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

  • Cristian Chifu1,
  • Gabriela Petruşel1Email author and
  • Monica-Felicia Bota2
Fixed Point Theory and Applications20132013:203

https://doi.org/10.1186/1687-1812-2013-203

Received: 8 May 2013

Accepted: 10 July 2013

Published: 29 July 2013

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 pointstrict fixed pointmetric space endowed with a graphwell-posed problem

1 Preliminaries

A new approach in the theory of fixed points was recently given by Jachymski [1] and Gwóźdź-Lukawska and Jachymski [2] 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 [3] and by Beg, Butt and Radojevic in [4].

Let ( X , d ) be a metric space and let Δ be the diagonal of X × X . Let G be a directed graph such that the set V ( G ) of its vertices coincides with X and Δ E ( G ) , where E ( G ) is the set of the edges of the graph. Assume also that G has no parallel edges and, thus, one can identify G with the pair ( V ( G ) , E ( G ) ) .

If x and y are vertices of G, then a path in G from x to y of length k N is a finite sequence ( x n ) n { 0 , 1 , 2 , , k } of vertices such that x 0 = x , x k = y and ( x i 1 , x i ) E ( G ) for i { 1 , 2 , , k } . Notice that a graph G is connected if there is a path between any two vertices and it is weakly connected if G ˜ is connected, where G ˜ denotes the undirected graph obtained from G by ignoring the direction of edges.

Denote by G 1 the graph obtained from G by reversing the direction of edges. Thus,
Since it is more convenient to treat G ˜ as a directed graph for which the set of its edges is symmetric, under this convention, we have that
If G is such that E ( G ) is symmetric, then for x V ( G ) , the symbol [ x ] G denotes the equivalence class of the relation defined on V ( G ) by the rule:
y z if there is a path in  G  from  y  to  z .
Let us consider the following families of subsets of a metric space ( X , d ) :
P ( X ) : = { Y P ( X ) Y } ; P b ( X ) : = { Y P ( X ) Y  is bounded } ; P cl ( X ) : = { Y P ( X ) Y  is closed } ; P cp ( X ) : = { Y P ( X ) Y  is compact } .
The gap functional between the sets A and B in 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, if x 0 X then D ( x 0 , B ) : = D ( { x 0 } , 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 by d is given by
δ : P ( X ) × P ( X ) R + { + } , δ ( A , B ) = sup { d ( a , b ) a A , b B } .

In particular, we denote by δ ( A ) : = δ ( A , A ) the diameter of the set A.

Let ( X , d ) be a metric space. If T : X P ( X ) is a multivalued operator, then x X is called a fixed point for T if and only if x T ( x ) . The set Fix ( T ) : = { x X x T ( x ) } is called the fixed point set of T, while S Fix ( T ) = { x X { x } = T x } is called the strict fixed point set of T. Graph ( T ) : = { ( x , y ) y T ( x ) } denotes the graph of T.

Definition 1.1 Let φ : R + R + be a mapping. Then φ is called a strong comparison function if the following assertions hold:
  1. (i)

    φ is increasing;

     
  2. (ii)

    φ n ( t ) 0 as n for all t R + ;

     
  3. (iii)

    n = 1 φ n ( t ) < for all t R + .

     
Definition 1.2 Let ( X , d ) be a complete metric space, let G be a directed graph, and let T : X P b ( X ) be a multivalued operator. By definition, T is called a ( δ , φ ) -G-contraction if there exists φ : R + R + , a strong comparison function, such that
δ ( T ( x ) , T ( y ) ) φ ( d ( x , y ) ) for all  ( x , y ) E ( G ) .

In this paper, we present some fixed point and strict fixed point theorems for multivalued operators satisfying a contractive condition of Reich type involving the functional δ (see [5, 6]). The equality between Fix ( T ) and S Fix ( 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 [7], Nicolae et al. [3], Nieto and Rodríguez-López [8] and [9], Nieto et al. [10], O’Regan and Petruşel [11], Petruşel and Rus [12], and Ran and Reurings [13].

2 Fixed point and strict fixed point theorems

We begin this section by presenting a strict fixed point theorem for a Reich type contraction with respect to the functional δ.

