Fixed points and strict fixed points for multivalued contractions of Reich type on metric spaces endowed with a graph

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R E S E A R C H

Open Access

Fixed points and strict ﬁxed points for

multivalued contractions of Reich type on

metric spaces endowed with a graph

Cristian Chifu

1

_{, Gabriela Petru¸sel}

1*

_{and Monica-Felicia Bota}

2

*_{Correspondence:}

gabi.petrusel@tbs.ubbcluj.ro 1_{Department of Business,}

Babe¸s-Bolyai University Cluj-Napoca, Horea street, No. 7, Cluj-Napoca, Romania Full list of author information is available at the end of the article

Abstract

The purpose of this paper is to present some strict ﬁxed point theorems for

multivalued operators satisfying a Reich-type condition on a metric space endowed with a graph. The well-posedness of the ﬁxed point problem is also studied.

MSC: 47H10; 54H25

Keywords: ﬁxed point; strict ﬁxed point; metric space endowed with a graph; well-posed problem

1 Preliminaries

A new approach in the theory of ﬁxed points was recently given by Jachymski [] and Gwóźdź-Lukawska and Jachymski [] by using the context of metric spaces endowed with a graph. Other recent results for single-valued and multivalued operators in such metric spaces are given by Nicolae, O’Regan and Petruşel in [] and by Beg, Butt and Radojevic in [].

Let (X,d) be a metric space and letbe the diagonal ofX×X. LetGbe a directed graph such that the setV(G) of its vertices coincides withXand⊆E(G), whereE(G) is the set of the edges of the graph. Assume also thatGhas no parallel edges and, thus, one can identifyGwith the pair (V(G),E(G)).

Ifxandyare vertices ofG, then a path inGfromxtoyof lengthk∈Nis a ﬁnite sequence (xn)n∈{,,,...,k} of vertices such thatx=x,xk=yand (xi–,xi)∈E(G) fori∈ {, , . . . ,k}.

Notice that a graphGis connected if there is a path between any two vertices and it is weakly connected ifG˜ is connected, whereG˜ denotes the undirected graph obtained from Gby ignoring the direction of edges.

Denote byG–_{the graph obtained from}_G_{by reversing the direction of edges. Thus,}

EG–=(x,y)∈X×X: (y,x)∈E(G). (∗)

Since it is more convenient to treatG˜ as a directed graph for which the set of its edges is symmetric, under this convention, we have that

E(G) =˜ E(G)∪EG–. (∗∗)

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If Gis such thatE(G) is symmetric, then forx∈V(G), the symbol [x]G denotes the

equivalence class of the relationdeﬁned onV(G) by the rule:

yz if there is a path inGfromytoz.

Let us consider the following families of subsets of a metric space (X,d):

P(X) :=Y∈P(X)|Y=∅; Pb(X) :=

Y∈P(X)|Yis bounded;

Pcl(X) :=

Y∈P(X)|Yis closed; Pcp(X) :=

Y∈P(X)|Yis compact.

The gap functional between the setsAandBin the metric space (X,d) is given by

D:P(X)×P(X)→R+∪ {+∞}, D(A,B) =inf

d(a,b)|a∈A,b∈B.

In particular, ifx∈XthenD(x,B) :=D({x},B). The Pompeiu-Hausdorﬀ functional is deﬁned by

H:P(X)×P(X)→R+∪ {+∞},

H(A,B) =max

sup

a∈A

D(a,B),sup

b∈B

D(A,b)

.

The diameter generalized functional generated bydis given by

δ:P(X)×P(X)→R+∪ {+∞},

δ(A,B) =supd(a,b)|a∈A,b∈B.

In particular, we denote byδ(A) :=δ(A,A) the diameter of the setA.

