Vector-valued metrics, fixed points and coupled fixed points for nonlinear operators

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R E S E A R C H

Open Access

Vector-valued metrics, fixed points and

coupled fixed points for nonlinear operators

Adrian Petru¸sel, Gabriela Petru¸sel and Cristina Urs

*

*Correspondence:

cristina.urs@math.ubbcluj.ro Babe¸s-Bolyai University Cluj-Napoca, Kog˘alniceanu Street no. 1, Cluj-Napoca, 400084, Romania

Abstract

In this paper, we will present fixed point theorems for singlevalued and multivalued operators in spaces endowed with vector-valued metrics, as well as a Gnana Bhaskar-Lakshmikantham-type theorem for the coupled fixed point problem, associated to a pair of singlevalued operators (satisfying a generalized mixed monotone property) in ordered metric spaces. The approach is based on Perov-type fixed point theorems in spaces endowed with vector-valued metrics. The Ulam-Hyers stability and the limit shadowing property of the fixed point problems are also discussed.

MSC: 47H10; 54H25

Keywords: singlevalued operator; multivalued operator; vector-valued metric; fixed point; ordered metric space; coupled fixed point; Ulam-Hyers stability; limit

shadowing property

1 Introduction

The classical Banach contraction principle is a very useful tool in nonlinear analysis with many applications to operatorial equations, fractal theory, optimization theory and other topics. Banach contraction principle was extended for singlevalued contraction on spaces endowed with vector-valued metrics by Perov [] and Perov and Kibenko []. For some other contributions to this topic, we also refer to [–],etc. The case of multivalued con-tractions on spaces endowed with vector-valued metrics is treated in [–],etc.

In the study of the fixed points for an operator, it is sometimes useful to consider a more general concept, namely coupled fixed points. The concept of coupled fixed point for nonlinear operators was introduced and studied by Opoitsev (see [–]) and then, in , by Guo and Lakshmikantham (see []) in connection with coupled quasisolutions of an initial value problem for ordinary differential equations. Later, a new research direction for the theory of coupled fixed points in ordered metric spaces was initiated by Gnana Bhaskar and Lakshmikantham in [] and by Lakshmikantham and Ćirić in []. Their approach is based on some contractive type conditions on the operator. For other results on coupled fixed point theory, see [–],etc.

Let us recall first some important preliminary concepts and results.

LetXbe a nonempty set. A mappingd:X×X→Rmis called a vector-valued metric

onXif the following properties are satisfied:

(a) d(x,y)Ofor allx,yX; ifd(x,y) =O, thenx=y(whereO:= (, , . . . , )

m-times

);

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(b) d(x,y) =d(y,x)for allx,yX;

(c) d(x,y)d(x,z) +d(z,y)for allx,yX.

A nonempty setXendowed with a vector-valued metricdis called a generalized metric space in the sense of Perov (in short, a generalized metric space), and it will be denoted by (X,d). The notions of convergent sequence, Cauchy sequence, completeness, open and closed subset, open and closed ball, . . . are similar to those for usual metric spaces.

Notice that the generalized metric space in the sense of Perov is a particular case of Riesz spaces (see [, ]) and of so-called cone metric spaces (orK-metric space) (see [, ]). We denote byMmm(R+) the set of allm×mmatrices with positive elements and byIthe identitym×mmatrix. Ifx,y∈Rm,x= (x, . . . ,xm) andy= (y, . . . ,ym), then, by definition

xy if and only if xiyi fori∈ {, , . . . ,m}.

Notice that through this paper, we will make an identification between row and column vectors inRm.

Definition . A square matrix of real numbers is said to be convergent to zero if and only if its spectral radiusρ(A) is strictly less than . In other words, this means that all the eigenvalues ofAare in the open unit disc,i.e.,|λ|< , for everyλ∈Cwithdet(A–λI) = , whereIdenotes the unit matrix ofMm,m(R) (see []).

A classical result in matrix analysis is the following theorem (see [, ]).

Theorem . Let AMmm(R+).The following assertions are equivalent

(i) Ais convergent towards zero;

(ii) AnOasn→ ∞;

(iii) The matrix(I–A)is nonsingular and

(I–A)–=I+A+· · ·+An+· · ·; ()

(iv) The matrix(I–A)is nonsingular and(I–A)–has nonnegative elements;

(v) AnqOandqAnOasn→ ∞,for eachqRm;

(vi) The matricesqAandAqconverge toOfor eachq∈(,Q),whereQ:=ρ(A).

IfXis a nonempty set andf :XXis an operator, then

Fix(f) :=xX;x=f(x).

We recall now Perov’s fixed point theorem (see [], see also []).

Theorem .(Perov) Let(X,d)be a complete generalized metric space,and let the oper-ator f :XX be with the property that there exists a matrix AMmm(R+)convergent towards zero such that

df(x),f(y) ≤Ad(x,y), for all x,yX.

