1. Introduction and Preliminary Notions and Results
Let
be a metric space and
be the set of all nonempty subsets of
X. We denote
We recall first the following notions:
- (1)
The distance from a point
to a set
:
- (2)
The excess of
Y over
Z (where
):
- (3)
The Hausdorff–Pompeiu distance between two sets
:
Notice that H is a generalized metric (in the sense that ) on , and it is a classical metric on .
For
and
, we denote by
the open ball (respectively, the closed ball) centered in
x with radius
r.
If
X is a nonempty set and
is a multi-valued operator, then
is called a fixed point for
T if
. The set
is the fixed point set of
T, while
is the strict fixed point set of
T. We also denote by
the graph of the multi-valued operator
T.
Remark 1. If X is a nonempty set and , then the sequence satisfyingis called an iterative sequence of the Picard type for T starting from . The multi-valued contraction principle was proved in 1969 by Nadler (see [
1]), while a slight extension of it was presented by Covitz and Nadler in 1970 (see [
2]).
There are several generalizations of the above multi-valued contraction principle of Nadler/Covitz–Nadler. A consistent extension of it appeared in a paper of Feng and Liu (see [
3]), as follows.
Definition 1. Let be a metric space, be a multi-valued operator, and . Consider the setThen, T is called a multi-valued α-contraction of the Feng–Liu type if there exists , such that for each there is , for which the following assumption holds Theorem 1. ([3]) Let be a complete metric space and be a multi-valued α-contraction of the Feng–Liu type. Suppose that the mapping defined by is lower and semi-continuous. Then, . Another generalization of the multi-valued contraction principle involves the notion of the multi-valued graph contraction.
Definition 2. Ref. [4] Let be a metric space and be a multi-valued operator. Then, T is called a multi-valued graph α-contraction if and The following theorem, proved in [
4], is the main result for multi-valued graph contractions.
Theorem 2. ([4]) Let be a complete metric space and be a multi-valued graph α-contraction with a closed graph. Then, T is a -multi-valued weak Picard operator; i.e., for each , there exists in X a sequence , such that: - (i)
, ;
- (ii)
, for each ;
- (iii)
is convergent to a fixed point of T;
- (iv)
.
Notice that any multi-valued
-contraction is a multi-valued graph
-contraction and any multi-valued graph
-contraction is a multi-valued
-contraction of the Feng–Liu type. For examples and applications of the fixed point theory for the multi-valued graph contraction, see [
4].
In this paper, we will prove several new results related to the concept of the multi-valued Feng–Liu contraction. An existence, approximation and localization fixed point theorem for a generalized multi-valued nonself Feng–Liu contraction and a new fixed point theorem for multi-valued Feng–Liu contractions in vector-valued metric spaces are proved. Stability results and an application to a system of operatorial inclusions are also given. For the role and the importance of vector-valued metrics in a nonlinear analysis, see [
5,
6,
7,
8].
2. A Local Fixed Point Theorem for a Generalized Multi-Valued Feng–Liu Operator
We start this section by proving a local fixed point theorem for a generalized multi-valued -contraction of the Feng–Liu type. For this purpose, we adapt the notion of the multi-valued -contraction of the Feng–Liu type to a nonself setting.
Definition 3. Let be a metric space, and and be a multi-valued operator. Consider and . DefineThen, T is called a generalized multi-valued nonself -contraction of the Feng–Liu type if there exist and , such that, for each there is , for which the following implication holds Notice that for and , we obtain the classical concept of the multi-valued -contraction of the Feng–Liu type.
We present now an existence, approximation and localization fixed point theorem for a generalized multi-valued nonself contraction of the Feng–Liu type on a closed ball.
Theorem 3. Let be a complete metric space, let and . We consider a multi-valued nonself -contraction of the Feng–Liu type, such that . Suppose that the mapping of defined by is lower and semi-continuous on , or that T has a closed graph. Then, there exists a sequence of Picard iterates for T, starting from , which converges to a fixed point of T. Moreover, if , then the following relations hold:
- (a)
, for each ;
- (b)
.