Let (X,d) be a metric space. IfT :X→P(X) is a multivalued operator, thenx∈Xis called a fixed point forT if and only ifx∈T(x). The setFix(T) :={x∈X|x∈T(x)}is called the fixed point set ofT, whileSFix(T) ={x∈X| {x}=Tx}is called the strict fixed point set ofT.Graph(T) :={(x,y)|y∈T(x)}denotes the graph ofT.

Deﬁnition . Letϕ:R+→R+be a mapping. Thenϕis called a strong comparison func-tion if the following asserfunc-tions hold:

(i) ϕis increasing;

(ii) ϕn(t)→asn→ ∞for allt∈R+;

(iii) ∞_n_=ϕn(t) <∞for allt∈R+.

Deﬁnition . Let (X,d) be a complete metric space, letGbe a directed graph, and let T:X→Pb(X) be a multivalued operator. By deﬁnition,T is called a (δ,ϕ)-G-contraction if there existsϕ:R+→R+, a strong comparison function, such that

δT(x),T(y)≤ϕd(x,y) for all (x,y)∈E(G).

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[, ]). The equality betweenFix(T) andSFix(T) and the well-posedness of the ﬁxed point problem are also studied.

Our results also generalize and extend some ﬁxed point theorems in partially ordered complete metric spaces given in Harjani and Sadarangani [], Nicolaeet al.[], Nieto and Rodríguez-López [] and [], Nietoet al.[], O’Regan and Petruşel [], Petruşel and Rus [], and Ran and Reurings [].

2 Fixed point and strict ﬁxed point theorems

We begin this section by presenting a strict ﬁxed point theorem for a Reich type contrac-tion with respect to the funccontrac-tionalδ.

Theorem . Let(X,d)be a complete metric space and let G be a directed graph such that the triple(X,d,G)satisﬁes the following property:

(P) for any sequence(xn)n∈N⊂X with xn→x as n→ ∞,

there exists a subsequence(xkn)n∈Nof (xn)n∈Nsuch that(xkn,x)∈E(G).

Let T:X→Pb(X)be a multivalued operator.Suppose that the following assertions hold:

(i) There existsa,b,c∈R+withb= anda+b+c< such that

δT(x),T(y)≤ad(x,y) +bδx,T(x)+cδy,T(y)

for all(x,y)∈E(G). (ii) For eachx∈X,the set

˜

XT(x) := y∈T(x) : (x,y)∈E(G)andδ

x,T(x)≤qd(x,y)

for someq∈

, –a–c b

is nonempty.

Then we have:

(a) Fix(T) =SFix(T)=∅;

(b) If we additionally suppose that

x∗,y∗∈Fix(T) ⇒ x∗,y∗∈E(G),

thenFix(T) =SFix(T) ={x∗}.

Proof (a) Letx∈X. SinceX˜T(x)=∅, there existsx∈T(x) and  <q<–a_b–c such that (x,x)∈E(G) and

δx,T(x)

≤qd(x,x).

By (i) we have that

δx,T(x)

≤δT(x),T(x)

≤ad(x,x) +bδ

x,T(x)

+cδx,T(x)

≤ad(x,x) +bqd(x,x) +cδ

x,T(x)

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Hence,

δx,T(x)

≤a+bq

 –c d(x,x). (.)

For x ∈X, since X˜T(x)=∅, we get again that there exists x ∈ T(x) such that

δ(x,T(x))≤qd(x,x) and (x,x)∈E(G). Then

d(x,x)≤δ

x,T(x)

≤a+bq

 –c d(x,x). (.)

On the other hand, by (i), we have that

δx,T(x)

≤δT(x),T(x)

≤ad(x,x) +bδ

x,T(x)

+cδx,T(x)

≤ad(x,x) +bqd(x,x) +cδ

x,T(x)

.

Using (.) we obtain

δx,T(x)

≤a+bq

 –c d(x,x)≤

a+bq  –c



d(x,x). (.)

Forx∈X, we haveX˜T(x)=∅, and so there existsx∈T(x) such thatδ(x,T(x))≤ qd(x,x) and (x,x)∈E(G).