Then

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() the sequence of successive approximations(xn)n∈N,xn:=fn(x)is convergent inXto x∗,for allx∈X;

() one has the following estimation

dxn,x∗ ≤An(I–A)–d(x,x); ()

() ifg:XXis an operator such that there existsy∗∈Fix(g),and there exists

η:= (η, . . . ,ηm)∈R+mwithηi> for eachi∈ {, , . . . ,m}such that

df(x),g(x)η, for eachxX,

then

dx∗,y∗ ≤(I–A)–η;

() ifg:XXis an operator,yn:=gn(x),and there existsη:= (η, . . . ,ηm)∈Rm+ with

ηi> for eachi∈ {, , . . . ,m}such that

df(x),g(x)η, for allxX,

we have the following estimation

dyn,x∗ ≤(I–A)–η+An(I–A)–d(xo,x). ()

Notice that in Precup [], as well as in [, ] and [] are pointed out the advantages of working with vector-valued norm with respect to the usual scalar norms.

There is a vast literature concerning this approach, see also, for example, [, , , , ], etc.

We will focus our attention to the following system of operatorial equations ⎧

⎨ ⎩

x=T(x,y),

y=T(x,y),

whereT,T:X×XXare two given operators.

By definition, a solution (x,y)X×Xof the above system is called a coupled fixed point for the operatorsTandT. Notice that ifS:X×XXis an operator and we define

T(x,y) :=S(x,y) and T(x,y) :=S(y,x),

then we get the classical concept of a coupled fixed point for the operatorSintroduced by Opoitsev and then intensively studied in some papers by Guo and Lakshmikantham, Gnana Bhaskar and Lakshmikantham, Lakshmikantham and Ćirić,etc.

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First aim of this work is to present some existence and stability results for fixed point equations and inclusions in generalized metric spaces in the sense of Perov. Our sec-ond purpose is to present, in the setting of an ordered metric space, a Gnana Bhaskar-Lakshmikantham-type theorem for the coupled fixed point problem associated to a pair of singlevalued operators satisfying a generalized mixed monotone assumption. The ap-proach is based on an abstract fixed point theorem in ordered complete metric spaces. Our results are related with other existence and stability results for the a coupled fixed point problem for singlevalued operators proved in [] by the support of Perov’s fixed point theorem.

2 Existence, uniqueness and stability for fixed point equations and inclusions We start this section by an extension of Perov’s theorem. At the same time, the result is a generalization to vector-valued metric spaces of the main theorem in [].

Theorem . Let(X,d)be a generalized complete metric space,and let f :XX be an almost contraction with matrices A,B and C,i.e.,the matrix A+CMmm(R+)converges to zero,BMmm(R+)and

df(x),f(y) ≤Ad(x,y) +Bdy,f(x) +Cdx,f(x) , for all x,yX.

Then,the following conclusions hold

. f has at least one fixed point inXand,for eachx∈X,the sequencexn:=fn(x)of

successive approximations off starting fromxconverges tox∗(x)∈Fix(f)asn→ ∞;

. For eachx∈X,we have

dxn,x∗(x) ≤An(I–A)–d

x,f(x) , for alln∈N

and

dx,x∗(x) ≤(I–A)–d

x,f(x);

. If,additionally,the matrixA+Bconverges to zero,thenf has a unique fixed point

inX.

Proof Letx∈Xbe arbitrary, and consider (xn)n∈N the sequence of successive approxi-mations forf starting fromx,i.e.,

xn+=f(xn), for alln∈N.

We have thatd(x,x) =d(f(x),f(x))≤Ad(x,x) +Bd(x,f(x)) +Cd(x,f(x)) = (A+ C)d(x,x).

Inductively, we get that the sequence (xn)n∈Nsatisfies, for alln∈N, the following esti-mation

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Hence, for alln∈Nandp∈N,p≥, we get that

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

≤(A+C)nd(x,x) + (A+C)n+d(x,x) +· · ·+ (A+C)n+p–d(x,x)

≤(A+C)nI+ (A+C) + (A+C)+· · ·+ (A+C)p–+· · ·d(x ,x)

≤(A+C)nI– (A+C)–d(x,x).

Thus,d(xn,xn+p)→O, asn→ ∞. Hence (xn)n∈N is a Cauchy sequence in the complete metric space (X,d). Thus, (xn)n∈Nconverges to a certain elementx∗(x)∈X.

Next, we show thatx∗:=x∗(x)∈Fix(f). Indeed, we have the following estimation

dx∗,fx∗ ≤dx∗,xn +d

xn,f

x∗ =dx∗,xn +d

f(xn–),f

x

dx∗,xn +Ad

xn–,x∗ +Bd

x∗,xn +Cd(xn–,xn)→O, asn→ ∞.

Hencex∗∈Fix(f). In addition, lettingp→ ∞in the estimation ofd(xn,xn+p), we get

dxn,x∗(x) ≤(A+C)n

I– (A+C)–dx,f(x) , for alln∈N.

Forn= , we obtain

dx,x∗(x) ≤(A+C)n

I– (A+C)–dx,f(x).

We show now the uniqueness of the fixed point. Letx∗,y∗∈Fix(f) withx∗=y∗. Then

dx∗,y∗ =dfx∗ ,fy∗ ≤Adx∗,y∗ +Bdy∗,fx∗ +Cdx∗,fx

= (A+B)dx∗,y∗ .

Thus (I–AB)d(x∗,y∗)≤O∈Rm. SinceA+Bconverges to zero, we get thatIABis non-singular and (I–AB)–M

m,m(R+). Henced(x∗,y∗)≤Oand sox∗=y∗.

Remark . The result above extends Corollary . in [], where the case of almost con-tractions with matricC=Ois treated.

