Existence of a solution of integral equations via fixed point theorem

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R E S E A R C H

Open Access

Existence of a solution of integral equations

via ﬁxed point theorem

Selma Gülyaz

1

_{, Erdal Karapınar}

2*

_{, Vladimir Rakocevi´c}

3

_{and Peyman Salimi}

4

*_{Correspondence:}

[email protected]; [email protected]

2_{Department of Mathematics,}

Atilim University, ˙Incek, Ankara 06836, Turkey

Full list of author information is available at the end of the article

Abstract

In this paper, we establish a solution to the following integral equation:

u(t) =

T

0

G(t,s)f

(s,

u(s))ds for allt∈[0,T], ()

whereT> 0,f: [0,T]×R→RandG: [0,T]×[0,T]→[0,∞) are continuous functions. For this purpose, we also obtain some auxiliary ﬁxed point results which generalize, improve and unify some ﬁxed point theorems in the literature.

MSC: 47H10; 54H25

Keywords: cyclic representation; ﬁxed point

1 Introduction and preliminaries

Fixed point theory is one of the most efficient tools in nonlinear functional analysis to solve the nonlinear differential and integral equations. The existence/uniqueness of a so-lution of differential/integral equations turns into the existence/uniqueness of a (common) fixed point of the operators which are obtained after suitable substitutions and elementary calculations; see,e.g., [–].

In this paper, we first obtain some fixed point theorems to solve the integral equa-tion menequa-tioned above. For the sake of completeness, we recollect some basic defi-nitions and elementary results. Let X be a nonempty set and T be a self-mapping on X. Then, the set of all fixed points of T on X is denoted by Fix(T)X. Let be

the set of all functions ψ : [,∞)×[,∞)→ [,∞) satisfying the following condi-tions:

() ψis continuous,

() ψ(t,t) = if and only ift=t= ,

() ψ(t,t)≤_(t+t).

Cyclic mapping and cyclic contraction were introduced by Kirk-Srinavasan-Veeramani to improve the well-known Banach ﬁxed point theorem. Later, various types of cyclic con-traction have been investigated by a number of authors; see,e.g., [, –].

Deﬁnition .[] Suppose that (X,d) is a metric space andTis a self-mapping onX. Let

mbe a natural number andXi,i= , . . . ,m, be nonempty sets. ThenY=

m

i=Xiis called a

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cyclic representation ofXwith respect toT if

T(X)⊂X, . . . , T(Xm–)⊂Xm, T(Xm)⊂Xm+,

whereXm+=X.

Deﬁnition .[] LetT:X→X,r>  andη,ξ:X→[, +∞) be two functions. We say thatTisr-(η,ξ)-admissible if

(i) η(x)≥rfor somex∈Ximpliesη(Tx)≥r, (ii) ξ(x)≤rfor somex∈Ximpliesξ(Tx)≤r.

Deﬁnition . Let (X,d) be a metric space andT:Y→Ybe a self-mapping, whereY=

m

i=Xiis a cyclic representation ofY with respect toT. Letη,ξ :Y →[, +∞) be two

functions. An operatorT:Y→Yis called:

• a cyclic weakr-(η,ξ)-C-contractive mapping of the ﬁrst kind if

η(x)η(y)d(Tx,Ty)≤ξ(x)ξ(y)

 

d(x,Ty) +d(y,Tx)–ψd(x,Ty),d(y,Tx) ()

holds for allx∈Xiandy∈Xi+, whereψ∈.

• a cyclic weakr-(η,ξ)-C-contractive mapping of the second kind if

η(x)η(y) +rd(Tx,Ty)≤ξ(x)ξ(y) +r[[d(x,Ty)+d(y,Tx)]–ψ(d(x,Ty),d(y,Tx))] _()

such thatr₊_r_{> }_{holds for all}_x_∈_X

iandy∈Xi+, whereψ∈. 2 Auxiliary ﬁxed point results

We state the main result of this section as follows.

