A solution for the non cooperative equilibrium problem of two person via fixed point theory

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R E S E A R C H

Open Access

A solution for the non-cooperative

equilibrium problem of two person via ﬁxed

point theory

Tran Duc Thanh

1

_{, Aatef Hobiny}

2

_{and Erdal Karapınar}

3,4*

*_{Correspondence:}

[email protected]

3_{Department of Mathematics,}

Atilim University, ˙Incek, Ankara, 06836, Turkey

4_{Nonlinear Analysis and Applied}

Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article

Abstract

In this paper, we investigate the non-cooperative equilibrium problem of two person games in the setting of game theory and propose a solution via coupled ﬁxed point results in the context of partial metric spaces. We also realize that our coupled ﬁxed point results can be applied to get a solution of a class of nonlinear Fredholm type integral equations.

MSC: 46J10; 46J15; 47H10

Keywords: coupled ﬁxed point; partial metric;F-contractions; non-cooperative equilibrium

1 Introduction

It is very well known fact that real world problem can be modeled as a mathematical equation. The existence of a solution of such problems has been investigated in several branches of mathematics, such as differential equations, integral equations, functional equations, partial differential equations, random differential equations,etc.and one has proposed solutions for such problems via fixed point theory. But the application area of fixed point theory is not only limited to mathematics, but also occurs in other quantita-tive sciences, such as, computer science, economics, biology, physics,etc.Game theory, a branch of economics, has used fixed point theory techniques and approaches to solve its own problems.

Game theory can be regarded as a formal (mathematical) way to study games. Indeed, we consider the games as conflicts where some number of individuals (called players) take part and each one tries to maximize his utility in taking part in the conflict. Games can be classified in many ways, but here we focus on the following classification: Cooperative games, in which, players are allowed to cooperate and non-cooperative games, in which players are not allowed to cooperate. In the sequel, we shall demonstrate how the question of the existence of equilibria is related to the question of the existence of a fixed point. Throughout the paper, we follow the notion and notation in []. We recall some basic concepts.

Atwo person gameGin normal form consists of the following data:

() topological spacesSandS, the so called strategies for player  resp. player ,

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() a topological subspaceU⊂S×Sof an allowed strategy pair,

() a biloss operator

L:U→R

(s,s)→L(s,s);L(s,s),

()

whereLi(s,s)is the loss of playeriif the strategiessandsare played.

A pair (s,s)∈Uis called anon-cooperative equilibriumif

L(s,s)≤L(s,s), ∀s∈S,

L(s,s)≤L(s,s), ∀s∈S.

()

Assume that there exist mappings

C:S→S,

D:S→S,

()

such that the following equations hold:

L

C(s),s

=min

s∈SL(s,s), ∀s∈S,

Ls,D(s)=min

s∈SL(s,s), ∀s∈S.

()

Such mappingsCandDare calledoptimal decision rules. Then any solution (s,s) of the system

C(s) =s,

D(s) =s,

()

is a non-cooperative equilibrium. Denoting byFthe function

F:S×S→S×S

(s,s)→

C(s);D(s)

,

()

any coupled ﬁxed point (s,s) ofFis a non-cooperative equilibrium. Hence, the inves-tigation of the existence of a solution for a non-cooperative equilibrium is equivalent to searching for the existence of a couple ﬁxed point. More details as regards game theory can be found in [].

The main goal of the present work is to solve the problem of the non-cooperative equi-librium of two person games. For this purpose, we shall present some coupled ﬁxed point theorems in partial metric spaces. Our aim is to explore not only the results themselves but also their applications to nonlinear integral equations.

2 Preliminaries

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and computer domain. Indeed, a partial metric is a function that is obtained from the metric by replacing the conditiond(x,x) =  with the conditiond(x,x)≤d(x,y) for allx,y. In the last decade, a number of authors have brought into focus ﬁxed point problems in the context of partial metric spaces as well as topological properties of a partial metric space; seee.g.[–] and the related references given therein.

We ﬁrst need to recall some basic concepts and necessary results. Throughout the paper,

NandNdenote the set of positive integers and the set of nonnegative integers, respec-tively. Similarly,R,R+_{, and}_R+

represent the set of reals, positive reals, and nonnegative reals, respectively.