Theorem 2.1 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 ( x n ) n N X with x n x as n , there exists a subsequence ( x k n ) n N of ( x n ) n N such that ( x k n , x ) E ( G ) .
Let T : X P b ( X ) be a multivalued operator. Suppose that the following assertions hold:
  1. (i)
    There exists a , b , c R + with b 0 and a + b + c < 1 such that
    δ ( T ( x ) , T ( y ) ) a d ( x , y ) + b δ ( x , T ( x ) ) + c δ ( y , T ( y ) )

    for all ( x , y ) E ( G ) .

     
  2. (ii)
    For each x X , the set
    X ˜ T ( x ) : = { y T ( x ) : ( x , y ) E ( G ) and δ ( x , T ( x ) ) q d ( x , y ) for some q ] 1 , 1 a c b [ }

    is nonempty.

     
Then we have:
  1. (a)

    Fix ( T ) = S Fix ( T ) ;

     
  2. (b)
    If we additionally suppose that
    x , y Fix ( T ) ( x , y ) E ( G ) ,

    then Fix ( T ) = S Fix ( T ) = { x } .

     
Proof (a) Let x 0 X . Since X ˜ T ( x 0 ) , there exists x 1 T ( x 0 ) and 1 < q < 1 a c b such that ( x 0 , x 1 ) E ( G ) and
δ ( x 0 , T ( x 0 ) ) q d ( x 0 , x 1 ) .
By (i) we have that
δ ( x 1 , T ( x 1 ) ) δ ( T ( x 0 ) , T ( x 1 ) ) a d ( x 0 , x 1 ) + b δ ( x 0 , T ( x 0 ) ) + c δ ( x 1 , T ( x 1 ) ) a d ( x 0 , x 1 ) + b q d ( x 0 , x 1 ) + c δ ( x 1 , T ( x 1 ) ) .
Hence,
δ ( x 1 , T ( x 1 ) ) a + b q 1 c d ( x 0 , x 1 ) .
(2.1)
For x 1 X , since X ˜ T ( x 1 ) , we get again that there exists x 2 T ( x 1 ) such that δ ( x 1 , T ( x 1 ) ) q d ( x 1 , x 2 ) and ( x 1 , x 2 ) E ( G ) . Then
d ( x 1 , x 2 ) δ ( x 1 , T ( x 1 ) ) a + b q 1 c d ( x 0 , x 1 ) .
(2.2)
On the other hand, by (i), we have that
δ ( x 2 , T ( x 2 ) ) δ ( T ( x 1 ) , T ( x 2 ) ) a d ( x 1 , x 2 ) + b δ ( x 1 , T ( x 1 ) ) + c δ ( x 2 , T ( x 2 ) ) a d ( x 1 , x 2 ) + b q d ( x 1 , x 2 ) + c δ ( x 2 , T ( x 2 ) ) .
Using (2.2) we obtain
δ ( x 2 , T ( x 2 ) ) a + b q 1 c d ( x 1 , x 2 ) ( a + b q 1 c ) 2 d ( x 0 , x 1 ) .
(2.3)

For x 2 X , we have X ˜ T ( x 2 ) , and so there exists x 3 T ( x 2 ) such that δ ( x 2 , T ( x 2 ) ) q d ( x 2 , x 3 ) and ( x 2 , x 3 ) E ( G ) .

Then
d ( x 2 , x 3 ) δ ( x 2 , T ( x 2 ) ) ( a + b q 1 c ) 2 d ( x 0 , x 1 ) .
(2.4)
By these procedures, we obtain a sequence ( x n ) n N with the following properties:
  1. (1)

    ( x n , x n + 1 ) E ( G ) for each n N ;

     
  2. (2)

    d ( x n , x n + 1 ) ( a + b q 1 c ) n d ( x 0 , x 1 ) for each n N ;

     
  3. (3)

    δ ( x n , T ( x n ) ) ( a + b q 1 c ) n d ( x 0 , x 1 ) for each n N .

     

From (2) we obtain that the sequence ( x n ) n N is Cauchy. Since the metric space X is complete, we get that the sequence is convergent, i.e., x n x as n . By the property (P), there exists a subsequence ( x k n ) n N of ( x n ) n N such that ( x k n , x ) E ( G ) for each n N .