Then

d(x,x)≤δ

x,T(x)

≤

a+bq

 –c 

d(x,x). (.)

By these procedures, we obtain a sequence (xn)n∈Nwith the following properties:

() (xn,xn+)∈E(G)for eachn∈N;

() d(xn,xn+)≤(a_–+bq_c )nd(x,x)for eachn∈N;

() δ(xn,T(xn))≤(a_–+bq_c)nd(x,x)for eachn∈N.

From () we obtain that the sequence (xn)n∈N is Cauchy. Since the metric spaceXis complete, we get that the sequence is convergent,i.e.,xn→x∗asn→ ∞. By the property

(P), there exists a subsequence (xkn)n∈Nof (xn)n∈Nsuch that (xkn,x∗)∈E(G) for eachn∈N.

We have

δx∗,Tx∗≤dx∗,xkn+

+δxkn+,T

x∗≤dx∗,xkn+

+δT(xkn),T

x∗

≤dx∗,xkn+

+adxkn,x∗

+bδxkn,T(xkn)

+cδx∗,Tx∗

≤dx∗,xkn+

+adxkn,x∗

+b

a+bq

 –c kn

d(x,x) +cδ

x∗,Tx∗,

δx∗,Tx∗≤   –cd

x∗,xkn+

+ a

 –cd

xkn,x∗

+ b

 –c

a+bq  –c

kn

d(x,x). (.)

Butd(x∗,xkn+)→ asn→ ∞andd(xkn,x∗)→ asn→ ∞. Hence,δ(x∗,T(x∗)) = ,

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BecauseSFix(T)⊂Fix(T), we need to show thatFix(T)⊂SFix(T).

Letx∗∈Fix(T)⇒x∗∈T(x∗). Because⊂E(G), we have that (x∗,x∗)∈E(G). Using (ii) withx=y=x∗, we obtain

δTx∗≤adx∗,x∗+bδx∗,Tx∗+cδx∗,Tx∗.

So,δ(T(x∗))≤(b+c)δ(x∗,T(x∗)). Becausex∗∈T(x∗), we get thatδ(x∗,T(x∗))≤δ(T(x∗)). Hence, we have

δTx∗≤(b+c)δTx∗. (.)

Suppose thatcard(T(x∗)) > . This implies thatδ(T(x∗)) > . Thus from (.) we obtain thatb+c> , which contradicts the hypothesisa+b+c< .

Thusδ(T(x∗)) = ⇒T(x∗) ={x∗},i.e.,x∗∈SFix(T) andFix(T)⊂SFix(T). Hence,Fix(T) =SFix(T)=∅.

(b) Suppose that there existx∗,y∗∈Fix(T) =SFix(T) withx∗=y∗. We have that

• x∗∈SFix(T)⇒δ(x∗,T(x∗)) = ; • y∗∈SFix(T)⇒δ(y∗,T(y∗)) = ; • (x∗,y∗)∈E(G).

Using (i) we obtain

dx∗,y∗=δTx∗,Ty∗≤adx∗,y∗+bδx∗,Tx∗+cδy∗,Ty∗.

Thus,d(x∗,y∗)≤ad(x∗,y∗), which implies thata≥, which is a contradiction.

Hence,Fix(T) =SFix(T) ={x∗}.

Next we present some examples and counterexamples of multivalued operators which satisfy the hypothesis in Theorem ..

Example . LetX:={(, ), (, ), (, ), (, )}andT:X→Pcl(X) be given by

T(x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

{(, )}, x= (, ),

{(, )}, x= (, ),

{(, ), (, )}, x= (, ),

{(, ), (, )}, x= (, ).

(.)

LetE(G) :={((, ), (, )), ((, ), (, )), ((, ), (, ))} ∪.

Notice that all the hypotheses in Theorem . are satisﬁed (the condition (i) is veriﬁed fora=c= .,b= . and soFix(T) =SFix(T) ={(, )}.