Deﬁnition .(Seee.g.[, ]) LetXbe a nonempty set. The mappingp:X×X→[,∞) is said to be apartial metriconXif for anyx,y,z∈Xthe following conditions hold true:

(P) x=yif and only ifp(x,x) =p(y,y) =p(x,y). (P) p(x,x)≤p(x,y).

(P) p(x,y) =p(y,x).

(P) p(x,z)≤p(x,y) +p(y,z) –p(y,y).

The pair (X,p) is then called apartial metric space(in short, PMS).

Let (X,p) be a partial metric space. Then the functionsdp,dm:X×X→[,∞) given by

dp(x,y) = p(x,y) –p(x,x) –p(y,y)

and

dm(x,y) =maxp(x,y) –p(x,x),p(x,y) –p(y,y)

are the (usual) metrics onX. It is easy to check thatdpanddmare equivalent. Note that each partial metricponXgenerates aT-topologyτpwith as a base the family of open p-balls{Bp(x,ε) :x∈X,ε> }, whereBp(x,ε) ={y∈X:p(x,y) <p(x,x) +ε}.

Deﬁnition .(Seee.g.[, ]) Let (X,p) be a partial metric space.

() A sequence{xn}inXconvergestox∈Xif and only ifp(x,x) =limn→∞p(xn,x).

() A sequence{xn}inXis called aCauchy sequenceif and only iflimn,m→∞p(xn,xm)

exists (and is ﬁnite).

() (X,p)is calledcompleteif every Cauchy sequence{xn}inXconverges tox∈X. () A mappingf :X→Xis said to becontinuousatx∈Xif, for everyε> , there

existsδ> such thatf(B(x,δ))⊂B(f(x),ε).

Referring to [], we say that a sequence{xn}in (X,p) is called a-Cauchysequence if limn,m→∞p(xn,xm) = . Also, we say that (X,p) is-completeif every -Cauchy sequence

inXconverges, with respect to the partial metricp, to a pointx∈Xsuch thatp(x,x) = . Notice that if (X,p) is complete, then it is -complete, but the converse does not hold. Moreover, every -Cauchy sequence in (X,p) is Cauchy in (X,dp).

Example .(Seee.g.[, ])

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() LetX= [, +∞)∩Q, whereQis the set of rational numbers. Deﬁne

p(x,y) =max{x,y}, for allx,y∈X. Then(X,p)is a -complete partial metric space which is not complete.

Proposition .(Seee.g.[, ]) Let(X,p)be a partial metric space.

() A sequence{xn}is a Cauchy sequence in(X,p)if and only if{xn}is a Cauchy sequence in(X,dp).

() (X,p)is complete if and only if(X,dp)complete.Moreover, lim

n→∞dp(xn,x) =  ⇔ nlim→∞p(x,x) =nlim→∞p(xn,x) =n,mlim→∞p(xm,xn).

The following lemmas have an important role to play in the proofs of the theorems.

Lemma .(Seee.g.[, ]) Assume xn→z as n→ ∞in a PMS(X,p)such that p(z,z) = . Thenlim_n_→∞p(xn,y) =p(z,y)for every y∈X.

Lemma .(Seee.g.[, ]) Let(X,p)be a complete PMS.Then:

() Ifp(x,y) = thenx=y. () Ifx=y,thenp(x,y) > .

Lemma .(Seee.g.[, ]) Let(X,p)be a PMS.If xn→x and yn→y as n→ ∞for all xn,yn,x,y∈X then p(xn,yn)→p(x,y)as n→ ∞.

The existence and uniqueness of fixed points of contractive type mappings in partially ordered metric spaces have been considered recently by several authors: Ran and Reurings [], Nieto and Rodriguez-Lopez [, ]. On the other hand, the notion of a coupled fixed point was suggested by Guo and Lakshmikantham in []. Following this initial result, Gnana Bhaskar and Lakshmikantham [] proposed the notion of mixed monotone prop-erty and get coupled fixed point results in the setting of partially ordered metric spaces (see also [–] and the related references therein.) Later, it was reported that most of the coupled fixed point results can be derived from the existence results, and vice versa; seee.g.[–]. On the other hand, coupled fixed point results still have worth regarding their applications. Most of the times, using coupled fixed point theory is the most eco-nomical way to solve problems (regarding time and speed of the process). This paper can be considered as an example.