We have
δ ( x , T ( x ) ) d ( x , x k n + 1 ) + δ ( x k n + 1 , T ( x ) ) d ( x , x k n + 1 ) + δ ( T ( x k n ) , T ( x ) ) δ ( x , T ( x ) ) d ( x , x k n + 1 ) + a d ( x k n , x ) + b δ ( x k n , T ( x k n ) ) + c δ ( x , T ( x ) ) δ ( x , T ( x ) ) d ( x , x k n + 1 ) + a d ( x k n , x ) + b ( a + b q 1 c ) k n d ( x 0 , x 1 ) + c δ ( x , T ( x ) ) , δ ( x , T ( x ) ) 1 1 c d ( x , x k n + 1 ) + a 1 c d ( x k n , x ) + b 1 c ( a + b q 1 c ) k n d ( x 0 , x 1 ) .
(2.5)

But d ( x , x k n + 1 ) 0 as n and d ( x k n , x ) 0 as n . Hence, δ ( x , T ( x ) ) = 0 , which implies that x S Fix ( T ) . Thus S Fix ( T ) .

We shall prove now that Fix ( T ) = S Fix ( T ) .

Because S Fix ( T ) Fix ( T ) , we need to show that Fix ( T ) S Fix ( T ) .

Let x Fix ( T ) x T ( x ) . Because Δ E ( G ) , we have that ( x , x ) E ( G ) . Using (ii) with x = y = x , we obtain
δ ( T ( x ) ) a d ( x , x ) + b δ ( x , T ( x ) ) + c δ ( x , T ( x ) ) .
So, δ ( T ( x ) ) ( b + c ) δ ( x , T ( x ) ) . Because x T ( x ) , we get that δ ( x , T ( x ) ) δ ( T ( x ) ) . Hence, we have
δ ( T ( x ) ) ( b + c ) δ ( T ( x ) ) .
(2.6)

Suppose that card ( T ( x ) ) > 1 . This implies that δ ( T ( x ) ) > 0 . Thus from (2.6) we obtain that b + c > 1 , which contradicts the hypothesis a + b + c < 1 .

Thus δ ( T ( x ) ) = 0 T ( x ) = { x } , i.e., x S Fix ( T ) and Fix ( T ) S Fix ( T ) .

Hence, Fix ( T ) = S Fix ( T ) .

(b) Suppose that there exist x , y Fix ( T ) = S Fix ( T ) with x y . We have that

  • x S Fix ( T ) δ ( x , T ( x ) ) = 0 ;

  • y S Fix ( T ) δ ( y , T ( y ) ) = 0 ;

  • ( x , y ) E ( G ) .

Using (i) we obtain
d ( x , y ) = δ ( T ( x ) , T ( y ) ) a d ( x , y ) + b δ ( x , T ( x ) ) + c δ ( y , T ( y ) ) .

Thus, d ( x , y ) a d ( x , y ) , which implies that a 1 , which is a contradiction.

Hence, Fix ( T ) = S Fix ( T ) = { x } . □

Next we present some examples and counterexamples of multivalued operators which satisfy the hypothesis in Theorem 2.1.

Example 2.1 Let X : = { ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) } and T : X P cl ( X ) be given by
T ( x ) = { { ( 0 , 0 ) } , x = ( 0 , 0 ) , { ( 0 , 0 ) } , x = ( 0 , 1 ) , { ( 0 , 0 ) , ( 0 , 1 ) } , x = ( 1 , 0 ) , { ( 0 , 0 ) , ( 0 , 1 ) } , x = ( 1 , 1 ) .
(2.7)

Let E ( G ) : = { ( ( 0 , 1 ) , ( 0 , 0 ) ) , ( ( 1 , 0 ) , ( 0 , 1 ) ) , ( ( 1 , 1 ) , ( 0 , 0 ) ) } Δ .

Notice that all the hypotheses in Theorem 2.1 are satisfied (the condition (i) is verified for a = c = 0.01 , b = 0.97 and so Fix ( T ) = S Fix ( T ) = { ( 0 , 0 ) } .

The following remarks show that it is not possible to have elements in F T S F T .