Two important abstract concepts are given now.

Definition .(see [, ]) If (X,d) is a generalized metric space, thenf :XXis called a weakly Picard operator if and only if the sequence (fn(x))

n∈N of successive approxima-tions off converges for allxXand the limit (which may depend onx) is a fixed point off.

Iff is weakly Picard operator, then we define the operatorf∞:XXby

f∞(x) := lim

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Notice that, in this case,f∞(X) =Fix(f). Moreover,f∞is a set retraction ofXtoFix(f). Iff is weakly Picard operator andFix(f) ={x∗}, then by definitionf is a Picard operator. In this case,f∞is the constant operator,i.e.,f∞(x) =x∗for allxX.

Definition .(see []) Let (X,d) be a generalized metric space, and letf :XXbe an operator. Then,fis said to be aψ-weakly Picard operator if and only iffis a weakly Picard operator andψ:Rm

+ →Rm+ is an increasing operator, continuous inOwithψ(O) =Osuch that

dx,f∞(x) ≤ψ(dx,f(x) , for allxX.

Moreover, aψ-weakly Picard operatorf :XX with a unique fixed point is said to be a ψ-Picard operator. In particular, ifψ :Rm+ →Rm+ is given by ψ(t) =M·t (with CMmm(R+)), then we say thatf isM-weakly Picard operator (respectively aM-Picard operator).

From Theorem ., we get the following example.

Example . If (X,d) is a generalized complete metric space andf :XXis an almost contraction with matricesA,B, andC, thenf is aψ-weakly Picard operator with the func-tionψ(t) := [I– (A+C)]–t. In particular, iff :XXis a contraction with matrixA, then f is aψ-Picard operator withψ(t) := (I–A)–t.

For the proof of our next theorems we need the following notion.

Definition . Let (X,d) be a generalized metric space, and letf :XXbe an operator. Then, the fixed point equation

x=f(x) ()

is said to be generalized Ulam-Hyers stable if there exists an increasing functionψ:Rm+

Rm

+, continuous inOwithψ(O) =Osuch that for anyε:= (ε, . . . ,εm) withεi>  fori∈ {, . . . ,m}and anyε-solutiony∗∈Xof (),i.e.,

dy∗,fy∗ ≤ε, ()

there exists a solutionx∗of () such that

dx∗,y∗ ≤ψ(ε). ()

In particular, ifψ(t) =C·t,t∈Rm

+ (whereCMmm(R+)), then the fixed point equation () is called Ulam-Hyers stable.

We can prove now the following abstract result (see also Rus []) concerning the Ulam-Hyers stability of the fixed point equation ().

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Proof Letε:= (ε, . . . ,εm) (withεi>  fori∈ {, . . . ,m}), and lety∗∈X be aε-solution

of (),i.e.,d(y∗,f(y∗))≤ε. Sincef is aψ-Picard operator, we have that

dx,f∞(x) ≤ψ(dx,f(x) , for eachxX.

Hence, there existsx∗:=f∞(y∗)∈Fix(f) such that

dy∗,fy∗ ≤ψ(dy∗,fy∗ ≤ψ(ε).

Remark . In particular, if (X,d) is a generalized complete metric space, andf :XX is an almost contraction with matricesA,BandC, then, by Example . and Theorem ., we get that the fixed point equation () is Ulam-Hyers stable.

We now move our attention to the multivalued case. If (X,ρ) is a metric space, and we denote byP(X) the space of all nonempty subsets ofX, then the gap functional (generated byρ) onP(X) is defined as

:P(X)×P(X)→R+, (A,B) :=inf

ρ(a,b)|aA,bB.

In particular, ifx∈X, we putDρ(x,B) in place ofDρ({x},B).

We will denote bythe Pompeiu-Hausdorff functional onP(X), defined as

:P(X)×P(X)→R+∪ {+∞}, (A,B) =max

sup

a∈ADρ(a,B),

sup

b∈BDρ(b,A)

.

Let (X,d) be a generalized metric space withd(x,y) :=

d

(x,y)

··· dm(x,y)

.

Notice thatdis a generalized metric onXif and only ifdiare metrics onXfor each

i∈ {, , . . . ,m}. We denote by

D(A,B) := ⎛ ⎜ ⎝

Dd(A,B)

· · ·

Ddm(A,B) ⎞ ⎟

⎠, the generalized gap functional onP(X)

and by

H(A,B) := ⎛ ⎜ ⎝

Hd(A,B)

· · ·

Hdm(A,B) ⎞ ⎟

⎠, the generalized Pompeiu-Hausdorff functional onP(X).

It is well known that ifdiis a metric onX,xXandYP(X), thenDdi(x,Y) =  if and only ifxYdi, whereYdidenotes the closure ofYin (X,di). As a consequence,

Ydi=xX:Ddi(x,Y) = 

.

Hence, xYdi for each i∈ {, , . . . ,m} is equivalent with Ddi(x,Y) =  for eachi

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SinceYd={xX:D(x,Y) =O}, we have:

(a) xYdxYdifor eachi∈ {, , . . . ,m}.

(b) Yis closed with respect todYis closed with respect to eachdifor

i∈ {, , . . . ,m}.

We denote byPcl(X) the set of all nonempty closed (with respect tod) subsets ofX. We have the following two auxiliary results.