Recall that a pair (x,y)∈X×Xis called acoupled ﬁxed pointof the mappingT:X×X→ XifT(x,y) =x,T(y,x) =y(seee.g.[]).

Deﬁnition .([]) Let (X,≤) be a partially ordered set andT:X×X→X. The map-pingTis said to have themixed monotone propertyifT(x,y) is monotone non-decreasing inxand monotone non-increasing iny, that is, for anyx,y∈X

x,x∈X, x≤x ⇒ T(x,y)≤T(x,y)

and

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Next, we introduce a class of functions which plays a crucial role in this paper. LetF:

R+

→Rbe a mapping satisfying:

(F) Fis strictly increasing and continuous.

(F) For each sequence(an)⊂R+,limn→∞an= if and only iflimn→∞F(an) = –∞. We denote byF the family of all functionsF that satisfy the conditions (F)-(F) (see []). It is easy to check thatF(x) =lnxandG(x) =lnx+xfor allx∈R+

belong toF. In [], Wardowski introduced the new concept of anF-contraction and proved ﬁxed point theorems in the classical setting of metric spaces. In [], the authors introduced the concept of anF-contraction, a generalizedF-contraction, and they proved some ﬁxed point theorems for multi-valued mappings in the partial metric spaces (see also [, , –]).

Deﬁnition .([]) Let (X,p) be a partial metric space. A mappingT :X×X→Xis

called anF-contraction if there existF∈Fandτ∈R+_such that

τ+Fp(Tx,Ty)≤Fp(x,y)

for allx,y∈X.

3 Auxiliary results: coupled ﬁxed points in partial metric spaces

In this section we state and prove some new coupled ﬁxed point results forF-contractive mappings in the context of complete partial metric spaces.

Theorem . Let(X,≤)be a partially ordered set and suppose there exists a partial met-ric p on X such that(X,p)is a-complete partial metric space.Let T:X×X→X be a continuous mapping having the mixed monotone property on X.Suppose also that

()

τ+FpT(x,y),T(u,v)≤Fmaxp(x,u),p(y,v) ()

for allx≤u,y≥v,for someF∈Fandτ> .

() There arex,y∈Xsuch thatx≤T(x,y),y≥T(y,x).

Then T has a coupled ﬁxed point,that is,there exist x,y∈X such that x=T(x,y),y= T(y,x).

Proof Letx,y∈Xbe such thatx≤T(x,y),y≥T(y,x). Letx=T(x,y) andy= T(y,x). Thenx≤xandy≥y. Again, letx=T(x,y) andy=T(y,x). SinceThas the mixed monotone property, we havex≤xandy≥y. Continuing this way, we get two sequences{xn}and{yn}inXsuch thatxn+=T(xn,yn),yn+=T(yn,xn) and

x≤x≤x≤ · · · ≤xn≤xn+· · ·, y≥y≥y≥ · · · ≥yn≥yn+≥ · · ·.

Now, for eachn= , , , . . . , we have

τ+Fp(xn,xn+)=τ+FpT(xn–,yn–),T(xn,yn)

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and

τ+Fp(yn,yn+)=τ+FpT(yn–,xn–),T(yn,xn)

≤Fmaxp(yn–,yn),p(xn–,xn)

. ()

Since (), () hold andFis increasing we get

τ+Fmaxp(xn,xn+),p(yn,yn+)≤Fmaxp(yn–,yn),p(xn–,xn). ()

It follows that

maxp(xn,xn+),p(yn,yn+)≤maxp(yn–,yn),p(xn–,xn)

for alln= , , . . . . Hence, the sequencern:=max{p(xn,xn+),p(yn,yn+)}is a non-increasing. Thus, there isr≥ such thatlimn→∞rn=r. SinceFis continuous, lettingn→ ∞in (),

we arrive at

τ+F(r)≤F(r).