Remark 2.1 If we suppose that there exists x F T S F T , then, since ( x , x ) Δ , we get (using the condition (i) in the above theorem with y = x ) that δ ( T ( x ) ) ( b + c ) δ ( T ( x ) ) , which is a contradiction with a + b + c < 1 .

Remark 2.2 If, in the previous theorem, instead of the property (P), we suppose that T has a closed graph, then we obtain again the conclusion Fix ( T ) = S Fix ( T ) .

Remark 2.3 If, in the above remark, we additionally suppose that
x , y Fix ( T ) ( x , y ) E ( G ) ,

then Fix ( T ) = S Fix ( T ) = { x } .

The next result presents a strict fixed point theorem where the operator T satisfies a ( δ , φ ) -G-contractive condition on E ( G ) .

Theorem 2.2 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 ( x n ) n N X with x n x as n , there exists a subsequence ( x k n ) n N of ( x n ) n N such that ( x k n , x ) E ( G ) .
Let T : X P b ( X ) be a multivalued operator. Suppose that the following assertions hold:
  1. (i)

    T is a ( δ , φ ) -G-contraction.

     
  2. (ii)
    For each x X , the set
    X ˜ T : = { y T ( x ) : ( x , y ) E ( G ) and δ ( x , T ( x ) ) q d ( x , y ) for some q ] 1 , 1 a c b [ }
     

is nonempty.

Then we have:
  1. (a)

    Fix ( T ) = S Fix ( T ) ;

     
  2. (b)
    If, in addition, the following implication holds:
    x , y Fix ( T ) ( x , y ) E ( G ) ,
     

then Fix ( T ) = S Fix ( T ) = { x } .

Proof (a) Let x 0 X . Then, since X ˜ T ( x 0 ) is nonempty, there exist x 1 T ( x 0 ) and q ] 1 , 1 a c b [ such that ( x 0 , x 1 ) E ( G ) and
δ ( x 0 , T ( x 0 ) ) q d ( x 0 , x 1 ) .
By (i) we have that
δ ( x 1 , T ( x 1 ) ) δ ( T ( x 0 ) , T ( x 1 ) ) φ ( d ( x 0 , x 1 ) ) .

For x 1 X , by the same approach as before, there exists x 2 T ( x 1 ) such that δ ( x 1 , T ( x 1 ) ) q d ( x 1 , x 2 ) and ( x 1 , x 2 ) E ( G ) .

We have
d ( x 1 , x 2 ) δ ( x 1 , T ( x 1 ) ) δ ( T ( x 0 ) , T ( x 1 ) ) φ ( d ( x 0 , x 1 ) ) .
(2.8)
On the other hand, by (i) we have that
δ ( x 2 , T ( x 2 ) ) δ ( T ( x 1 ) , T ( x 2 ) ) φ ( d ( x 1 , x 2 ) ) φ 2 ( d ( x 0 , x 1 ) ) .
By the same procedure, for x 2 X there exists x 3 T ( x 2 ) such that δ ( x 2 , T ( x 2 ) ) q d ( x 2 , x 3 ) and ( x 2 , x 3 ) E ( G ) . Thus
d ( x 2 , x 3 ) δ ( x 2 , T ( x 2 ) ) φ 2 ( d ( x 0 , x 1 ) ) .
(2.9)
We have
δ ( x 3 , T ( x 3 ) ) δ ( T ( x 2 ) , T ( x 3 ) ) φ ( d ( x 2 , x 3 ) ) φ 3 ( d ( x 0 , x 1 ) ) .
By these procedures, we obtain a sequence ( x n ) n N with the following properties:
  1. (1)

    ( x n , x n + 1 ) E ( G ) for each n N ;

     
  2. (2)

    d ( x n , x n + 1 ) φ n ( d ( x 0 , x 1 ) ) for each n N ;

     
  3. (3)

    δ ( x n , T ( x n ) ) φ n ( d ( x 0 , x 1 ) ) for each n N .

     

By (2), using the properties of φ, we obtain that the sequence ( x n ) n N is Cauchy. Since the metric space is complete, we have that the sequence is convergent, i.e., x n x as n . By the property (P), we get that there exists a subsequence ( x k n ) n N of ( x n ) n N such that ( x k n , x ) E ( G ) for each n N .

