) <t for all t> ;

(iii) I–ϕis nondecreasing and bijective,where I is identity mapping on R+;

(iv)

∞

n=

ϕn(t) <∞ for each t∈(,∞), (.)

whereϕnis n-times repeated composition ofϕwith itself. Then:

() For eachx,y∈Xandn∈N,

maxpTix,Tjx,pTix,Tjy,pTiy,Tjx,pTiy,Tjy|i,j∈N,i,j≤n ≤ϕδO(x,y,n).

() For eachx,y∈X,n∈Nand for eachi∈Nwithi≤,there existsli∈Nwithli≤n

such that

δO(x,y,n)=maxp(x,x),p(x,y),p(y,x),p(y,y),px,Tl_x_,_p_x,_Tl_y_,

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() For eachx,y∈X,

δO(x,y,∞)≤(I–ϕ)–b(x,y),

whereb(x,y) =p(x,x) +p(y,y) +p(x,y) +p(y,x) +p(x,Tx) +p(Tx,x) +p(y,Ty) + p(Ty,y).

() For eachx∈X,{Tn_x}_{is a Cauchy sequence}_.

() For eachx,y∈Xandn∈N,

pTnx,Tny≤ϕn–(I–ϕ)–b(x,y). () For eachx,y∈X,limn→∞p(Tnx,Tny) = .

Proof Let x,y∈X and n∈N, and leti andj be natural numbers withi,j≤n. Then Ti–_x,_Ti_x,_Tj–_x,_Tj_x,_Ti–_y,_Ti_y,_Tj–_y,_Tj_y_∈_O(x,_y,_n).

From (.) and hypothesis (ii), we have

pTix,Tjx=pTTi–x,TTj–x

≤ϕmaxpTi–x,Tj–x,pTi–x,Tix,pTj–x,Tjx,pTi–x,Tjx, pTj–x,Tix,pTj–x,Ti–x,pTix,Ti–x,

pTjx,Tj–x,pTjx,Ti–x,pTix,Tj–x ≤ϕδO(x,y,n),

pTix,Tjy=pTTi–x,TTj–y

≤ϕmaxpTi–x,Tj–y,pTi–x,Tix,pTj–y,Tjy,pTi–x,Tjy, pTj–y,Tix,pTj–y,Ti–x,pTix,Ti–x,

pTjy,Tj–y,pTjy,Ti–x,pTix,Tj–y ≤ϕδO(x,y,n),

pTiy,Tjx=pTTi–y,TTj–x

≤ϕmaxpTi–y,Tj–x,pTi–y,Tiy,pTj–x,Tjx,pTi–y,Tjx, pTj–x,Tiy,pTj–x,Ti–y,pTiy,Ti–y,

pTjx,Tj–x,pTjx,Ti–y,pTiy,Tj–x ≤ϕδO(x,y,n),

pTiy,Tjy=pTTi–y,TTj–y

≤ϕmaxpTi–_y,_Tj–_y_,_p_Ti–_y,_Ti_y_,_p_Tj–_y,_Tj_y_,_p_Ti–_y,_Tj_y_,

pTj–y,Tiy,pTj–y,Ti–y,pTiy,Ti–y, pTjy,Tj–y,pTjy,Ti–y,pTiy,Tj–y ≤ϕδO(x,y,n),

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From (), it follows that for eachx,y∈X,n∈Nand for eachi∈Nwithi≤, there exists li∈Nwithli≤nsuch that

δO(x,y,n)=maxp(x,x),p(x,y),p(y,x),p(y,y),px,Tl_x_,_p_x,_Tl_y_,_p_y,_Tl_x_, py,Tl_y_,_p_Tl_x,_x_,_p_Tl_x,_y_,_p_Tl_y,_x_,_p_Tl_y,_y_,

which proves ().