Lemma . Let(X,d)be a generalized metric space,let xX,and let YPcl(X).Then,

xY if and only if D(x,Y) =O.

Lemma . Let(X,d)be a generalized metric space,Y,ZP(X),q> .Then, for any yY,there exists zZ such that

d(y,z)qH(Y,Z).

If (X,d) is a nonempty set, andF:XP(X) is a multivalued operator, then the graph of the operatorFis denoted by

Graph(F) :=(x,y)X×X:yF(x),

while the fixed point set and, respectively, the strict fixed point set ofFare denoted by the symbols

Fix(F) :=xX:xF(x), respectively SFix(F) :=xX:F(x) ={x}.

We will present now an extension of the Nadler fixed point theorem in a space endowed with a vector-valued metric, which is also a multivalued version of Perov’s theorem.

Theorem . Let(X,d)be a generalized complete metric space,and let F:XPcl(X) be a multivalued almost contraction with matrices A and B,i.e.,there exists two matrices A,BMmm(R+)such that A converges to zero and

HF(x),F(y)Ad(x,y) +BDy,F(x), for all x,yX.

Then

(i) Fix(F)=∅;

(ii) for each(x,y)∈Graph(F),there exists a sequence(xn)n∈N(withx=x,x=yand

xn+∈F(xn)for eachn∈N∗)such that(xn)n∈Nis convergent to a fixed point

x∗:=x∗(x,y)ofF,and the following relations hold

dxn,x∗ ≤An(I–A)–d(x,x), for eachn∈N∗

and

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Proof Letx∈Xandx∈F(x) be arbitrarily chosen. Letq∈(,Q), whereQis defined by Theorem .. Then, by Lemma ., there existsx∈F(x) such that

d(x,x)≤qH

F(x),F(x) ≤q

Ad(x,x) +BD

x,F(x)

=qAd(x,x).

Inductively, there existsxn+∈F(xn) such that

d(xn,xn+)≤(qA)nd(x,x), for anyn∈N∗.

We have

d(xn,xn+p)≤d(xn,xn+) +· · ·+d(xn+p–,xn+p) ≤(qA)nd(x,x) +· · ·+ (qA)n+p–d(x,x)

≤(qA)nI+qA+· · ·+ (qA)p–+· · ·d(x ,x)

= (qA)n(I–qA)–d(x,x).

Thus

d(xn,xn+p)≤(qA)n(I–qA)–d(x,x), forn∈N∗andp∈N∗. ()

Lettingn→ ∞, we get that (xn) is a Cauchy sequence inX. Since (X,d) is complete, it

follows that there existsx∗∈Xsuch thatxn d

−→x∗,n→ ∞. Thus,

Dx∗,Fx∗ = ⎛ ⎜ ⎝

D(x∗,F(x∗))

· · ·

Dm(x∗,F(x∗))

⎞ ⎟ ⎠ ≤ ⎛ ⎜ ⎝

d(x∗,xn+) +D(xn+,F(x∗))

· · ·

dm(x∗,xn+) +Dm(xn+,F(x∗)) ⎞ ⎟ ⎠ ≤ ⎛ ⎜ ⎝

d(x∗,xn+) +H(F(xn),F(x∗)) · · ·

dm(x∗,xn+) +Hm(F(xn),F(x∗))

⎞ ⎟ ⎠

=dx∗,xn+ +H

F(xn),F

x

dx∗,xn+ +Ad

xn,x∗ +BD

x∗,F(xn)

dx∗,xn+ +Ad

xn,x∗ +B

dx∗,xn+ +D

xn+,F(xn)

=dx∗,xn+ +Ad

xn,x∗ +Bd

x∗,xn+ .

Lettingn→ ∞, we get thatD(x∗,F(x∗)) =O. By Lemma ., we get thatx∗∈F(x∗). More-over, lettingp→ ∞in (), we obtain

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Thus,

dx,x∗ ≤d(x,x) +d

x,x

d(x,x) +qA(IqA)–d(x,x)

=I+qA(IqA)–d(x,x)

=I+qAI+qA+· · ·+ (qA)n+· · · d(x,x)

= (I–qA)–d(x,x).

Lettingq, we get thatd(x,x∗)≤(I–A)–d(x,x).

For the following notions see Ruset al.[] and A. Petruşel [].

Definition . Let (X,d) be a generalized metric space, and letF:XPcl(X) be a mul-tivalued operator. By definition,Fis a multivalued weakly Picard (briefly MWP) operator if for each (x,y)∈Graph(F), there exists a sequence (xn)n∈Nsuch that

(i) x=x,x=y;

(ii) xn+∈F(xn), for eachn∈N;

(iii) the sequence(xn)n∈Nis convergent, and its limit is a fixed point ofF.

Remark . A sequence (xn)n∈N satisfying the condition (i) and (ii) in the

defini-tion above is called a sequence of successive approximadefini-tions ofF starting from (x,y)∈ Graph(F).

IfF:XP(X) is an MWP operator, then we defineF∞:Graph(F)→P(Fix(F)) by the formulaF∞(x,y) :={z∈Fix(F)|there exists a sequence of successive approximations ofF starting from (x,y) that converges toz}.