Sinceτ>  and the deﬁnition ofF, we can deduce thatF(r) = –∞; this implies thatr= . Therefore

lim n→∞max

p(xn,xn+),p(yn,yn+)= . ()

Next, we claim that

lim m,n→∞max

p(xm,xn),p(yn,ym)

= . ()

Suppose, to the contrary, that there existsε>  for which we can seek two subsequences

{xm(k)}and{xn(k)}of, respectively,{xm}and{xn}such thatn(k) is the smallest index for which

n(k) >m(k) >k, maxp(xm(k),xn(k)),p(yn(k),ym(k))≥ε. ()

This means that

maxp(xm(k),xn(k)–),p(ym(k),yn(k)–)<ε, ()

and we obtain

p(xm(k),xn(k))≤p(xm(k),xn(k)–) +p(xn(k)–,xn(k)) –p(xn(k)–,xn(k)–)

≤p(xm(k),xn(k)–) +p(xn(k)–,xn(k)) <ε+p(xn(k)–,xn(k)). ()

Similarly, we get

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Combining (), (), and (), we obtain

ε≤maxp(xm(k),xn(k)),p(ym(k),yn(k))≤ε+maxp(xn(k)–,xn(k)),p(yn(k)–,yn(k)). ()

Lettingk→ ∞in () and using (), we have

lim k→∞max

p(xm(k),xn(k)),p(ym(k),yn(k))

=ε. ()

Now, by the facts that

p(xm(k),xn(k))≤p(xm(k),xn(k)–) +p(xn(k)–,xn(k))

and

p(ym(k),yn(k))≤p(ym(k),yn(k)–) +p(yn(k)–,yn(k))

we obtain

maxp(xm(k),xn(k)),p(ym(k),yn(k))≤maxp(xm(k),xn(k)–),p(ym(k),yn(k)–)

+maxp(xn(k)–,xn(k)),p(yn(k)–,yn(k))

. ()

By the same argument, we also have

maxp(xm(k),xn(k)–),p(ym(k),yn(k)–)≤maxp(xm(k),xn(k)),p(ym(k),yn(k))

+maxp(xn(k)–,xn(k)),p(yn(k)–,yn(k)). ()

Lettingk→ ∞in (), () and using (), (), we have

lim k→∞max

p(xm(k),xn(k)–),p(ym(k),yn(k)–)=ε. ()

Next, sincexm(k)≤xn(k)–andym(k)≥yn(k)–, we have

τ+Fp(xm(k)+,xn(k))=τ+FpT(xm(k),ym(k)),T(xn(k)–,xn(k)–)

≤Fmaxp(xm(k),xn(k)–),p(yn(k)–,ym(k)).

For the same reason, we also have

τ+Fp(yn(k),ym(k)+)

=τ+FpT(ym(k),xm(k)),T(yn(k)–,xn(k)–)

≤Fmaxp(xn(k)–,xm(k)),p(yn(k)–,ym(k)).

Therefore

τ+maxFp(xm(k)+,xn(k)),Fp(yn(k),ym(k)+)

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Lettingk→ ∞and using (), we arrive at

τ+F(ε)≤F(ε).

This yieldsε= , this is a contradiction. Hence, we have proved that

lim m,n→∞max

p(xm,xn),p(yn,ym)

= .

This implies that

lim

m,n→∞p(xm,xn) =  and m,nlim→∞p(ym,yn) = . ()

Since (X,p) is -complete partial metric space, we can ﬁndu,v∈Xsuch that

lim

n→∞p(u,xn) =p(u,u) = 

and

lim

n→∞p(v,yn) =p(v,v) = .

Now, we show thatu=T(u,v) andv=T(v,u). Indeed, sinceu≤uandv≥v, we have

τ+FpT(u,v),T(u,v)≤Fmaxp(u,u),p(v,v)=F() = –∞.

This implies thatp(T(u,v),T(u,v)) = .

Sincexn→u,yn→vasn→ ∞in (X,p) and T is continuous, we have T(xn,yn)→ T(u,v) in (X,p), this means that

lim n→∞p

xn+,T(u,v)= lim n→∞p

T(xn,yn),T(u,v)= .

Now, we have

pu,T(u,v)≤p(u,xn+) +pxn+,T(u,v)–p(xn+,xn+).

Lettingn→ ∞, we getp(u,T(u,v)) = , and sou=T(u,v). By the same argument, we also

havep(v,T(v,u)) = , and sov=T(v,u).