Applying the triangle inequality, hypothesis (iii), (), and (), we have

px,Tl_x≤_p(x,_{Tx) +}_p_Tx,_Tl_x≤_p(x,_{Tx) +}_ϕ_δ_O(x,_y,_n)_, px,Tl_y≤_p(x,_{Tx) +}_p_Tx,_Tl_y≤_p(x,_{Tx) +}_ϕ_δ_O(x,_y,_n)_, py,Tl_x≤_p(y,_{Ty) +}_p_Ty,_Tl_x≤_p(y,_{Ty) +}_ϕ_δ_O(x,_y,_n)_, py,Tl_x≤_p(y,_{Ty) +}_p_Ty,_Tl_x≤_p(y,_{Ty) +}_ϕ_δ_O(x,_y,_n)_, pTl_x,_x≤_p_Tl_x,_Tx₊_p(Tx,_x)≤_p(Tx,_{x) +}_ϕ_δ_O(x,_y,_n)_, pTl_x,_y≤_p_Tl_x,_Ty₊_p(Ty,_y)≤_p(Ty,_{y) +}_ϕ_δ_O(x,_y,_n)_, pTl_y,_x≤_p_Tl_y,_Tx₊_p(Tx,_x)≤_p(Tx,_{x) +}_ϕ_δ_O(x,_y,_n)_, pTl_y,_y≤_p_Tl_y,_Ty₊_p(Ty,_y)≤_p(Ty,_{y) +}_ϕ_δ_O(x,_y,_n)_.

Thereforeδ(O(x,y,n))≤(I–ϕ)–_(b(x,_y)).

Sincenis arbitrary, the proof of () is complete.

To prove (), letxbe an arbitrary point ofXand deﬁnexn=Tnxfor everyn∈N. On

account of (.) and hypothesis (ii), we have

p(xn,xn+) =p(Txn–,Txn)

≤ϕmaxp(xn–,xn),p(xn–,xn),p(xn,xn+),p(xn–,xn+),p(xn,xn),

p(xn,xn–),p(xn,xn–),p(xn+,xn),p(xn+,xn–),p(xn,xn)

, (.)

p(xn+,xn) =p(Txn,Txn–)

≤ϕmaxp(xn,xn–),p(xn,xn+),p(xn–,xn),p(xn,xn),p(xn–,xn+),

p(xn–,xn),p(xn+,xn),p(xn,xn–),p(xn,xn),p(xn+,xn–)

, (.)

p(xn–,xn) =p(Txn–,Txn–)

≤ϕmaxp(xn–,xn–),p(xn–,xn–),p(xn–,xn),p(xn–,xn),p(xn–,xn–),

p(xn–,xn–),p(xn–,xn–),p(xn,xn–),p(xn,xn–),p(xn–,xn–)

, (.)

p(xn,xn–) =p(Txn–,Txn–)

≤ϕmaxp(xn–,xn–),p(xn–,xn),p(xn–,xn–),p(xn–,xn–),p(xn–,xn),

p(xn–,xn–),p(xn,xn–),p(xn–,xn–),p(xn–,xn–),p(xn,xn–)

, (.)

p(xn–,xn+) =p(Txn–,Txn)

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p(xn,xn–),p(xn,xn–),p(xn–,xn–),

p(xn+,xn),p(xn+,xn–),p(xn–,xn)

, (.)

p(xn+,xn–) =p(Txn,Txn–)

≤ϕmaxp(xn,xn–),p(xn,xn+),p(xn–,xn–),p(xn,xn–),

p(xn–,xn+),p(xn–,xn),p(xn+,xn),

p(xn–,xn–),p(xn–,xn),p(xn+,xn–)

, (.)

p(xn,xn) =p(Txn–,Txn–)

≤ϕmaxp(xn–,xn–),p(xn–,xn),p(xn,xn–)

. (.)

Substituting (.)∼(.) into (.), proceeding in this manner and by hypotheses, (), (), and () of Lemma ., we have

p(xn,xn+)≤ϕ

maxp(xi,xj) :n– ≤i,j≤n+ 

≤ϕmaxp(xi,xj) :n– ≤i,j≤n+ 

.. .