The following remarks show that it is not possible to have elements inFT\SFT.

Remark . If we suppose that there existsx∈FT\SFT, then, since (x,x)∈, we get

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Remark . If, in the previous theorem, instead of the property (P), we suppose thatT has a closed graph, then we obtain again the conclusionFix(T) =SFix(T)=∅.

Remark . If, in the above remark, we additionally suppose that

x∗,y∗∈Fix(T) ⇒ x∗,y∗∈E(G),

thenFix(T) =SFix(T) ={x∗}.

The next result presents a strict ﬁxed point theorem where the operatorT satisﬁes a (δ,ϕ)-G-contractive condition onE(G).

Theorem . Let(X,d)be a complete metric space and let G be a directed graph such that the triple(X,d,G)satisﬁes the following property:

(P) for any sequence(xn)n∈N⊂X with xn→x as n→ ∞,

there exists a subsequence(xkn)n∈Nof (xn)n∈Nsuch that(xkn,x)∈E(G).

Let T:X→Pb(X)be a multivalued operator.Suppose that the following assertions hold:

(i) T is a(δ,ϕ)-G-contraction. (ii) For eachx∈X,the set

˜

XT:= y∈T(x) : (x,y)∈E(G)andδ

x,T(x)≤qd(x,y)for someq∈

, –a–c b

is nonempty.

Then we have:

(a) Fix(T) =SFix(T)=∅;

(b) If,in addition,the following implication holds:

x∗,y∗∈Fix(T) ⇒ x∗,y∗∈E(G),

thenFix(T) =SFix(T) ={x∗}.

Proof (a) Letx∈X. Then, since X˜T(x) is nonempty, there existx ∈T(x) and q∈ ],–a_b–c[ such that (x,x)∈E(G) and

δx,T(x)

≤qd(x,x).

By (i) we have that

δx,T(x)

≤δT(x),T(x)

≤ϕd(x,x)

.

Forx∈X, by the same approach as before, there existsx∈T(x) such thatδ(x,T(x))≤ qd(x,x) and (x,x)∈E(G).

We have

d(x,x)≤δ

x,T(x)

≤δT(x),T(x)

≤ϕd(x,x)

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On the other hand, by (i) we have that

δx,T(x)

≤δT(x),T(x)

≤ϕd(x,x)

≤ϕd(x,x)

.

By the same procedure, for x ∈X there exists x ∈T(x) such that δ(x,T(x))≤ qd(x,x) and (x,x)∈E(G). Thus

d(x,x)≤δ

x,T(x)

≤ϕd(x,x)

. (.)

We have

δx,T(x)

≤δT(x),T(x)

≤ϕd(x,x)

≤ϕd(x,x)

.

() (xn,xn+)∈E(G)for eachn∈N;

() d(xn,xn+)≤ϕn(d(x,x))for eachn∈N;

() δ(xn,T(xn))≤ϕn(d(x,x))for eachn∈N.

By (), using the properties ofϕ, we obtain that the sequence (xn)n∈Nis Cauchy. Since the

metric space is complete, we have that the sequence is convergent,i.e.,xn→x∗asn→ ∞.

By the property (P), we get that there exists a subsequence (xkn)n∈Nof (xn)n∈Nsuch that

(xkn,x∗)∈E(G) for eachn∈N.

We shall prove now thatx∗∈SFix(T). We have

δx∗,Tx∗≤dx∗,xkn+

+δxkn+,T

x∗≤dx∗,xkn+

+δT(xkn),T

x∗

≤dx∗,xkn+

+ϕdxkn,x∗

.

Sinced(x∗,xkn+)→ asn→ ∞andϕ is continuous in  withϕ() = , we get that δ(x∗,T(x∗)) = .

Hence,x∗∈SFix(T)⇒SFix(T)=∅. We shall prove now thatFix(T) =SFix(T).