Theorem . Let(X,d)be a complete metric space,m∈N,X,X, . . . ,Xm be nonempty closed subsets of(X,d)and Y=m_i=Xi.Suppose that T:Y→Y is a cyclic weak r-(η,ξ) -C-contractive mapping of the ﬁrst kind such that

(i) Tisr-(η,ξ)-admissible;

(ii) there existsx∈Ysuch thatη(x)≥randξ(x)≤r;

(iii) if{xn}is a sequence inYsuch thatη(xn)≥randξ(xn)≤rfor alln∈Nandxn→x

asn→ ∞,thenη(x)≥randξ(x)≤r.

Then T has a ﬁxed point x∈n_i=Xi.Moreover,ifη(x)≥r,η(y)≥r,ξ(x)≤r,ξ(y)≤r for all x,y∈Fix(T)Y,then T has a unique ﬁxed point.

Proof Let there exist x∈ Y such that η(x)≥r and ξ(x)≤ r. Since T is r-(η,ξ

)-admissible, thenη(Tx)≥randξ(Tx)≤r. Again, since T isr-(η,ξ)-admissible, then

η(Tx)≥randξ(Tx)≤r. By continuing this process, we get

ηTnx

≥r and ξTnx

≤r for alln∈N. ()

On the other hand, sincex∈Y, there exists someisuch thatx∈Xi. NowT(Xi)⊆

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Txn=xn+, wherexn∈Xin. Hence, forn≥, there existsin∈ {, , . . . ,m}such thatxn∈Xin

andxn+∈Xin+. In casexn=xn+for somen= , , , . . . , then it is clear thatxnis a ﬁxed point ofT. Now assume thatxn=xn+for alln. Hence, we haved(xn–,xn) >  for alln. Set dn:=d(xn,xn+). We shall show that the sequence{dn}is non-increasing. Due to () with x=xn–andy=xn, we get

rd(xn,xn+)

=rd(Txn–,Txn)≤η(xn–)η(xn)d(Txn–,Txn)

≤ξ(xn–)ξ(xn)

 

d(xn–,Txn) +d(xn,Txn–)

–ψd(xn–,Txn),d(xn,Txn–)

=ξ(xn–)ξ(xn)



d(xn–,xn+) –ψ

d(xn–,xn+), 

≤r



d(xn–,xn+) –ψ

d(xn–,xn+),  ,

which implies

d(xn,xn+)≤



d(xn–,xn+) –ψ

d(xn–,xn+), 

≤

d(xn–,xn+)

≤ 

d(xn–,xn) +d(xn,xn+)

, ()

and sodn≤dn–for alln∈N. Then there existd≥ such thatlimn→∞dn=d. Suppose,

on the contrary, thatd> . Also, taking limit asn→ ∞in (), we deduce

d≤

nlim→∞d(xn–,xn+)≤



(d+d) =d,

that is,

lim

n→∞d(xn–,xn+) = d. ()

Taking limit asn→ ∞in () and using (), we get

d≤

[d] –ψ(d, ).

Consequently, we haveψ(d, ) = , which yieldsd= . Hence

lim

n→∞d(xn,xn+) = . ()

We shall show that{xn}is a Cauchy sequence. To reach this goal, ﬁrst we prove the

fol-lowing claim:

(K) For everyε> , there existsn∈Nsuch that ifr,q≥nwithr–q≡(m), then

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Suppose, to the contrary, that there existsε>  such that for anyn∈N, we can ﬁnd rn>qn≥nwithrn–qn≡(m) satisfying

d(xqn,xrn)≥ε. ()

Now, we taken> m. Then, corresponding toqn≥n, one can choosernin such a way that

it is the smallest integer withrn>qnsatisfyingrn–qn≡(m) andd(xqn,xrn)≥ε. Therefore, d(xqn,xrn–m) <ε. By using the triangular inequality,

ε≤d(xqn,xrn)≤d(xqn,xrn–m) + m

i=

d(xrn–i,xrn–i–) <ε+ m

i=

d(xrn–i,xrn–i–).