Definition . Let (X,d) be a generalized metric space, and letF:XP(X) be an MWP operator. Then,Fis called aψ-multivalued weakly Picard operator (brieflyψ-MWP oper-ator) if and only ifψ:Rm

+ →Rm+ is an increasing operator, continuous inOwithψ(O) =O, and there exists a selectionf∞ofF∞such that

dx,f∞(x,y)ψd(x,y) , for all (x,y)∈Graph(F).

Example . Let (X,d) be a generalized complete metric space, and letF:XPcl(X) be a multivalued almost contraction with matricesAandB. Then, by Theorem . (see (i) and (ii)), we get thatFis a (I–A)–-MWP operator.

Two important stability concepts are given now.

Definition . Let (X,d) be a generalized metric space, and letF:XP(X) be a mul-tivalued operator. The fixed point inclusion

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is called generalized Ulam-Hyers stable if and only if there existsψ:Rm

+ →Rm+ increas-ing, continuous in Owith ψ(O) =Osuch that for eachε:= (ε, . . . ,εm) (withεi>  for

i∈ {, . . . ,m}) and for eachε-solutiony∗∈Xof (),i.e.,

Dy∗,Fy∗ ≤ε, ()

there exists a solutionx∗of the fixed point inclusion () such that

dy∗,x∗ ≤ψ(ε).

In particular, ifψ(t) =C·tfor eacht∈Rm

+ (whereCMmm(R+)), then the fixed point inclusion () is said to be Ulam-Hyers stable.

Definition . Let (X,d) be a generalized metric space, and letF:XP(X) be a mul-tivalued operator. Then, the mulmul-tivalued operatorFis said to have the limit shadowing property if for each sequence (yn)n∈NinXsuch thatD(yn+,F(yn))→Oasn→+∞, there

exists a sequence (xn)n∈N of successive approximations of F such thatd(xn,yn)→Oas

n→+∞.

An auxiliary result is as follows.

Cauchy-type lemma Let AMmm(R+)be a matrix convergent toward zero and(Bn)n∈N∈

Rm

+ be a sequence such thatlimn→+∞Bn=O.Then

lim

n→+∞

n

k= AnkBk

=O.

We can prove now the Ulam-Hyers stability of the fixed point inclusion () for the case of a multivalued contraction, which has at least one strict fixed point. The limit shadowing property is also established.

Theorem . Let(X,d)be a generalized complete metric space,and let F:XPcl(X)be a multivalued A-contraction,i.e.,there exists a matrix AMmm(R+)such that A converges to zero and

HF(x),F(y)Ad(x,y), for all x,yX.

Suppose also that SFix(F)=∅,i.e.,there exists x∗∈X such that{x∗}=F(x∗).Then

(a) the fixed point inclusion()is Ulam-Hyers stable;

(b) the multivalued operatorFhas the limit shadowing property.

Proof (a) Letε:= (ε, . . . ,εm) (withεi>  fori∈ {, . . . ,m}), and lety∗∈Xbe anε-solution

of (),i.e.,D(y∗,F(y∗))≤ε. Letx∗∈Xbe such that{x∗}=F(x∗). Then

dy∗,x∗ ≤Dy∗,Fy∗ +HFy∗ ,Fx∗ ≤ε+Ady∗,x∗ .

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(b) Let (yn)n∈N be a sequence inXsuch that D(yn+,F(yn))→Oasn→ ∞. We shall

prove first thatd(yn,x∗)→Oasn→+∞. We successively have

dx∗,yn+ ≤H

x∗,F(yn) +D

yn+,F(yn) ≤Ad

x∗,yn +D

yn+,F(yn)

AAdx∗,yn– +D

yn,F(yn–) +D

yn+,F(yn) ≤ · · · ≤An+dx∗,y +AnD

y,F(y) +· · ·+AD

yn,F(yn–) +D

yn+,F(yn) .

By Cauchy’s lemma, the right hand side tends toOasn→+∞. Thusd(x∗,yn+)→Oas n→+∞.

On the other hand, by Theorem .(i)-(ii), we know that there exists a sequence (xn)n∈N

of successive approximations forFstarting from arbitrary (x,x)∈Graph(F), which con-verge to a fixed pointx∗∈Xof the operatorF. Since, the fixed point is unique, we get that d(xn,x∗)→Oasn→+∞. Hence, for such a sequence (xn)n∈N, we have

d(yn,xn)≤d

yn,x∗ +d

x∗,xnO asn→+∞.

We also have the following abstract results concerning the Ulam-Hyers stability of the fixed point inclusion () for multivalued operators.

Theorem . Let(X,d)be a generalized metric space,and let F:XPcl(X)be a multi-valuedψ-weakly Picard operator.Suppose also that there exists a matrix C∈Mmm(R+) such that for any ε:= (ε, . . . ,εm) (with εi >  for i∈ {, . . . ,m}) and any zX with

D(z,F(z))εthere exists uF(z)such that d(z,u).Then,the fixed point inclusion ()is generalized Ulam-Hyers stable.

Proof Letε:= (ε, . . . ,εm) (withεi>  fori∈ {, . . . ,m}) andy∗∈Xbe aε-solution of (),

i.e.,D(y∗,F(y∗))≤ε. SinceFis a multivaluedψ-weakly Picard operator, for each (x,y)∈ Graph(F), we have

dx,f∞(x,y)ψd(x,y) .