BecauseSFix(T)⊂Fix(T), we need to show thatFix(T)⊂SFix(T).

Letx∗∈Fix(T). Because⊂E(G), we have that (x∗,x∗)∈E(G). Using (i) withx=y=x∗, we obtain

δTx∗≤ϕdx∗,x∗= .

Hence,x∗∈SFix(T) and the proof of this conclusion is complete.

(b) Suppose that there existx∗,y∗∈Fix(T) =SFix(T) withx∗=y∗. We have that

• x∗∈SFix(T)⇒δ(x∗,T(x∗)) = ; • y∗∈SFix(T)⇒δ(y∗,T(y∗)) = ; • (x∗,y∗)∈E(G).

Using (i) we obtain

dx∗,y∗=δTx∗,Ty∗≤ϕdx∗,y∗<dx∗,y∗.

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In the next result, the operatorTsatisﬁes another contractive condition with respect to

δonE(G)∩Graph(T).

Theorem . Let(X,d)be a complete metric space,let G be a directed graph,and let T: X→Pb(X)be a multivalued operator.Suppose that f:X→R+deﬁned f(x) :=δ(x,T(x))is a lower semicontinuous mapping.Suppose that the following assertions hold:

(i) There exista,b∈R+,withb= anda+b< ,such that

δy,T(y)≤ad(x,y) +bδx,T(x) for all(x,y)∈E(G)∩Graph(T).

(ii) For eachx∈X,the set

˜

XT(x) := y∈T(x) : (x,y)∈E(G)andδ

x,T(x)≤qd(x,y)for someq∈

, –a b

is nonempty.

ThenFix(T) =SFix(T)=∅.

Proof Letx∈X. Then, sinceX˜T(x) is nonempty, there existx∈T(x) and  <q<–_ba such that

δx,T(x)

≤qd(x,x)

and (x,x)∈E(G). Sincex∈T(x), we get that (x,x)∈E(G)∩Graph(T). By (i), takingy=xandx=x, we have that

δx,T(x)

≤ad(x,x) +bδ

x,T(x)

≤ad(x,x) +bqd(x,x).

Hence,

δx,T(x)

≤(a+bq)d(x,x). (.)

Forx∈X(sinceX˜T(x)=∅), there existsx∈T(x) such thatδ(x,T(x))≤qd(x,x) and (x,x)∈E(G). Butx∈T(x) and so (x,x)∈E(G)∩Graph(T).

Then

d(x,x)≤δ

x,T(x)

≤(a+bq)d(x,x). (.)

By (i), takingy=xandx=x, we have that

δx,T(x)

≤ad(x,x) +bδ

x,T(x)

≤ad(x,x) +bqd(x,x) = (a+bq)d(x,x)≤(a+bq)d(x,x).

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() d(xn,xn+)≤(a+bq)nd(x,x)for eachn∈N;

() δ(xn,T(xn))≤(a+bq)nd(x,x)for eachn∈N.

From () we obtain that the sequence (xn)n∈N is Cauchy. Since the metric spaceXis complete, we have that the sequence is convergent,i.e.,xn→x∗asn→ ∞. Now, by the

lower semicontinuity of the functionf, we have

≤fx∗≤lim inf

n→∞ f(xn) = .

Thusf(x∗) = , which means thatδ(x∗,T(x∗)) = . Thusx∗∈SFix(T).

Letx∗∈Fix(T). Then (x∗,x∗)∈Graph(T) and hence (x∗,x∗)∈E(G)∩Graph(T). Using (i) withx=y=x∗, we obtain

δTx∗=δx∗,Tx∗≤adx∗,x∗+bδx∗,Tx∗.

So,δ(T(x∗))≤bδ(T(x∗)). If we suppose thatcardT(x∗) > , thenδ(T(x∗)) > . Thus,b≥, which contradicts the hypothesis.

Thusδ(T(x∗)) =  and soT(x∗) ={x∗}. The proof is now complete.