Lettingn→ ∞in the last inequality, keeping () in mind, we derive that

lim

n→∞d(xqn,xrn) =ε. ()

Again,

ε≤d(xqn,xrn)

≤d(xqn,xqn+) +d(xqn+,xrn+) +d(xrn+,xrn)

≤d(xqn,xqn+) +d(xqn+,xqn) +d(xqn,xrn) +d(xrn,xrn+) +d(xrn+,xrn).

Taking () and () into account, we get

lim

n→∞d(xqn+,xrn+) =ε ()

asn→ ∞in ().

Also we have the following inequalities:

d(xqn,xrn+)≤d(xqn,xrn) +d(xrn,xrn+) ()

and

d(xqn,xrn)≤d(xqn,xrn+) +d(xrn,xrn+). ()

Lettingn→ ∞in () and (), we derive that

lim

n→∞d(xqn,xrn+) =ε. ()

Again, we have

d(xrn,xqn+)≤d(xrn,xrn+) +d(xrn+,xqn+) ()

and

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Lettingn→ ∞in () and (), we conclude that

lim

n→∞d(xrn,xqn+) =ε. ()

Sincexqnandxrnlie in diﬀerent adjacently labeled setsXiandXi+for certain ≤i≤m,

using the fact thatT is a cyclic weakr-(η,ξ)-C-contractive mapping of the ﬁrst kind, we have

rd(xrn+,xqn+)

=rd(Txrn,Txqn)≤η(xrn)η(xqn)d(Txrn,Txqn)

≤ξ(xrn)ξ(xqn)

 

d(xrn,Txqn) +d(xqn,Txrn)

–ψd(xrn,Txqn),d(xqn,Txrn)

≤r

 

d(xrn,xqn+) +d(xqn,xrn+)

–ψd(xrn,xqn+),d(xqn,xrn+) , ()

which implies

d(xrn+,xqn+)≤

 

–ψd(xrn,xqn+),d(xqn,xrn+)

.

Lettingn→ ∞in the inequality above and keeping the expressions (), (), (), (), () in mind, we conclude that

ε≤ε–ψ(ε,ε).

Thus, we haveψ(ε,ε) = , which yields thatε= . Hence, (K) is satisﬁed.

We shall show that the sequence{xn}is Cauchy. Fixε> . By the claim, we ﬁndn∈N

such that ifr,q≥nwithr–q≡(m), then

d(xr,xq)≤

ε

. ()

Sincelim_n_→∞d(xn,xn+) = , we also ﬁndn∈Nsuch that

d(xn,xn+)≤

ε

m ()

for anyn≥n. Suppose thatr,s≥max{n,n}ands>r. Then, there existsk∈ {, , . . . ,m}

such thats–r≡k(m). Therefore, s–r+ϕ≡(m) forϕ=m–k+ . So, we have, for

j∈ {, . . . ,m},s+j–r≡(m)

d(xr,xs)≤d(xr,xs+j) +d(xs+j,xs+j–) +· · ·+d(xs+,xs).

By () and () and from the last inequality, we get

d(xr,xs)≤

ε +j×

ε m≤

ε +m×

ε m=ε.

This proves that{xn}is a Cauchy sequence. SinceYis closed in (X,d), then (Y,d) is also

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we prove thatxis a ﬁxed point ofT. In fact, sincelimn→∞xn=xand, asY=

m

i=Xiis a

cyclic representation ofYwith respect toT, the sequence{xn}has inﬁnite terms in each Xifori∈ {, , . . . ,m}. Suppose thatx∈Xi,Tx∈Xi+ and we take a subsequencexnk of

{xn}withxnk∈Xi–. Now from (iii) we haveη(x)≥randξ(x)≤r. By using the contractive

condition, we can obtain

rd(Tx,Txnk)≤η(x)η(xnk)d(Tx,Txnk)

≤ξ(x)ξ(xnk)

 

d(x,Txnk) +d(xnk,Tx)

–ψd(x,Txnk),d(xnk,Tx)

≤r

 

d(x,Txnk) +d(xnk,Tx)

–ψd(x,Txnk),d(xnk,Tx) , ()

which implies

d(Tx,xnk+)≤

 

d(x,xnk+) +d(xnk,Tx)

–ψd(x,xnk+),d(xnk,Tx)

.