In the next theorem, we omit the continuity hypothesis ofT.

Theorem . Let(X,≤)be a partially ordered set and suppose there exists a partial met-ric p on X such that(X,p)is a-complete partial metric space.Let T:X×X→X be a mapping having the mixed monotone property on X.Assume that:

()

τ+FpT(x,y),T(u,v)≤Fmaxp(x,u),p(y,v) ()

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() There arex,y∈Xsuch thatx≤T(x,y),y≥T(y,x).

Also,assume that X has the properties:

(i) If a non-decreasing sequence{xn}inXconverges toxthenxn≤xfor alln. (ii) If a non-increasing sequence{yn}inXconverges toythenyn≥yfor alln.

Then T has a coupled ﬁxed point,that is,there exist x,y∈X such that x=T(x,y),y= T(y,x).

Proof We follow the line of the proof of Theorem .. Hence, we only need to show that

lim n→∞p

xn+,F(u,v)= lim n→∞p

T(xn,yn),T(u,v)= ,

under conditions (i) and (ii). Indeed, we havexn≤uandyn≥vfor alln. Applying (), we have

τ+Fpxn+,T(u,v)=τ+FpF(xn,yn),F(u,v)≤Fmaxp(xn,u),p(yn,v).

Lettingn→ ∞, we obtain

lim n→∞F

pxn+,T(u,v)= –∞.

Hence

lim n→∞p

xn+,F(u,v)= lim n→∞p

T(xn,yn),T(u,v)= .

We easily get the following corollary.

Corollary . Let(X,≤)be a partially ordered set and suppose there exists a partial

met-ric p on X such that(X,p)is a-complete partial metric space.Let T:X×X→X be a mapping having the mixed monotone property on X.Assume that:

()

τ+FpT(x,y),T(u,v)≤F

p(x,u) +p(y,v) 

()

for allx≤u,y≥v,for someF∈Fandτ> .

() There arex,y∈Xsuch thatx≤T(x,y),y≥T(y,x). Also,assume that either

(a) Tis continuous;or

(b) Xhas the properties:

(i) If a non-decreasing sequence{xn}inXconverges toxthenxn≤xfor alln. (ii) If a non-increasing sequence{yn}inXconverges toythenyn≥yfor alln. Then T has a coupled ﬁxed point,that is,there exist x,y∈X such that x=T(x,y),y= T(y,x).

Proof By the fact that

p(x,u) +p(y,v)

 ≤max

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for allx,y,u,v∈X, the condition () implies the condition (). Therefore, the result as desired follows from Theorem . and Theorem ..

The following corollary states thatThas a ﬁxed point under a certain condition.

Corollary . In addition to the hypotheses of Corollary.,if xand yare comparable, then T has a unique ﬁxed point,that is,there exists x∈X such that T(x,x) =x.

Proof Sincex,yare comparable, we havex≥yorx≤y. Suppose we are in the ﬁrst case. Then, by the mixed monotone property ofT, we have

x=T(x,y)≥T(y,y)≥T(y,x) =y,

and, hence, by induction one obtains

xn≥yn for alln≥.

Now, sincex=limn→∞xn+,y=limn→∞yn+, we havep(x,y) =limn→∞p(xn+,yn+). On the

other hand, we have

τ+Fp(xn+,yn+)=τ+FpT(xn,yn),T(yn,xn)

≤Fmaxp(xn,xn),p(yn,yn). ()

Following Lemma ., we also have

lim

n→∞p(xn,xn) =nlim→∞p(yn,yn) = .

Lettingn→ ∞in (), we arrive atlimn→∞p(xn+,yn+) = . Thereforep(x,y) = , orx=y.

HenceT(x,x) =x.

Remark . We underline the fact that the coupled ﬁxed point theorem in this paper can be observed from the ﬁxed point result of a single mapping by using the techniques in [–]. On the other hand, we prefer to keep the proofs for the sake of completeness.

4 Main result: non-cooperative equilibrium problem for two players

In this section, by using coupled ﬁxed point theorems, we shall show that a two person game has a non-cooperative equilibrium. The reader may consult the excellent sources on general concepts of two person games in [] and [].