Remark . Example . satisﬁes the conditions from Theorem . fora= . andb= ..

3 Well-posedness of the ﬁxed point problem

In this section we present some well-posedness results for the ﬁxed point problem. We consider both the well-posedness and the well-posedness in the generalized sense for a multivalued operatorT.

We begin by recalling the deﬁnition of these notions from [] and [].

Definition . Let (X,d) be a metric space and letT:X→P(X) be a multivalued opera-tor. By definition, the fixed point problem is well posed forTwith respect toHif:

(i) SFix(T) ={x∗};

(ii) If(xn)n∈Nis a sequence inXsuch thatH(xn,T(xn))→asn→ ∞, thenxn d

→x∗as

n→ ∞.

Definition . Let (X,d) be a metric space and letT:X→P(X) be a multivalued opera-tor. By definition, the fixed point problem is well posed in the generalized sense forTwith respect toHif:

(i) SFixT=∅;

(ii) If(xn)n∈Nis a sequence inXsuch thatH(xn,T(xn))→asn→ ∞, then there exists

a subsequence(xkn)n∈Nof(xn)n∈Nsuch thatxkn

d

→x∗asn→ ∞.

In our ﬁrst result we will establish the well-posedness of the ﬁxed point problem for the operatorT, whereTis a Reich-typeδ-contraction.

Theorem . Let(X,d)be a complete metric space and let G be a directed graph such that the triple(X,d,G)satisﬁes the property(P).

Let T:X→Pb(X)be a multivalued operator.Suppose that

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(ii) ifx∗,y∗∈Fix(T),then(x∗,y∗)∈E(G);

(iii) for any sequence(xn)n∈N,xn∈XwithH(xn,T(xn))→asn→ ∞,we have

(xn,x∗)∈E(G).

In these conditions the ﬁxed point problem is well posed for T with respect to H.

Proof From (i) and (ii) we obtain thatSFix(T) ={x∗}. Let (xn)n∈N⊂Xbe a sequence which

satisﬁes (iii). It is obvious thatH(xn,T(xn)) =δ(xn,T(xn)),

dxn,x∗

≤δxn,T

x∗≤δxn,T(xn)

+δT(xn),T

x∗

≤δxn,T(xn)

+adxn,x∗

+bδxn,T(xn)

+cδx∗,Tx∗.

Thus

dxn,x∗

≤  +b

 –aδ

xn,T(xn)

→ asn→ ∞.

Hence,xn→x∗asn→ ∞.

Remark . If we replace the property (P) with the condition thatT has a closed graph, we reach the same conclusion.

The next result deals with the well-posedness of the ﬁxed point problem in the general-ized sense.

Theorem . Let(X,d)be a complete metric space and let G be a directed graph such that the triple(X,d,G)satisﬁes the property(P).

(i) conditions(i)and(ii)in Theorem.hold;

(ii) for any sequence(xn)n∈N,xn∈XwithH(xn,T(xn))→asn→ ∞,there exists a

subsequence(xkn)n∈Nsuch that(xkn,x∗)∈E(G)andH(xkn,T(xkn))→.

In these conditions the ﬁxed point problem is well posed in the generalized sense for T with respect to H.

Proof From (i) we have thatSFix(T)=∅. Let (xn)n∈N⊂Xbe a sequence which satisﬁes

(ii). Then there exists a subsequence (xkn)n∈Nsuch that (xkn,x∗)∈E(G).

We haveH(xkn,T(xkn)) =δ(xkn,T(xkn)),

dxkn,x∗

≤δxkn,T

x∗≤δxkn,T(xkn)

+δT(xkn),T

x∗

≤δxkn,T(xkn)

+adxkn,x∗

+bδxkn,T(xkn)

+cδx∗,Tx∗.

Thus

dxkn,x∗

≤ +b

 –aδ

xkn,T(xkn)

→ asn→ ∞.

Hence,xkn→x∗.