We shall prove now that x S Fix ( T ) . We have
δ ( x , T ( x ) ) d ( x , x k n + 1 ) + δ ( x k n + 1 , T ( x ) ) d ( x , x k n + 1 ) + δ ( T ( x k n ) , T ( x ) ) d ( x , x k n + 1 ) + φ ( d ( x k n , x ) ) .

Since d ( x , x k n + 1 ) 0 as n and φ is continuous in 0 with φ ( 0 ) = 0 , we get that δ ( x , T ( x ) ) = 0 .

Hence, x S Fix ( T ) S Fix ( T ) .

We shall prove now that Fix ( T ) = S Fix ( T ) .

Because S Fix ( T ) Fix ( T ) , we need to show that Fix ( T ) S Fix ( T ) .

Let x Fix ( T ) . Because Δ E ( G ) , we have that ( x , x ) E ( G ) . Using (i) with x = y = x , we obtain
δ ( T ( x ) ) φ ( d ( x , x ) ) = 0 .

Hence, x S Fix ( T ) and the proof of this conclusion is complete.

(b) Suppose that there exist x , y Fix ( T ) = S Fix ( T ) with x y . We have that

  • x S Fix ( T ) δ ( x , T ( x ) ) = 0 ;

  • y S Fix ( T ) δ ( y , T ( y ) ) = 0 ;

  • ( x , y ) E ( G ) .

Using (i) we obtain
d ( x , y ) = δ ( T ( x ) , T ( y ) ) φ ( d ( x , y ) ) < d ( x , y ) .

This is a contradiction. Hence, Fix ( T ) = S Fix ( T ) = { x } . □

In the next result, the operator T satisfies another contractive condition with respect to δ on E ( G ) Graph ( T ) .

Theorem 2.3 Let ( X , d ) be a complete metric space, let G be a directed graph, and let T : X P b ( 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:
  1. (i)
    There exist a , b R + , with b 0 and a + b < 1 , such that
    δ ( y , T ( y ) ) a d ( x , y ) + b δ ( x , T ( x ) ) for all ( x , y ) E ( G ) Graph ( T ) .
     
  2. (ii)
    For each x X , the set
    X ˜ T ( x ) : = { y T ( x ) : ( x , y ) E ( G ) and δ ( x , T ( x ) ) q d ( x , y ) for some q ] 1 , 1 a b [ }

    is nonempty.

     

Then Fix ( T ) = S Fix ( T ) .

Proof Let x 0 X . Then, since X ˜ T ( x 0 ) is nonempty, there exist x 1 T ( x 0 ) and 1 < q < 1 a b such that
δ ( x 0 , T ( x 0 ) ) q d ( x 0 , x 1 )

and ( x 0 , x 1 ) E ( G ) . Since x 1 T ( x 0 ) , we get that ( x 0 , x 1 ) E ( G ) Graph ( T ) .

By (i), taking y = x 1 and x = x 0 , we have that
δ ( x 1 , T ( x 1 ) ) a d ( x 0 , x 1 ) + b δ ( x 0 , T ( x 0 ) ) a d ( x 0 , x 1 ) + b q d ( x 0 , x 1 ) .
Hence,
δ ( x 1 , T ( x 1 ) ) ( a + b q ) d ( x 0 , x 1 ) .
(2.10)

For x 1 X (since X ˜ T ( x 1 ) ), there exists x 2 T ( x 1 ) such that δ ( x 1 , T ( x 1 ) ) q d ( x 1 , x 2 ) and ( x 1 , x 2 ) E ( G ) . But x 2 T ( x 1 ) and so ( x 1 , x 2 ) E ( G ) Graph ( T ) .

Then
d ( x 1 , x 2 ) δ ( x 1 , T ( x 1 ) ) ( a + b q ) d ( x 0 , x 1 ) .
(2.11)
By (i), taking y = x 2 and x = x 1 , we have that
δ ( x 2 , T ( x 2 ) ) a d ( x 1 , x 2 ) + b δ ( x 1 , T ( x 1 ) ) a d ( x 1 , x 2 ) + b q d ( x 1 , x 2 ) = ( a + b q ) d ( x 1 , x 2 ) ( a + b q ) 2 d ( x 0 , x 1 ) .
By these procedures, we obtain a sequence ( x n ) n N with the following properties:
  1. (1)

    ( x n , x n + 1 ) E ( G ) Graph ( T ) for each n N ;

     
  2. (2)

    d ( x n , x n + 1 ) ( a + b q ) n d ( x 0 , x 1 ) for each n N ;

     
  3. (3)

    δ ( x n , T ( x n ) ) ( a + b q ) n d ( x 0 , x 1 ) for each n N .