Let (S,p) be a -complete partial metric space. Suppose thatShas a partially order re-lation≤. We consider a two person gameGin normal form that consists of the following data:

() S=SandS=Sare strategies for playerand, respectively, player; () the setU=S×Sof allowed strategies pairs;

() we have the biloss operator

L:U→R

(s,s)→

L(s,s);L(s,s)

,

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whereLi(s,s) is the loss of playeriif the strategiessandsare played. A pair (s,s)∈U is called a non-cooperative equilibrium if

L(s,s)≤L(s,s), ∀s∈S,

L(s,s)≤L(s,s), ∀s∈S.

()

This means that

L(s,s) =min

s∈SL(s,s),

L(s,s) =min

s∈SL(s,s).

()

To see what strategy pairs are non-cooperative equilibria, one considers the optimal deci-sion rulesC,Dsuch that

LC(s),s=min

s∈SL(s,s),

Ls,D(s)=min

s∈SL(s,s).

()

Then any ﬁxed point of the map

(s,s)→C(s),D(s)

is a non-cooperative equilibrium.

In this section, we shall consider thatD(s) =C(s) for all s∈S. It is easy to see that if L(s,s) =L(s,s) for all (s,s)∈S×SthenD(s) =C(s) and it is not diﬃcult to give an example thatD(s) =C(s) in the caseL(s,s)=L(s,s). LetT:S×S→Rbe the map deﬁned by

T(x,y) =C(y)

for allx,y∈S. Suppose thatThas coupled ﬁxed point (a,b)∈R. It follows that

a=T(a,b) =C(b),

b=T(b,a) =C(a),

()

and (a,b) is fixed point of the map (s,s)→(C(s),C(s)). Therefore, the existence of the coupled fixed point ofTimplies a non-cooperative equilibrium. Hence, we can reduce the process of proving the existence of a non-cooperative equilibrium to giving the existence of a coupled fixed point ofT.

Theorem . Let S andGbe as mentioned above.Suppose that the optimal decision rule is a monotone continuous function C which satisﬁes:

()

τ+FpC(x),C(y)≤Fp(x,y) ()

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() There arex,y∈R+_such thatx≤C(y),y≥C(x).

Then the two person gameGhas a non-cooperative equilibrium.

Proof LetT:S×S→Sbe deﬁned by

T(x,y) =C(y)

for allx,y∈S. SinceCis continuous, we see thatTis continuous. SinceCis monotone, it is easy to check thatThas the mixed monotone property onX. For allx,y,u,v∈R+, with x≤u,y≥vwe have

pT(x,y),T(u,v)=pC(y),C(v).

Therefore, the condition () reduces to

τ+FpC(y),C(v)≤Fmaxp(x,u),p(y,v), ()

for everyx≤u,y≥v. Since

maxp(x,u),p(y,v)≥p(y,v)

andFis increasing, we see that the condition () implies (). Applying Theorem ., we conclude thatThas a coupled ﬁxed point. This implies that the two person gameGhas a

non-cooperative equilibrium.

Since every metric is partial metric, we immediately obtain the following corollary.

Corollary . LetGbe as mentioned above.Suppose that(S,d)is a metric space and the optimal decision rule is a monotone continuous function C which satisﬁes:

()

τ+FdC(x),C(y)≤Fd(x,y) ()

for allx,y∈Sandx<y,for someF∈Fandτ> . () There arex,y∈R+

such thatx≤C(y),y≥C(x). Then the two person gameGhas a non-cooperative equilibrium.

Now we shall give an example to show that Corollary . is eﬀective.

Example . ConsiderS=R+

endowed with the metricd(x,y) =|x–y|for allx,y∈S. Let Gbe a two person game with biloss operator

L(s,s) =s( +s)e–τ– s,

L(s,s) =s_( +s)e–τ– s,

wheres,s∈R+

and a givenτ> . It is easy to compute the optimal decision rulesC,D such that forG

C(s) = e –τ

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and

D(s) = e –τ

 +s,

wheres,s∈R+_. We haveD(s) =C(s) for alls∈R+_, andCis continuous map. We need show thatCsatisﬁes all conditions of Corollary .. We have

dC(x),C(y)=e–τ   +x–

  +y

≤e–τ|x–y|=e–τd(x,y)

for allx,y∈R+

. By passing to logarithms, we arrive at

τ+lndC(x),C(y)≤lnd(x,y)

for allx=y. SinceF(x) =lnx∈Fwe can deduce thatCsatisﬁes () in Corollary .. Choos-ingx= , we have

C(x) =e–τ.