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Next we consider the case where the operatorTsatisﬁes aϕ-contraction condition.

Theorem . Let(X,d)be a complete metric space and let G be a directed graph such that the triple(X,d,G)satisﬁes the property(P).

(i) conditions(i)and(ii)in Theorem.hold;

(ii) the following implication holds:x∗,y∗∈Fix(T)implies(x∗,y∗)∈E(G);

(iii) the functionψ:R+→R+,given byψ(t) =t–ϕ(t),has the following property:if

ψ(tn)→asn→ ∞,thentn→asn→ ∞;

(iv) for any sequence(xn)n∈N⊂XwithH(xn,T(xn))→asn→ ∞,we have

(xn,x∗)∈E(G)for alln∈N.

In these conditions the ﬁxed point problem is well posed for T with respect to H.

Proof From (i) and (ii) we obtain thatSFix(T) ={x∗}. Let (xn)n∈N,xn∈Xbe a sequence

which satisﬁes (iv). It is obvious thatH(xn,T(xn)) =δ(xn,T(xn)),

dxn,x∗

≤δxn,T

x∗≤δxn,T(xn)

+δT(xn),T

x∗

≤δxn,T(xn)

+ϕdxn,x∗

.

Thus

dxn,x∗

–ϕdxn,x∗

≤δxn,T(xn)

→ asn→ ∞.

Using condition (iii), we get thatd(xn,x∗)→ asn→ ∞. Hence,xn→x∗.

Remark . If we replace the property (P) with the condition thatThas a closed graph, we reach the same conclusion.

The next result gives a well-posedness (in the generalized sense) criterion for the ﬁxed point problem.

Theorem . Let(X,d)be a complete metric space and let G be a directed graph such that

the triple(X,d,G)satisﬁes the property(P).

(i) the conditions(i)and(ii)in Theorem.hold;

(ii) the functionψ:R+→R+,givenψ(t) =t–ϕ(t),has the following property:for any sequence(tn)n∈N,there exists a subsequence(xkn)n∈Nsuch that ifψ(tkn)→as

n→ ∞,thentkn→asn→ ∞;

(iii) for any sequence(xn)n∈N,xn∈XwithH(xn,T(xn))→asn→ ∞,there exists a

subsequence(xkn)n∈Nsuch that(xkn,x∗)∈E(G)andH(xkn,T(xkn))→.

In these conditions the ﬁxed point problem is well posed in the generalized sense for T with respect to H.

Proof From (i) we have thatSFix(T)=∅. Let (xn)n∈N,xn∈Xbe a sequence which satisﬁes

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We haveH(xkn,T(xkn)) =δ(xkn,T(xkn)),

dxkn,x∗

≤δxkn,T

x∗≤δxkn,T(xkn)

+δT(xkn),T

x∗

≤δxkn,T(xkn)

+ϕdxkn,x∗

.

Thus

dxkn,x

∗_–_ϕ_d_x

kn,x

∗_≤_δ_x

kn,T(xkn)

→ asn→ ∞.

Using condition (ii), we get thatd(xkn,x∗)→ asn→ ∞. Hence,xn→x∗.

Remark . If we replace the property (P) with the condition thatThas a closed graph, we reach the same conclusion.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript. The authors have equal contributions to this paper.

Author details

1_{Department of Business, Babe¸s-Bolyai University Cluj-Napoca, Horea street, No. 7, Cluj-Napoca, Romania.}2_Department

of Mathematics, Babe¸s-Bolyai University Cluj-Napoca, Kogalniceanu street, No. 1, Cluj-Napoca, Romania.

Acknowledgements

The third author is supported by a grant of the Romanian National Authority for Scientiﬁc Research, CNCS UEFISCDI, project number PN-II-ID-PCE-2011-3-0094.

Received: 8 May 2013 Accepted: 10 July 2013 Published: 29 July 2013

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doi:10.1186/1687-1812-2013-203