Passing to the limit ask→ ∞in the last inequality, we get

d(x,Tx)≤ 

d(x,Tx) –ψ

,d(x,Tx)

≤ 

d(x,Tx),

which impliesd(x,Tx) = ,i.e.,x=Tx. Finally, to prove the uniqueness of the ﬁxed point, suppose thatx,y∈Fix(T)Y such thatη(x)≥r,η(y)≥r,ξ(x)≤r,ξ(y)≤r, wherex=y. The

cyclic character ofTand the fact thatx,y∈Xare ﬁxed points ofTimply thatx,y∈m_i=Xi.

Suppose thatx=y. That is,d(x,y) > . Using the contractive condition, we obtain

rd(Tx,Ty)≤η(x)η(y)d(Tx,Ty)

≤ξ(x)ξ(y)

 

d(x,Ty) +d(y,Tx)–ψd(x,Ty),d(y,Tx)

≤r

 

d(x,Ty) +d(y,Tx)–ψd(x,Ty),d(y,Tx) ,

which implies

d(x,y)≤d(x,y) –ψd(x,y),d(x,y).

Thenψ(d(x,y),d(x,y)) =  and sod(x,y) = ,i.e.,x=y, which is a contradiction. This

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Example . LetX=Rwith the metric d(x,y) =|x–y|for all x,y∈X. SupposeA=

(–∞, ] andA= [,∞) andY=



i=Ai. DeﬁneT:Y→Yandη,ξ:Y→[,∞) by

Tx= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x+ √

x_+ ifx∈(–∞, –],

sin_x

x ifx∈[–, –), –x ifx∈[–, –),  ifx∈[–, ], –lnx ifx∈(, ),

–x

(–x)(–x) ifx∈[, ),

 √

 –x ifx∈[,∞),

η(x) =

 ifx∈[–, ],  otherwise,

ξ(x) =

 ifx∈[–, ],  otherwise.

Also, deﬁneψ : [,∞)→[,∞) byψ(t,s) =_(t+s). Clearly,TA⊆A,TA⊆Aand

η()≥ andξ()≤. Letη(x)≥, thenx∈[–, ]. On the other hand,Tw∈[–, ] for all

w∈[–, ],i.e.,η(Tx)≥. Similarly,ξ(x)≤ impliesξ(Tx)≤. Therefore,Tis anr-(η,ξ )-admissible mapping. Let{xn}be a sequence inXsuch thatη(xn)≥,ξ(xn)≤ andxn→x

asn→ ∞. Thenxn∈[–, ]. So,x∈[–, ],i.e.,η(x)≥ andξ(x)≤.

Letx∈Aandy∈A. Now, ifx∈/[–, ] ory∈/[, ], thenη(x)η(y) = . Also, ifx∈[–, ]

andy∈[, ], thend(Tx,Ty) = . That is,η(x)η(y)d(Tx,Ty) =  for allx∈Aand ally∈A.

Hence,

η(x)η(y)d(Tx,Ty) = ≤ξ(x)ξ(y)

 

d(x,Ty) +d(y,Tx)–ψd(x,Ty),d(y,Tx)

for allx∈Aandy∈A. ThenTis a cyclic weakr-(η,ξ)-C-contractive mapping of the ﬁrst

kind. Therefore all the conditions of Theorem . hold andThas a ﬁxed point inA∩A.

Here,x=  is a ﬁxed point ofT.