≤ϕn–maxp(xi,xj) : ≤i,j≤n+ 

≤ϕn–δO(x,x,∞)

≤ϕn–(I–ϕ)–a(x), (.)

wherea(x) = [p(x,x) +p(x,Tx) +p(Tx,x)]. Ifn<m, then, by (.),

p(xn,xm)≤p(xn,xn+) +p(xn+,xn+) +· · ·+p(xm–,xm)

=

m–

k=n

p(xk,xk+)

≤

m–

k=n

ϕk–(I–ϕ)–_a(x)

≤

m

k=n–

ϕk(I–ϕ)–a(x). (.)

Combining (.) and (.), we get

lim

n→∞sup

p(xn,xm) :m>n

= . (.)

By means of Lemma . and (.),

{xn}is a Cauchy sequence,i.e.,

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To prove (), letx,y∈Xand deﬁnexn=Tnxandyn=Tnyfor everyn∈N. By the same

method as in (.)∼(.), we get

p(xn,yn)≤ϕ

maxp(xi,xj),p(xi,yj),p(yi,xj),p(yi,yj)|n– ≤i,j≤n

≤ϕmaxp(xi,xj),p(xi,yj),p(yi,xj),p(yi,yj)|n– ≤i,j≤n

.. .

≤ϕn–maxp(xi,xj),p(xi,yj),p(yi,xj),p(yi,yj)|≤i,j≤n

≤ϕn–δO(x,y,n)

≤ϕn–(I–ϕ)–b(x,y), (.)

which proves ().

By virtue of (.) and (.), we deduce that

lim

n→∞p

Tnx,Tny=  (.)

for eachx,y∈X. This is the proof of ().

Deﬁnition . Let (X,d) be a metric space, a mappingT :X→Xis called Ćirić type ϕ-generalized single-valuedp-contractive if it satisﬁes the following:

(c) There exist au-distanceponXandϕ: [,∞)→[,∞)such that

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

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

for allx,y∈X.

(c) For eachx∈Xwithlimn→∞Tnx=cx∈X, there existsy∈Xsuch that

limn→∞Tny=Tcx.

Theorem . Let(X,d)be a complete metric space with a u-distance p.Let T:X→X be Ćirić typeϕ-generalized single-valued p-contractive satisfying(ii)∼(iv)of Lemma.. Then:

() limn→∞Tnx=zfor eachx∈X.

() p(Tn_x,_z)_≤ ∞

k=n–ϕk((I–ϕ)–(a(x)))for eachx∈X,where

a(x) = [p(x,x) +p(x,Tx) +p(Tx,x)]×.

() Thas a unique ﬁxed pointzinXandp(z,z) = .

Proof Let x,y∈X and let xn =Tnx and yn=Tny for every n∈ N. Then, by () of

Lemma .,{xn}is a Cauchy sequence.

SinceXis complete,{xn}converges to somez∈X. This is the proof of (). Due to (.),

(iv) of Lemma ., Lemma ., Deﬁnition ., and (u), we have

p(xn,z)≤ lim

m→∞infp(xn,xm)≤

∞

k=n–

ϕk(I–ϕ)–a(x),

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By () and (c) of Deﬁnition ., there existsy∈Xsuch that

lim

n→∞T

n_y₌_Tz. _(.)

In view of (.) and (.), we get

lim

n→∞sup

suppTnx,Tmy:m>n ≤ lim

n→∞sup

suppTnx,Tmx+pTmx,Tmy:m>n ≤ lim

n→∞sup

suppTnx,Tmx:m>n+ lim

n→∞sup

suppTmx,Tmy:m>n

= . (.)

Due to (.), we obtain

lim

n→∞sup

pTnx,Tmy:m>n= . (.)

In terms of (.), (.), and Lemma ., there exist a subsequence{xkn}of{xn}and a

subsequence{ykn}of{yn}such that

lim

n→∞d(xkn,ykn) = . (.)

From (), (.), and (.), we have

d(z,Tz) = .

Thuszis a ﬁxed point ofT.