Lety= , we havey≥C(x). On the other handx= ≤C(y) =e–_τ. Therefore,C sat-isﬁes all conditions of Corollary .. Applying this corollary, we see that the two person gameGhas a non-cooperative equilibrium.

5 Application to nonlinear integral equations

In this section, we study the existence of unique solution of nonlinear integral equations, as an application of the ﬁxed point theorem proved in Section .

Let us consider the following integral equation:

x(t) =h(t) + t



K(t,s) +K(t,s)fs,x(s)+gs,x(s)ds, ()

where the unknown functionx(t) takes real values.

LetX=C([,K]) be the space of all real continuous functions deﬁned on [,K]. It well known thatC([,K]) endowed with the metric

d(x,y) =x–y= max t∈[,K]

_{x(t) –}_y(t)

is a complete metric space. By a solution of (), we mean a continuous functionx∈X that satisﬁes () on [,K]. By certain conditions onK,K,f,g, and using the results of the previous section, we will prove that () has a unique solution. For this, note thatX can be equipped with the partial ordergiven by

x,y∈X, xy

⇐⇒ x(t)≤y(t)∀t∈[,K] andx,y ≤orx(t) =y(t)∀t∈[,K].

()

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Assumption .

(A) f,g∈C([,K]×R),h∈X, andK,K∈C([,K]×[,K])such thatK(t,s)≥and

K(t,s)≤for allt,s≥;

(B) f(t,·) :R→Ris increasing for allt∈[,K];g(t,·) :R→Ris decreasing for all

t∈[,K];

(C) there existsτ∈[,∞)such that

≤f(t,x) –f(t,y)≤τe–τx–y

 , ∀x≥y

and

–τe–τx–y

 ≤g(t,x) –g(t,y)≤, ∀x≥y;

(D) maxt,s∈[,K]|K(t,s) –K(t,s)| ≤. DeﬁneT:X×X→Xby

T(x,y)(t) =h(t) + t

 K(t,s)

fs,x(s)+gs,y(s)ds

+ t



K(t,s)fs,y(s)+gs,x(s)ds

for allt∈[,K].

Deﬁnition . An element (α,β)∈C([,K]×C[,K]) is acoupled normal lowerand a normal upper solutionof the integral equation () ifαβand

αT(α,β) and βT(β,α).

Theorem . Suppose that Assumption.is fulﬁlled.Then the existence of a coupled normal lower and normal upper solution for()provides the existence of a unique solution of()in C([,K]).

Proof Suppose{un}is a monotone non-decreasing sequence inXthat converges tou∈X. Then, for everyt∈[,K], the sequence of real numbersu(t)≤u(t)≤ · · · ≤un(t)≤ · · · converges tou(t). Moreover, since the normed map is continuous, we can deduce that

u ≤ providedun ≤ for alln. Therefore, for everyt∈[,K],n∈N,un(t)≤u(t). Henceun≤u, for alln∈N.

Similarly, we can verify that the limitv(t) of a monotone non-increasing sequencevn(t) inXis a lower bound for all elements in the sequence. That is,v≤vnfor alln. Hence, the condition (b) in Corollary . holds.

Forx∈X, we deﬁnedxτ =maxt∈[,K]|x(t)|e–τt, whereτ≥ is chosen arbitrarily. It is easy to check that · τ is a norm equivalent to the maximum norm inXandXendowed with the metricdτ deﬁned by

dτ(x,y) =x–yτ= max t∈[,K]

x(t) –y(t)e–τt

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Now, considerXendowed with the partial metric given by

pτ(x,y) = ⎧ ⎨ ⎩

dτ(x,y) ifxτ≤,yτ≤,

dτ(x,y) +τ otherwise. ()

It is easy to see that (X,pτ) is a -complete partial metric space but is not complete (see []). We recall thatT:X×X→Xby

T(x,y)(t) =h(t) + t



K(t,s)fs,x(s)+gs,y(s)ds

+ t

 K(t,s)

fs,y(s)+gs,x(s)ds

for allt∈[,K].