Theorem . Let(X,d)be a complete metric space,m∈N,X,X, . . . ,Xm be nonempty closed subsets of(X,p)and Y =m_i=Xi.Suppose that T:Y→Y is a cyclic weak r-(η,ξ) -C-contractive mapping of the second kind such that

(i) Tisr-(η,ξ)-admissible;

(ii) there existsx∈Ysuch thatη(x)≥randξ(x)≤r;

(iii) if{xn}is a sequence inYsuch thatη(xn)≥randξ(xn)≤rfor alln∈Nandxn→x

asn→ ∞,thenη(x)≥randξ(x)≤r.

Then T has a ﬁxed point x∈n_i=Xi.Moreover,ifη(x)≥r,η(y)≥r,ξ(x)≤r,ξ(y)≤r for all x,y∈Fix(T)Y,then T has a unique ﬁxed point.

Proof By a similar method as in the proof of Theorem ., we have

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We shall show that the sequence{dn:=d(xn,xn+)}is non-increasing. Due to () withx= xn–andy=xn, we get

r+rd(xn,xn+)=r+rd(Txn–,Txn)≤η(xn–)η(xn) +r

d(Txn–,Txn)

≤ξ(xn–)ξ(xn) +r [

[d(xn–,Txn)+d(xn,Txn–)]–ψ(d(xn–,Txn),d(xn,Txn–))]

=ξ(xn–)ξ(xn) +r [

d(xn–,xn+)–ψ(d(xn–,xn+),)]

≤r+r[d(xn–,xn+)–ψ(d(xn–,xn+),)] ,

which implies

d(xn,xn+)≤



d(xn–,xn+) –ψ

d(xn–,xn+), 

≤

d(xn–,xn+)

≤ 

d(xn–,xn) +d(xn,xn+)

, ()

and sodn≤dn–for alln∈N. Then there existsd≥ such thatlimn→∞dn=d. We shall

show thatd=  by the method ofreductio ad absurdum. Suppose thatd> . By letting

n→ ∞in (), we deduce

d≤

nlim→∞d(xn–,xn+)≤



(d+d) =d,

that is,

lim

n→∞d(xn–,xn+) = d. ()

Taking limit asn→ ∞in () and using (), we get

d≤

[d] –ψ(d, ).

Thus, we haveψ(d, ) =  and henced= , which is a contradiction. Consequently, we have

lim

n→∞dn=nlim→∞d(xn,xn+) = . ()

We shall show that{xn}is a Cauchy sequence. To reach this goal, ﬁrst we prove the

fol-lowing claim:

(K) For everyε> , there existsn∈Nsuch that ifr,q≥nwithr–q≡(m), then

d(xr,xq) <ε.

Suppose, to the contrary, that there exists ε>  such that for anyn∈Nwe can ﬁnd

rn>qn≥nwithrn–qn≡(m) satisfying

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Following the related lines in Theorem ., we deduce

lim

n→∞d(xqn,xrn) =ε, ()

lim

n→∞d(xqn+,xrn+) =ε, ()

lim

n→∞d(xqn,xrn+) =ε ()

and

lim

n→∞d(xrn,xqn+) =ε. ()

Sincexqn andxrnlie in diﬀerent adjacently labeled setsXiandXi+for certain ≤i≤m,

using the fact that a cyclic weak r-(η,ξ)-C-contractive mapping of the second kind, we have

r₊_rd(xrn+,xqn+)

=r₊_rd(Txrn,Txqn)_≤_η₍_x

rn)η(xqn) +r

d(Tx_rn,Tx_qn)

≤ξ(xrn)ξ(xqn) +r [

[d(xrn,Txqn)+d(xqn,Txrn)]–ψ(d(xrn,Txqn),d(xqn,Txrn))]

≤r+r[



[d(xrn,xqn+)+d(xqn,xrn+)]–ψ(d(x_rn,x_qn+),d(xqn,Txrn+))],

which implies

d(xrn+,xqn+)≤

 

–ψd(xrn,xqn+),d(xqn,Txrn+)

.

Lettingn→ ∞in the inequality above and by applying () (), (), (), (), we de-duce that

ε≤ε–ψ(ε,ε).