To prove the unique ﬁxed point ofT, letz=Tzandw=Tw. Then, by hypothesis, we obtain

p(w,z) =p(Tw,Tz)≤ϕmaxp(w,z),p(w,w),p(z,z),p(z,w),

p(z,w) =p(Tz,Tw)≤ϕmaxp(w,z),p(w,w),p(z,z),p(z,w),

p(z,z) =p(Tz,Tz)≤ϕmaxp(w,z),p(w,w),p(z,z),p(z,w),

p(w,w) =p(Tw,Tw)≤ϕmaxp(w,z),p(w,w),p(z,z),p(z,w).

(.)

By (.) and the hypothesis

maxp(w,z),p(w,w),p(z,z),p(z,w)= . (.)

From Lemma . and (.), we have

w=z.

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Corollary . Let(X,d)be a complete metric space with a u-distance p on X.Let T:X→ X be a mapping that satisﬁes the following conditions:

() p(Tx,Ty)≤kmaxp(x,y),p(x,Tx),p(y,Ty),p(x,Ty),p(y,Tx),

p(y,x),p(Tx,x),p(Ty,y),p(Ty,x),p(Tx,y) (.)

for all x,y∈X and for some k∈(, );

() for each x∈X with lim

n→∞T

n_x₌_c

x∈X,there exists y∈X,

such that lim

n→∞T n_y₌_Tc

x.

Then T has a unique ﬁxed point z in X and p(z,z) = .

Proof Letϕ:R+→R+be deﬁned by

ϕ(t) =kt,  <k< . (.)

Then, by (.), all the conditions of Theorem . are satisﬁed.

ThusThas a unique ﬁxed pointzinXandp(z,z) = .

Lemma . Let(X,d)be a complete metric space with a u-distance p on X and let T:X→ X be a mapping satisfying(.)and

infp(x,y) +p(x,Tx) :x∈X>  (.)

for every y∈X with y=Ty.

Then,for each x∈X withlim_n_→∞Tn_x₌_c

x∈X,there exists y∈X such thatlimn→∞Tny=

Tcx.

Proof Suppose that there exists somex∈Xwithlimn→∞Tnx=cx∈Xsuch that

lim

n→∞T n_y₌_Tc

x for ally∈X. (.)

From (.) we get

lim

n→∞T

n_x₌_c

x∈X and lim n→∞T

n_{(Tx) =} _lim n→∞T

n+_x₌_Tc

x. (.)

Then, by (.), the same method as in Theorem . and simple calculations, we have

cx=Tcx, lim n→∞p

Tnx,cx

=  and lim

n→∞p

Tnx,Tn+x= . (.) On account of (.) and the hypotheses of Lemma ., we obtain

 < infp(x,cx) +p(x,Tx) :x∈X

≤infpTnx,cx

+pTnx,Tn+x:n∈N= .

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From Corollary . and Lemma . we have the following corollary.

Corollary .([]) Let(X,d)be a complete metric space with a u-distance p on X.Let T:X→X be a mapping satisfying(.)and(.).Then T has a unique ﬁxed point z in X and p(z,z) = .

Proof Since all the conditions of Corollary . satisfy all the conditions of Corollary .,

we obtain result of Corollary ..

In the next example we shall show that all the conditions of Theorem . are satisﬁed, but condition (.) in Corollary . and condition (.) in Lemma . are not satis-ﬁed.

Example . Letk∈(, ) and letX= [, ] be closed interval with the usual metric, and

p:X×X→R+,T:X→Xandϕ:R+→R+be mappings deﬁned as follows:

p(x,y) =

 –k  +k

x, (.)

Tx=

 +k 

x, (.)

ϕ(t) =

(+_k)t, ≤t≤–_+k_k,

t

+t,

–k

+k<t.

(.)

Deﬁneθ:X×X×R+×R+→R+by

θ(x,y,s,t) =s (.)

for allx,y∈Xands,t∈R+.