Now, by our additional assumption, for y∗∈X there exists u∗ ∈F(y∗) such that d(y∗, u∗)≤. Thus, definex∗:=f∞(y∗,u∗)∈Fix(F), and we get

dy∗,x∗ ≤ψdy∗,u∗ ≤ψ(Cε).

As an exemplification of the previous theorem, we have the following result.

Let us recall first an important notion. A subsetUof a (generalized) metric space (X,d) is called proximinal if for eachxXthere existsuUsuch thatd(x,y) =D(x,U).

As a consequence of Theorem . and of the abstract result above, we obtain the fol-lowing theorem.

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Proof Letε:= (ε, . . . ,εm) (withεi>  fori∈ {, . . . ,m}), and lety∗∈Xbe aε-solution of (),

i.e.,D(y∗,F(y∗))≤ε. SinceFis an (I–A)–-MWP operator, for each (x,y)∈Graph(F), we have

dx,f∞(x,y) ≤(I–A)–d(x,y).

SinceF(y∗) is a proximinal set, there existsu∗∈F(y∗) such thatd(y∗,u∗) =D(y∗,F(y∗)). Thus, if we considerx∗:=f∞(y∗,u∗)∈Fix(F), we get

dy∗,x∗ ≤(I–A)–dy,u(IA)–ε.

Remark . It is an open question to give other examples of how Theorem . can be applied. A more general open question is to give similar results for multivalued almost contractions with matricesAandB.

For other examples and results regarding the Ulam-Hyers stability and the limit shad-owing property of the operatorial equations and inclusions, see Bota-Petruşel [], Petru-Petruşel-Yao [], Petruşel-Rus [] and Rus [, ].

3 An application to coupled fixed point results for singlevalued operators without mixed monotone property

LetXbe a nonempty set endowed with a partial order relation denoted by≤. Then we denote

X≤:=(x,x)∈X×X:x≤xorx≤x

.

Iff:XXis an operator, then we denote the Cartesian product offwith itself as follows

f ×f :X×XX×X, given by (f ×f)(x,x) :=

f(x),f(x) .

Definition . LetXbe a nonempty set. Then (X,d,≤) is called an ordered generalized metric space if

(i) (X,d)is a generalized metric space in the sense of Perov; (ii) (X,≤)is a partially ordered set.

The following result will be an important tool in our approach.

Theorem . Let(X,d,≤)be an ordered generalized metric space,and let f :XX be an operator.We suppose that

() for each(x,y) /Xthere existsz(x,y) :=zXsuch that(x,z), (y,z)X≤; () X≤∈I(f ×f);

() f : (X,d)→(X,d)is continuous;

() the metricdis complete;

() there existsx∈Xsuch that(x,f(x))∈X≤;

() there exists a matrixAMmm(R+),which converges to zero such that

df(x),f(y) ≤Ad(x,y) for each(x,y)X≤.

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Proof LetxXbe arbitrary. Since (x,f(x))∈X≤, by () and (), we get that there exists

x∗∈Xsuch that (fn(x

))n∈N→x∗asn→+∞. By (), we get thatx∗∈Fix(f).

If (x,x)∈X≤, then by (), we have that (fn(x),fn(x))∈X≤for eachn∈N. Thus, by (), we get that (fn(x))n∈N→x∗asn→+∞.

If (x,x) /∈X≤, then by (), it follows that there exists z(x,x) :=zX≤ such that (x,z), (x,z)X≤. By the fact that (x,z)X≤, as before, we get that (fn(z))n∈N →x∗ asn→+∞. This together with the fact that (x,z)X≤implies that (fn(x))n∈N→x∗as n→+∞.

Finally, the uniqueness of the fixed point follows by the contraction condition () using

again the assumption ().

Remark . The conclusion of the theorem above holds if instead of hypothesis () we put

() f : (X,≤)→(X,≤)is monotone increasing

or

() f : (X,≤)→(X,≤)is monotone decreasing.

Of course, it is easy to remark that assertion () in Theorem . is more general. For example, if we consider the ordered metric space (R,d,), thenf :RR,f(x

,x) := (g(x,x),g(x,x)) satisfies () for anyg:R→R.

Remark . Condition () from the theorem above is equivalent with

() f has a lower or an upper fixed point inX.

Remark . For some similar results, see Theorem . and Theorem . in [].

We will apply the above result for the coupled fixed point problem generated by two operators.

LetX be a nonempty set endowed with a partial order relation denoted by ≤. If we considerz:= (x,y),w:= (u,v) two arbitrary elements ofZ:=X×X, then, by definition,

zw if and only if (x≥uandyv).

Notice thatis a partial order relation onZ. We denote

Z=(z,w) :=(x,y), (u,v)Z×Z:zworwz.

LetT:ZZbe an operator defined by

T(x,y) :=

T(x,y) T(x,y)

=T(x,y),T(x,y) . ()

The Cartesian product ofT andT will be denoted byT×T, and it is defined in the following way

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The first main result of this section is the following theorem.