Consequently, we haveψ(ε,ε) = , and henceε= . As a result, we conclude that (K) is satisﬁed. We assert that the sequence{xn}is Cauchy. Fixε> . By the claim, we ﬁndn∈N

such that ifr,q≥nwithr–q≡(m), then

d(xr,xq)≤

ε

. ()

Sincelimn→∞d(xn,xn+) = , we also ﬁndn∈Nsuch that

d(xn,xn+)≤

ε

m ()

for anyn≥n. Suppose thatr,s≥max{n,n}ands>r. Then there existsk∈ {, , . . . ,m}

such thats–r≡k(m). Therefore, s–r+ϕ≡(m) forϕ=m–k+ . So, we have, for

j∈ {, . . . ,m},s+j–r≡(m),

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By () and () and from the last inequality, we get

d(xr,xs)≤

ε +j×

ε m

≤ε +m×

ε m=ε.

This proves that{xn}is a Cauchy sequence. SinceYis closed in (X,d), then (Y,d) is also

complete, there existsx∈Y=m_i=Xisuch thatlimn→∞xn=xin (Y,d). In what follows,

we prove thatxis a ﬁxed point ofT. In fact, sincelim_n_→∞xn=xand, asY= m

i=Xiis a

cyclic representation ofYwith respect toT, the sequence{xn}has inﬁnite terms in each Xifori∈ {, , . . . ,m}. Suppose thatx∈Xi,Tx∈Xi+ and we take a subsequencexnk of

{xn}withxnk∈Xi–. Now from (iii) we haveη(x)≥randξ(x)≤r. By using the contractive

condition, we can obtain

r+rd(Tx,Txnk)_≤_η₍_x₎_η₍_x nk) +r

d(Tx,Tx_nk)

≤ξ(x)ξ(xnk) +r [

[d(x,Txnk)+d(xnk,Tx)]–ψ(d(x,Txnk),d(xnk,Tx))]

≤r+r[



[d(x,Txnk)+d(xnk,Tx)]–ψ(d(x,Txnk),d(xnk,Tx))]_, _()

which implies

d(Tx,xnk+)≤

 

d(x,xnk+) +d(xnk,Tx)

–ψd(x,xnk+),d(xnk,Tx)

.

Passing to the limit ask→ ∞in the last inequality, we get

d(x,Tx)≤ 

d(x,Tx) –ψ

,d(x,Tx)≤

d(x,Tx),

which impliesd(x,Tx) = ,i.e.,x=Tx. Finally, to prove the uniqueness of the ﬁxed point, suppose thatx,y∈Fix(T)Y such thatη(x)≥r,η(y)≥r,ξ(x)≤r,ξ(y)≤r, wherex=y. The

cyclic character ofTand the fact thatx,y∈Xare ﬁxed points ofTimply thatx,y∈m_i=Xi.

Suppose thatx=y. That is,d(x,y) > . Using the contractive condition, we obtain

r+rd(Tx,Ty)≤η(x)η(y) +rd(Tx,Ty)

≤ξ(x)ξ(y) +r[[d(x,Ty)+d(y,Tx)]–ψ(d(x,Ty),d(y,Tx))]

≤r+r[[d(x,Ty)+d(y,Tx)]–ψ(d(x,Ty),d(y,Tx))],

which implies

d(x,y)≤d(x,y) –ψd(x,y),d(x,y).

Hence, we obtainψ(d(x,y),d(x,y)) = , which impliesd(x,y) = , that is,x=ya

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3 Existence of solutions of an integral equation

ForT> , we denote byX=C([,T]) the set of real continuous functions on [,T]. We endowXwith the metric

d∞(u,v) =u–v∞ for allu,v∈X.