Then, by (.)∼(.) and simple calculations, we know thatpis au-distance onX and ϕ satisﬁes (ii) and (iii) in Lemma .. We now show that ϕ satisﬁes (iv) in Lem-ma ..

On account of (.), if ≤t≤–k

+k, thenϕ

n_{(t) = (}+k

 )

n_t_{for all}_n_∈_N_{and so (iv) holds}

for allt∈[,–_+_kk]. Ift>–k

+k, then there existsM∈Nsuch that

ϕM(t)≤ –k

 +k. (.)

Suppose thatϕn(t) >–_+k_kfor alln∈Nandt∈(–_+k_k,∞).

Then, by (.), –_+k_k <ϕn(t) = _+t_nt for all n∈N and t∈ (–_+k_k,∞). Thus  < –_+k_k ≤

limn→∞ϕn(t) =limn→∞+tnt= , a contradiction.

Hence (.) holds.

By virtue of (.) and (.), we get

ϕn(t) =ϕn–MϕM(t)=

 +k 

n–M

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for alln∈Nwithn>M. Thusϕsatisﬁes (iv) in Lemma . fort∈(–_+k_k,∞). Therefore (iv) in Lemma . holds. Using (.)∼(.), we have

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

=ϕ

max

 –k  +k

x,

 –k  +k

y

, (.)

p(Tx,Ty) =

 –k  +k

Tx=

 –k  +k

 +k 

x=

 –k 

x and

ϕ

 –k  +k

x

=

 +k 

 –k  +k

x=

 –k 

x

for allx,y∈X.

By (.), (c) of Deﬁnition . is satisﬁed. Due to (.), since lim_n_→∞Tn_x₌_lim

n→∞(+_k)nx=  for each x∈X, there exists

y=x

 ∈Xsuch thatlimn→∞T

n_y₌_lim

n→∞(+_k)n·x_=  =T.

This implies (c) of Deﬁnition ..

Therefore all the conditions of Theorem . are satisﬁed.

By means of (.) and (.), there exista= ∈Xandb= ∈Xsuch that

k·maxp(a,b),p(a,Ta),p(b,Tb),p(a,Tb),p(b,Ta),

p(b,a),p(Ta,a),p(Tb,b),p(Tb,a),p(Ta,b)

=k·

 –k  +k

and (.)

p(Ta,Tb) =

 –k  +k

Ta=

 –k  +k

 +k 

=

 –k 

and

 –k  >k

 –k  +k

fork∈(, ).

On account of (.), (.) in Corollary . is not satisﬁed. In terms of (.) and (.) we obtain

≤infp(x,y) +p(x,Tx) :x∈X

≤infpTnx,y+pTnx,Tn+x:n∈N =inf

 –k  +k

Tnx+

 –k  +k

Tnx:n∈N

=inf



 –k  +k

·

 +k 

n

x:n∈N

= 

for ally∈Xwithy=Ty. This means that (.) in Lemma . is not satisﬁed.

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The following theorem is a generalization of Suzuki’s ﬁxed point theorem [].

Theorem . Let(X,d)be a complete metric space with a u-distance p on X.Letϕ:R+→

R+be a mapping satisfying conditions(ii)∼(iv)of Lemma..

Let T:X→X be a mapping that satisﬁes the following conditions:

(i) pTx,Tx≤ϕp(x,Tx) for all x∈X;

(ii) If lim

n→∞sup

p(xn,xm) :m>n

= , lim

n→∞p(xn,Txn) = and

lim

n→∞p(xn,y) = ,then Ty=y.

(.)

Then there exists z∈X such that Tz=z and p(z,z) = .

Proof By (.), the same methods in Theorem . and simple calculations, we deduce that

lim

n→∞sup

pTnx,Tmx:m>n= , lim

n→∞T

n_x₌_z,

lim

n→∞p

Tnx,z=  and lim

n→∞p

Tnx,Tn+x= .

(.)

By means of (.) and hypotheses (i), (ii), we obtain

Tz=z and p(z,z) =  and zis a unique ﬁxed point ofT.