     
From (2) we obtain that the sequence ( x n ) n N is Cauchy. Since the metric space X is complete, we have that the sequence is convergent, i.e., x n x as n . Now, by the lower semicontinuity of the function f, we have
0 f ( x ) lim inf n f ( x n ) = 0 .

Thus f ( x ) = 0 , which means that δ ( x , T ( x ) ) = 0 . Thus x S Fix ( T ) .

Let x Fix ( T ) . Then ( x , x ) Graph ( T ) and hence ( x , x ) E ( G ) Graph ( T ) .

Using (i) with x = y = x , we obtain
δ ( T ( x ) ) = δ ( x , T ( x ) ) a d ( x , x ) + b δ ( x , T ( x ) ) .

So, δ ( T ( x ) ) b δ ( T ( x ) ) . If we suppose that card T ( x ) > 1 , then δ ( T ( x ) ) > 0 . Thus, b 1 , which contradicts the hypothesis.

Thus δ ( T ( x ) ) = 0 and so T ( x ) = { x } . The proof is now complete. □

Remark 2.4 Example 2.1 satisfies the conditions from Theorem 2.3 for a = 0.01 and b = 0.97 .

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 operator T.

We begin by recalling the definition of these notions from [14] and [15].

Definition 3.1 Let ( X , d ) be a metric space and let T : X P ( X ) be a multivalued operator. By definition, the fixed point problem is well posed for T with respect to H if:
  1. (i)

    S Fix ( T ) = { x } ;

     
  2. (ii)

    If ( x n ) n N is a sequence in X such that H ( x n , T ( x n ) ) 0 as n , then x n d x as n .

     
Definition 3.2 Let ( X , d ) be a metric space and let T : X P ( X ) be a multivalued operator. By definition, the fixed point problem is well posed in the generalized sense for T with respect to H if:
  1. (i)

    S Fix T ;

     
  2. (ii)

    If ( x n ) n N is a sequence in X such that H ( x n , T ( x n ) ) 0 as n , then there exists a subsequence ( x k n ) n N of ( x n ) n N such that x k n d x as n .

     

In our first result we will establish the well-posedness of the fixed point problem for the operator T, where T is a Reich-type δ-contraction.

Theorem 3.1 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 P b ( X ) be a multivalued operator. Suppose that
  1. (i)

    conditions (i) and (ii) in Theorem  2.1 hold;

     
  2. (ii)

    if x , y Fix ( T ) , then ( x , y ) E ( G ) ;

     
  3. (iii)

    for any sequence ( x n ) n N , x n X with H ( x n , T ( x n ) ) 0 as n , we have ( x n , 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 that S Fix ( T ) = { x } . Let ( x n ) n N X be a sequence which satisfies (iii). It is obvious that H ( x n , T ( x n ) ) = δ ( x n , T ( x n ) ) ,
d ( x n , x ) δ ( x n , T ( x ) ) δ ( x n , T ( x n ) ) + δ ( T ( x n ) , T ( x ) ) δ ( x n , T ( x n ) ) + a d ( x n , x ) + b δ ( x n , T ( x n ) ) + c δ ( x , T ( x ) ) .
Thus
d ( x n , x ) 1 + b 1 a δ ( x n , T ( x n ) ) 0 as  n .

Hence, x n x as n . □

Remark 3.1 If we replace the property (P) with the condition that T has a closed graph, we reach the same conclusion.

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

Theorem 3.2 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 P b ( X ) be a multivalued operator. Suppose that
  1. (i)

    conditions (i) and (ii) in Theorem  2.1 hold;

     
  2. (ii)

    for any sequence ( x n ) n N , x n X with H ( x n , T ( x n ) ) 0 as n , there exists a subsequence ( x k n ) n N such that ( x k n , x ) E ( G ) and H ( x k n , T ( x k n ) ) 0 .