It is evident that (X,d∞) is a complete metric space. Consider the integral equation

u(t) =

T



G(t,s)fs,u(s)ds for allt∈[,T], ()

() f: [,T]×R→RandG: [,T]×[,T]→[,∞)are continuous functions. () Let(α,β)∈X_,₍_α

,β)∈Rsuch that

α≤α(t)≤β(t)≤β for allt∈[,T]. ()

Assume that for allt∈[,T], we have

α(t)≤

T



G(t,s)fs,β(s)ds ()

and

β(t)≥

T



G(t,s)fs,α(s)ds. ()

Let for alls∈[,T],f(s,·)be a decreasing function, that is,

x,y∈R, x≥y ⇒ f(s,x)≤f(s,y). ()

LetZ:={u∈X:u≤β} ∪ {u∈X:u≥α}. There exist≤r< andθ,π:Z→R

such that ifθ(x)≥andθ(y)≥with (x≤βandy≥α) or (x≥αandy≤β),

then for everys∈[,T], we have

fs,x(s)–fs,y(s)≤r|π(y)|

 x(s) –Ty(s)+y(s) –Tx(s). ()

() Assume that

_Tπ(y)G(t,s)ds ∞≤

 ()

for allx∈Z, whereθ(x)≥. Suppose that

θ(x)≥ ⇒ θ(Tx)≥ forx∈ {u∈X:u≤β} ∪ {u∈X:u≥α}. ()

() If{xn}is a sequence in{u∈X:u≤β} ∪ {u∈X:u≥α}such thatθ(xn)≥for all n∈Nandxn→xasn→ ∞, thenθ(x)≥.

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Theorem . Under assumptions()-(), integral equation()has a solution in {u∈ C([,T]) :α(t)≤u(t)≤β(t)for all t∈[,T]}.

Proof Deﬁne the closed subsets ofX,AandAby

A={u∈X:u≤β}

and

A={u∈X:u≥α}.

Also deﬁne the mappingT:X→Xby

Tu(t) =

T



G(t,s)fs,u(s)ds for allt∈[,T].

Let us prove that

T(A)⊆A and T(A)⊆A. ()

Supposeu∈A, that is,

u(s)≤β(s) for alls∈[,T].

Applying condition (), sinceG(t,s)≥ for allt,s∈[,T], we obtain that

G(t,s)fs,u(s)≥G(t,s)fs,β(s) for allt,s∈[,T].

The above inequality with condition () imply that

T



G(t,s)fs,u(s)ds≥

T



G(t,s)fs,β(s)ds≥α(t)

for allt∈[,T]. Then we haveTu∈A.

Similarly, letu∈A, that is,

u(s)≥α(s) for alls∈[,T].

Using condition (), sinceG(t,s)≥ for allt,s∈[,T], we obtain that

G(t,s)fs,u(s)≤G(t,s)fs,α(s) for allt,s∈[,T].

The above inequality with condition () imply that

T



G(t,s)fs,u(s)ds≤

T



G(t,s)fs,α(s)ds≤β(t)

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Now, let (u,v)∈A×A, that is, for allt∈[,T],

u(t)≤β(t), v(t)≥α(t).

This implies from condition () that for allt∈[,T],

u(t)≤β, v(t)≥α.

Now, by conditions () and (), we have, for alls∈[,T],

Tu(t) –Tv(t)=

T



G(t,s)fs,u(s)–fs,v(s)ds

≤

T



G(t,s)fs,u(s)–fs,v(s)ds

≤

T



G(t,s)r|π(y)|

 u(s) –Tv(s)+v(s) –Tu(s)ds

≤ r 

u–Tv∞+v–Tu∞

T



π(v)G(t,s)ds ∞

≤ r 

u–Tv∞+v–Tu∞,

which implies

Tu–Tv∞≤ r 

u–Tv∞+v–Tu∞.

Deﬁneη,ξ:Z→[,∞) byη(u) =,, otherwiseθ(u)≥, andξ(u) = . Further,ψ(t,t) =(–r)_ (t+t).

Hence,

η(u)η(v)d∞(Tu,Tv)≤ r 

d∞(u,Tv) +d∞(v,Tu)

for all (u,v)∈A×A. By a similar method, we can show that the above inequality holds

if (u,v)∈A×A. Now, all the conditions of Theorem . hold andThas a ﬁxed pointz*

in

A∩A=

u∈C[,T]:α≤u(t)≤βfor allt∈[,T].