Theorem . Let(X,d,≤)be an ordered and complete metric space,and let T,T:X× XX be two operators.We suppose that

(i) for eachz= (x,y),w= (u,v)X×X,which are not comparable with respect to the

partial orderingonX×X,there existst:= (t,t)∈X×X(which may depend on

(x,y)and(u,v))such thattis comparable(with respect to the partial ordering)

with bothzandw,i.e.,

(x≥tandyt)or(x≤tandyt) and

(u≥tandvt)or(u≤tandvt);

(ii) for all(xuandyv)or(uxandvy),we have

⎧ ⎨ ⎩

T(x,y)T(u,v), T(x,y)T(u,v)

or ⎧ ⎨ ⎩

T(u,v)T(x,y), T(u,v)T(x,y);

(iii) T,T:X×XXare continuous;

(iv) there existsz:= (z,z)∈X×Xsuch that

⎨ ⎩

zT(z,z), z

≤T(z,z)

or ⎧ ⎨ ⎩

T(z,z)≥z, T(z,z)≤z;

(v) there exists a matrixA=kk

kk ∈M(R+)convergent toward zero such that

dT(x,y),T(u,v)kd(x,u) +kd(y,v),

dT(x,y),T(u,v)kd(x,u) +kd(y,v)

for all(xuandyv)or(uxandvy).

Then there exists a unique element(x∗,y∗)∈X×X such that

x∗=T

x∗,yand y∗=T

x∗,y∗ ,

and the sequence of the successive approximations(Tn

(w,w),Tn(w,w))converges to (x∗,y∗)as n→ ∞for all w= (w,w)∈X×X.

Proof DenoteZ:=X×X. We show that Theorem . is applicable for the operatorT : ZZdefined by

T(x,y) :=T(x,y),T(x,y) .

Notice first that by (i), we get that if (z:= (x,y),w:= (u,v)) /Z, there existstZsuch

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In order to prove () from Theorem ., letz= (x,y),w= (u,v) be arbitrary elements of Z(where (x≥uandyv) or (uxandvy)) such that

⎧ ⎨ ⎩

T(x,y)T(u,v), T(x,y)T(u,v)

or ⎧ ⎨ ⎩

T(u,v)T(x,y), T(u,v)T(x,y).

From

T(x,y)T(u,v),

T(x,y)T(u,v), we get that (T(x,y),T(x,y))(T(u,v),T(u,v)), that is

T(z)T(w). ()

By a similar approach, we alternatively have that

T(w)T(z). ()

Using () and (), we get that

T(z),T(w) ∈Z for all (z,w)Z.

Thus, we get (T×T)(Z)⊆Z, which implies that

ZI(T×T). ()

In order to obtain () from Theorem ., notice first that since there exists (z ,z)∈ X×Xsuch that

⎧ ⎨ ⎩

zT(z,z), z

≤T(z,z) or

⎧ ⎨ ⎩

T(z,z)≥z, T(z,z)≤z,

()

we obtain that

z,zT

z,z ,T

z,z . ()

Thus, we havezT(z). By a similar approach, we alternatively obtain thatT(z)z. Thus, (z,T(z))∈Z.

Finally, in order to prove hypotheses () and () from Theorem ., we define the map-pingd:Z×Z→R

+by

d(x,y), (u,v) :=

d(x,u) d(y,v)

.

Notice now that if (X,d,≤) is an ordered metric space, then (Z,d,) is an ordered gen-eralized metric space. The completeness ofdfollows from the completeness ofd. Notice also that the continuity ofTfollows by (iii). For hypothesis (), we successively have

dT(x,y),T(u,v)

=dT(x,y),T(x,y) ,

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=

d(T(x,y),T(u,v)) d(T(x,y),T(u,v))

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

=

kkkk

d(x,u) d(y,v)

=Ad(x,y), (u,v) .

Hence

dT(z),T(w) ≤Ad(z,w) for allz,wZ. ()

Thus, the triple (Z,d,) and the operatorT:ZZsatisfy all the hypothesis of Theo-rem .. Hence,Tis a Picard operator, and so, the equationz=T(z) has a unique solution z∗∈X×X, and the sequence of successive approximations of the operatorT, starting from anyw∈X×Xconverges toz∗. Thus, the unique elementz∗= (x∗,y∗) satisfies the system

⎧ ⎨ ⎩

x∗=T(x∗,y∗),

y∗=T(x∗,y∗)

andTn(w

)→z∗asn→ ∞, whereTn(w) = (Tn(w),Tn(w)) and

Tn(w) =Tn–

T(w),T(w),

Tn(w) =Tn–

T(w),T(w)

()

for alln∈N,n≥.

For the particular case of classical coupled fixed point problems (i.e.,T(x,y) :=S(x,y) andT(x,y) :=S(y,x), whereS:X×XXis a given operator) we get (by Theorem .) the following generalization of the Gnana Bhaskar-Lakshmikantham theorem in [].

Theorem . Let(X,d,≤)be an ordered and complete metric space,and let S:X×XX be an operator.We suppose that

(i) for eachz= (x,y),w= (u,v)X×X,which are not comparable with respect to the

partial orderingonX×X,there existst:= (t,t)∈X×X(which may depend on

(x,y)and(u,v))such thattis comparable(with respect to the partial ordering)

with bothzandw;

(ii) for all(xuandyv)or(uxandvy),we have

⎧ ⎨ ⎩

S(x,y)S(u,v), S(y,x)S(v,u) or

⎧ ⎨ ⎩

S(u,v)S(x,y), S(v,u)S(y,x);

(iii) S:X×XXis continuous;

(iv) there existsz:= (z,z)∈X×Xsuch that

⎨ ⎩ z

≥S(z,z), zS(z,z) or

⎧ ⎨ ⎩

S(z

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(v) there existk,k∈R+withk+k< such that

dS(x,y),S(u,v)kd(x,u) +kd(y,v)

for all(xuandyv)or(uxandvy).