Next, we show thatT has the mixed monotone property. Indeed, forx,x∈C([,K]) andx≤x, that is,x(t)≤x(t), for everyt∈[,K], we have

T(x,y)(t) –T(x,y)(t) = t

 K(t,s)

fs,x(s)

+gs,y(s)ds

+ t



K(t,s)fs,y(s)+gs,x(s)ds+h(t)

– t



K(t,s)fs,x(s)+gs,y(s)ds

– t



K(t,s)fs,y(s)+gs,x(s)ds–h(t)

= t



K(t,s)fs,x(s)–fs,x(s)ds

+ t



K(t,s)gs,x(s)–gs,x(s)ds≤,

for everyt∈[,K], by Assumption .. This yieldsT(x,y)(t)≤T(x,y)(t), for everyt∈ [,K], that is,T(x,y)≤T(x,y). By the same computation, we arrive atT(x,y)≤T(x,y) ify≥y. Hence,Thas the mixed monotone property.

Now, forx≥uandy≤v, we have

T(x,y)(t) –T(u,v)(t)

= t



K(t,s)fs,x(s)+gs,y(s)ds

+ t

 K(t,s)

fs,y(s)+gs,x(s)ds+h(t)

– t



K(t,s)fs,u(s)+gs,v(s)ds

+ t



K(t,s)fs,v(s)+gs,u(s)ds+h(t)

= t

 K(t,s)

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+ t



K(t,s)fs,y(s)–fs,v(s)+gs,x(s)–gs,u(s)ds

= t



K(t,s)fs,x(s)–fs,u(s)–gs,v(s)–gs,y(s)ds

– t

 K(t,s)

fs,v(s)–fs,y(s)–gs,x(s)–gs,u(s)ds

≤ t



K(t,s)τe–τ

x(s) –u(s)

 +

v(s) –y(s)  ds – t 

K(t,s)τe–τ

v(s) –y(s)

 +

x(s) –u(s) 

ds

≤τe–τ t



K(t,s) –K(t,s)

x(s) –u(s)

 +

v(s) –y(s) 

ds

=τe–τ t



K(t,s) –K(t,s)eτs

|x(s) –u(s)|e–τs

 +

|x(s) –u(s)|e–τs 

ds

≤τe–τ t

 max t,s∈[,K]

K(t,s) –K(t,s)eτs

x–uτ

 +

y–vτ 

ds

≤τe–τe τt

τ

x–uτ

 +

y–vτ 

.

It follows that

T(x,y)(t) –T(u,v)(t)e–τt≤e–τ

x–uτ

 +

y–vτ 

.

Hence, for allx,y,u,v∈Xsuch thatx≥uandy≤v, sincexτ,yτ,uτ,vτ ≤, we have

pτT(x,y),T(u,v)≤e–τ 

pτ(x,u) +pτ(y,v).

By passing to logarithms, we arrive at

τ+lnpτT(x,y),T(u,v)≤ln

pτ(x,u) +pτ(y,v) 

.

SinceF(x) =lnx∈F, we conclude thatT satisﬁes the condition (). Now, let (α,β) be a coupled normal lower and normal upper solution of the integral equation of (). Then we haveαβ,

αT(α,β) and βT(β,α).

Finally, applying Corollary ., we can conclude thatThas a ﬁxed pointx. HenceT(x,x) =

xandxis an unique solution of ().

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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Author details

1_{Department of Mathematics, Vinh University, 182 Le Duan, Vinh City, Vietnam.}2_{Nonlinear Analysis and Applied}

Mathematics Research Group (NAAM), Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.3_{Department of Mathematics, Atilim University, ˙Incek, Ankara, 06836, Turkey.} 4_{Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, 21589, Saudi}

Arabia.

Acknowledgements

The authors are grateful to the reviewers for their careful reviews and useful comments. This article was funded by the Deanship of Scientiﬁc Research (DSR), King Abdulaziz University. The authors, therefore, acknowledge with thanks DSR technical and ﬁnancial support.

Received: 3 March 2015 Accepted: 27 April 2015 References

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