That is,z*_∈_A

∩Ais the solution to ().

Example . In this example, we denote byX=C([, ]) the set of real continuous func-tions on [, ]. We endowXwith the metric

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Consider the following continuous functions:

f(t,x) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x _if_x_∈_(–_∞_{, ),}

 ifx∈[, ],

x_{– } _if_x_∈_{(, ),}

 ifx∈[,√e_{– ],} x_{+  –}_e _if_x_∈₍√_e_{– ,}_∞₎

for allt∈[, ]

and

G(t,s) = t  +te

s _{for all}_s_,_t_∈_{[, ]}_×_{[, ].}

Letα(t) =  andβ(t) = . Then, for (α,β) = (, )∈R, we have

α≤α(t)≤β(t)≤β;

α(t) = ≤





G(t,s)fs,β(s)ds= 

and

β(t) = ≥





G(t,s)fs,α(s)ds= .

Also,Z:={u∈X:u≤β} ∪ {u∈X:u≥α}=X. Deﬁneθ,π:Z→Rby

θx(t)=

 if ≤x(t)≤ for allt∈[, ]

–, otherwise and π(x) =



e– .

Clearly,θ()≥. Also, ifθ(x(t))≥, then ≤x(t)≤. On the other hand,

Tu(t) =





G(t,s)fs,u(s)ds= 

for all ≤u(t)≤. That is,θ(Tx(t))≥. Hence,θ(x)≥ impliesθ(Tx)≥.

Assumeθ(x(s))≥ andθ(y(s))≥ with (x≤βandy≥α) or (x≥αandy≤β).

Thus, ≤x(s)≤ and ≤y(s)≤, which impliesf(s,x(s)) =f(s,y(s)) = . That is,

fs,x(s)–fs,y(s)= ≤r|π(y)|

 x(s) –Ty(s)+y(s) –Tx(s)

for alls∈[, ], where ≤r< . Further,





π(y)G(t,s)ds=







e– 

t

 +te

s_ds₌ t

 +t≤,

and so

_Tπ(y)G(t,s)ds ∞≤

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Assume that{xn}is a sequence inXsuch thatθ(xn)≥ for alln∈Nandxn→xasn→ ∞.

Then ≤xn≤. So, ≤x≤. That is,θ(x)≥.

Therefore, all of the conditions of Theorem . are satisﬁed. Then the integral equation

u(t) = t  +t





esfs,u(s)ds

has a solution in{u∈C([, ]) : ≤u(t)≤ for allt∈[, ]}. Here,u(t) =  is a solution. But if we chosex(t) =  andy(t) =

√

e_{– , then}_f₍_s_,_x

(s)) =  andf(s,y(s)) = . That

is,

fs,x(s)

–fs,y(s)= .

Also,

lnx(s) –y(s) 

+ =

ln –√e_{– }_{+ }₌√_ln_e_{= ,}

and so

fs,x(s)

–fs,y(s)=  >  =

lnx(s) –y(s) 

+ .

That is, Theorem . of [] cannot be applied to this example.

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, Cumhuriyet University, Sivas, Turkey.}2_{Department of Mathematics, Atilim University, ˙Incek,}

Ankara 06836, Turkey. 3_{Faculty of Sciences and Mathematics, University of Nis, Visegradska 33, Nis, 18000, Serbia.}4_Young

Researchers and Elite Club, Rasht Branch, Islamic Azad University, Rasht, Iran.

Acknowledgements

The authors thank the anonymous referees for their remarkable comments, suggestions and ideas that helped to improve this paper. The third author (V Rakocevi´c) is supported by Grant No. 174025 of the Ministry of Science, Technology and Development, Republic of Serbia.

Received: 14 May 2013 Accepted: 18 October 2013 Published:11 Nov 2013 References

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