     

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 that S Fix ( T ) . Let ( x n ) n N X be a sequence which satisfies (ii). Then there exists a subsequence ( x k n ) n N such that ( x k n , x ) E ( G ) .

We have H ( x k n , T ( x k n ) ) = δ ( x k n , T ( x k n ) ) ,
d ( x k n , x ) δ ( x k n , T ( x ) ) δ ( x k n , T ( x k n ) ) + δ ( T ( x k n ) , T ( x ) ) δ ( x k n , T ( x k n ) ) + a d ( x k n , x ) + b δ ( x k n , T ( x k n ) ) + c δ ( x , T ( x ) ) .
Thus
d ( x k n , x ) 1 + b 1 a δ ( x k n , T ( x k n ) ) 0 as  n .

Hence, x k n x . □

Remark 3.2 If we replace the property (P) with the condition that T has a closed graph, we reach the same conclusion.

Next we consider the case where the operator T satisfies a φ-contraction condition.

Theorem 3.3 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 P b ( X ) be a multivalued operator. Suppose that
  1. (i)

    conditions (i) and (ii) in Theorem  2.2 hold;

     
  2. (ii)

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

     
  3. (iii)

    the function ψ : R + R + , given by ψ ( t ) = t φ ( t ) , has the following property: if ψ ( t n ) 0 as n , then t n 0 as n ;

     
  4. (iv)

    for any sequence ( x n ) n N X with H ( x n , T ( x n ) ) 0 as n , we have ( x n , x ) E ( G ) for all n N .

     

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

Proof From (i) and (ii) we obtain that S Fix ( T ) = { x } . Let ( x n ) n N , x n X be a sequence which satisfies (iv). It is obvious that H ( x n , T ( x n ) ) = δ ( x n , T ( x n ) ) ,
d ( x n , x ) δ ( x n , T ( x ) ) δ ( x n , T ( x n ) ) + δ ( T ( x n ) , T ( x ) ) δ ( x n , T ( x n ) ) + φ ( d ( x n , x ) ) .
Thus
d ( x n , x ) φ ( d ( x n , x ) ) δ ( x n , T ( x n ) ) 0 as  n .

Using condition (iii), we get that d ( x n , x ) 0 as n . Hence, x n x . □

Remark 3.3 If we replace the property (P) with the condition that T has 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 3.4 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 P b ( X ) be a multivalued operator. Suppose that
  1. (i)

    the conditions (i) and (ii) in Theorem  2.2 hold;

     
  2. (ii)

    the function ψ : R + R + , given ψ ( t ) = t φ ( t ) , has the following property: for any sequence ( t n ) n N , there exists a subsequence ( x k n ) n N such that if ψ ( t k n ) 0 as n , then t k n 0 as n ;

     
  3. (iii)

    for any sequence ( x n ) n N , x n X with H ( x n , T ( x n ) ) 0 as n , there exists a subsequence ( x k n ) n N such that ( x k n , x ) E ( G ) and H ( x k n , T ( x k n ) ) 0 .

     

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 that S Fix ( T ) . Let ( x n ) n N , x n X be a sequence which satisfies (iii). Then there exists a subsequence ( x k n ) n N such that ( x k n , x ) E ( G ) .

We have H ( x k n , T ( x k n ) ) = δ ( x k n , T ( x k n ) ) ,
d ( x k n , x ) δ ( x k n , T ( x ) ) δ ( x k n , T ( x k n ) ) + δ ( T ( x k n ) , T ( x ) ) δ ( x k n , T ( x k n ) ) + φ ( d ( x k n , x ) ) .
Thus
d ( x k n , x ) φ ( d ( x k n , x ) ) δ ( x k n , T ( x k n ) ) 0 as  n .

Using condition (ii), we get that d ( x k n , x ) 0 as n . Hence, x n x . □

Remark 3.4 If we replace the property (P) with the condition that T has a closed graph, we reach the same conclusion.

Declarations

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.

Authors’ Affiliations

(1)
Department of Business, Babeş-Bolyai University Cluj-Napoca, Cluj-Napoca, Romania
(2)
Department of Mathematics, Babeş-Bolyai University Cluj-Napoca, Cluj-Napoca, Romania

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