Then there exists a unique element(x∗,y∗)∈X×X such that

x∗=Sx∗,yand y∗=Sy∗,x

and the sequence of the successive approximations (Sn(w

,w),Sn(w,w))converges to (x∗,y∗)as n→ ∞for all w= (w,w)∈X×X.

As an application of the previous theorem, we get now an existence and uniqueness result for a system of functional-integral equations, which appears in some traffic flow models.

⎧ ⎨ ⎩

x(t) =f(t,x(t),Tk(t,s,x(s),y(s))ds),

y(t) =f(t,y(t),Tk(t,s,x(s),y(s))ds). ()

By a solution of the previous system, we understand a couple (x,y)C[,TC[,T], which satisfies the system for allt∈[,T].

As before, we consider onX:=C[,T] the following partial ordering relation

x≤Cy if and only if x(t)y(t) for allt∈[,T]

and the supremum norm

xC:= max

t∈[,T] x(t).

Notice that, as before, the partial ordering relation≤Cgenerates onX×Xa partial order-ingC.

If we define

S:X×XX, (x,y)−→S(x,y), whereS(x,y)(t) :=f

t,x(t),

T

kt,s,x(s),y(s) ds

,

then, the above system can be represented as a coupled fixed point problem

⎧ ⎨ ⎩

x=S(x,y),

y=S(y,x).

An existence and uniqueness result for the system () is the following theorem.

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(i) there existsz:= (z,z)∈C[,TC[,T]such that

⎨ ⎩

z(t)≥f(t,z(t), t

k(t,s,z

(t),z(t))ds),

z

(t)≤f(t,z(t), t

k(t,s,z(t),z(t))ds)

or

⎧ ⎨ ⎩

z(t)≤f(t,z(t),tk(t,s,z(t),z(t))ds), z

(t)≥f(t,z(t), t

k(t,s,z(t),z(t))ds);

(ii) (a) f(t,·,z)is increasing for allt∈[,T],z∈Randk(t,s,·,z)is increasing,

k(t,s,w,·)is decreasing andf(t,w,·)is increasing for allt,s∈[,T],w,z∈R or

(b) f(t,·,z)is decreasing for allt∈[,T],z∈Randk(t,s,·,z)is decreasing,

k(t,s,w,·)is increasing andf(t,w,·)is decreasing for allt,s∈[,T],w,z∈R; (iii) there existk,k∈R+such that

f(t,w,z) –f(t,w,z)≤k|w–w|+k|z–z|

for allt∈[,T]andw,w,z,z∈R;

(iv) there existα,β∈R+such that for allt,s∈[,T]andw,w,z,z∈R,we have k(t,s,w,z) –k(t,s,w,z)≤α|w–w|+β|z–z|;

(v) k+kT(α+β) < .

Then,there exists a unique solution(x∗,y∗)of system().

Proof From the hypotheses, we get that all the assumptions of Theorem . are satisfied for the operator

S:X×XX, (x,y)−→S(x,y), S(x,y)(t) :=f

t,x(t),

T

kt,s,x(s),y(s) ds

,

whereX:= (C[,T],≤C, · C).

We introduce now the concept of Ulam-Hyers stability for coupled fixed point problems.

Definition . Let (X,d) be a metric space, and letT,T:X×XXbe two operators. Then the operatorial equations system

⎧ ⎨ ⎩

x=T(x,y),

y=T(x,y)

()

is said to be Ulam-Hyers stable if there existc,c,c,c>  such that for eachε,ε>  and each pair (u∗,v∗)∈X×Xsuch that

du∗,T

u∗,v∗ ≤ε,

dv∗,T

u∗,v∗ ≤ε

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there exists a solution (x∗,y∗)∈X×Xof () such that

du∗,x∗ ≤cε+cε,

dv∗,y∗ ≤cε+cε.

()

It is an open problem to obtain Ulam-Hyers stability results for the coupled fixed point problem in the context of ordered metric spaces. For several results on this subject, see [, ].

Remark . The case of a system of operatorial inclusions of the form

⎧ ⎨ ⎩

xT(x,y), yT(x,y)

(whereT,T:X×XP(X) are two given multivalued operators) can be treated in a similar way.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the first two authors on one hand and the third author on the other hand have equal contributions to this paper. All authors read and approved the final manuscript.

Acknowledgements

(1) For the first author, this paper was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0094, while for the third author, this work was possible with the financial support of the Sectoral Operational Programme for Human Resources Development 2007-2013, co-financed by the European Social Fund, under the project number POSDRU/107/1.5/S/76841 with the titleModern Doctoral Studies: Internationalization and Interdisciplinarity. (2) The authors are thankful to the reviewers for the careful reading of the paper and fruitful comments which improved the quality of the paper in a significant way.

Received: 10 December 2012 Accepted: 29 July 2013 Published: 14 August 2013 References

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doi:10.1186/1687-1812-2013-218

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