Corollary .([]) Let(X,d)be a complete metric space with aτ-distance p on X.Let T:X→X be a mapping satisfying(ii)of Theorem.and

pTx,Tx≤kp(x,Tx)

for all x∈X and some k∈(, ).

Then T has a unique ﬁxed point z in X and p(z,z) = .

Proof Letϕ:R+→R+be deﬁned by

ϕ(t) =kt,  <k< .

Sincepis a τ-distance, pis a u-distance. Thus all the conditions of Theorem . are satisﬁed.

ThereforeThas a unique ﬁxed pointzinXandp(z,z) = .

4 Existence of a solution for an integral equation

In what follows, we assume thatX=C([, ]) is the set of all continuous functions deﬁned on [, ] andϕ:R+→R+satisfy conditions (ii), (iii), and (iv) of Lemma .. Letd,p:X×

X→R+andθ:X×X×R+×R+→R+be mappings deﬁned as follows:

d(x,y) = sup

t∈[,]

x(t) –y(t), p(x,y) = sup

t∈[,]

(15)

and

θ(x,y,s,t) =s

for all x,y∈X ands,t∈R+. Then clearly (X,d) is a complete metric space andpis a

u-distance onX. Now we prove the existence theorem for a solution of the following in-tegral equation by using Theorem .:

x(t) =r(x,t) +





G(t,s)fs,x(s)ds, (.)

wherex∈X,r:X×R→R,G: [, ]×[, ]→Randf: [, ]×R→Rare given mappings.

Theorem . Suppose that the following hypotheses hold:

(I) r:X×R→Ris a continuous mapping such that

r(x,t)≤

ϕx(t) for allx∈Xandt∈R.

(I) G: [, ]×[, ]→Ris a continuous mapping such that

G(t,s)≤

 for allt,s∈[, ].

(I) f : [, ]×R→Ris a continuous mapping such that

fs,x(s)≤ϕx(s) for allx∈Xands∈[, ].

(I) For eachx∈Xwithlimn→∞Tnx=cx∈X,there existsy∈Xsuch thatlimn→∞Tny=

Tcx.

Then the integral equation(.)has a solution x∈X.

Proof LetT:X→Xbe a mapping deﬁned by

(Tx)(t) =r(x,t) +





G(t,s)fs,x(s)ds

for allx∈Xandt∈[, ]. By conditions (I), (I), and (I), we have

(Tx)(t)=r(x,t) +





G(t,s)fs,x(s)ds

≤r(x,t)+





G(t,s)fs,x(s)ds

≤r(x,t)+





G(t,s)fs,x(s)ds

≤r(x,t)+ 





fs,x(s)ds

≤r(x,t)+ 





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≤ 

ϕx(t)+     ϕ sup

t∈[,]

x(t)ds

≤ 

ϕx(t)+  ϕ

sup

t∈[,]

x(t)

for allx∈Xandt∈[, ]. Then

p(Tx,Ty) = sup

t∈[,]

(Tx)(t)≤ sup

t∈[,]



ϕx(t)+  ϕ

sup

t∈[,]

x(t) ≤  ϕ sup

t∈[,]

x(t)+ ϕ

sup

t∈[,]

x(t)

=ϕ

sup

t∈[,]

x(t) ≤ϕ max sup

t∈[,]

_x(t), sup

t∈[,]

y(t), sup

t∈[,]

(Tx)(t), sup

t∈[,]

(Ty)(t)

=ϕmaxp(x,y),p(x,Tx),p(y,Ty),p(x,Ty),p(y,Tx),

for allx,y∈X.

Thus all of the hypotheses of Theorem . are satisﬁed. Hence the mappingThas a ﬁxed point that is a solution inX=C([, ]) of the integral equation (.).

Competing interests

The author declares that he has no competing interests.

Author’s contributions

The author completed the paper himself. The author read and approved the ﬁnal manuscript.

Acknowledgement

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2013R1A1A2057665).

Received: 23 February 2015 Accepted: 26 June 2015

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