Proof. We will show that there exists in
a sequence
of Picard iterates for
T, starting from
, which converges to a fixed point of
T. For
there exists
, having the properties that
and
. Then, we observe that
which shows that
Hence, since
, by the definition of the multi-valued nonself
-contraction of the Feng–Liu type, we obtain that
Thus, we have
For
, there exists
such that
and
. Additionally,
. As a consequence, we obtain
By our assumptions, we have that
.
Now, we observe that
As a consequence,
.
By this procedure, we obtain a sequence of Picard iterates for T, starting from with the following properties:
- (1)
;
- (2)
;
- (3)
.
By (2), we find that the sequence is Cauchy. Thus, the sequence converges to an element . We only need to demonstrate that is a fixed point of T. If T has a closed graph, the conclusion follows immediately by the fact that the sequence is of Picard iterates for T. If we suppose that the mapping is a lower semi-continuous on , then the conclusion follows by (3), observing that the sequence is convergent to 0.
The conclusions (a) and (b) follow (2), taking into account that, for
with
, we have
Letting
, we obtain (a). Then, taking
, we obtain
which immediately gives the conclusion (b). □
Remark 2. For related results, see [9,10,11,12,13]. For complementary results, see also [14,15,16,17,18]. 3. A Fixed Point Theorem for Multi-Valued Feng–Liu Contractions in Vector-Valued Metric Spaces
In this section, we will prove a fixed point result for multi-valued Feng–Liu contractions in complete vector-valued metric spaces. For this purpose, we recall some notions and results.
If
,
and
, then, by definition
We will make an identification between row and column vectors in
.
We now recall the concept of vector-valued metric space in the sense of Perov, see, e.g., [
19]. If
X is a nonempty set, then a functional
satisfying the usual axioms of a metric with respect to the above mentioned relation ⪯ is called a vector-valued metric in the sense of Perov. In this case, the pair
is a vector-valued metric space.
In this section, denotes the set of all matrices with positive elements, is the identity matrix and denotes the null matrix.
By definition, a matrix
is said to be convergent to zero if
as
. The following characterization theorem is useful for the proof of our main results, see, e.g., [
20].
Theorem 4. Let . The following assertions are equivalent:
- (i)
as ;
- (ii)
The spectral radius of K is strictly less than 1, i.e., the eigenvalues of K are in the open unit disc;
- (iii)
The matrix is nonsingular and - (iv)
The matrix is nonsingular and has nonnegative elements.
The following theorem is the main fixed point result for
K-contractions in complete vector-valued metric spaces, see [
19].
Theorem 5 (Perov)
. Let be a complete vector-valued metric space and let be an K-contraction; i.e., converges to zero andThen: Fix, i.e., there exists a unique solution of the fixed point equation ;
the sequence of successive approximations for f starting from any is convergent to ;
the following estimation holds A multi-valued variant of Perov’s theorem was given in [
21].
Theorem 6. Let be a complete vector-valued metric space and let be a multi-valued K-contraction, i.e., is convergent to zero andThen: - (i)
;
- (ii)
For each , there exists a sequence (with , and , for each ), such that is convergent to a fixed point of F, and the following relations hold:
Our next result is a generalization of the previous theorem in terms of a multi-valued Feng–Liu contraction.
We introduce first the following notion.
Definition 4. Let be a vector-valued metric space, be a multi-valued operator, be a diagonal matrix with elements and . Consider the setThen, F is called a multi-valued vectorial contraction of the Feng–Liu type if there exists a matrix , such that the matrix is convergent to zero, and for each , there is , for which the following relation holds Theorem 7. Let be a complete vector-valued metric space and be a multi-valued vectorial contraction of the Feng–Liu type. Suppose that F has closed graph. Then, for each , there exists an iterative sequence of the Picard type for F starting from with the following properties:
- (1)
converges to ;
- (2)
;
- (3)
.
Proof. Let
be arbitrarily chosen. Then, there exists
such that
For
, by the multi-valued vectorial contraction condition of the Feng–Liu type, there exists
, such that
Since
, we obtain that
Let us denote
. Notice that
. By the above procedure, there exists a sequence
in
X with the following properties:
- (a)
, for each ;
- (b)
, for each ;
- (c)
, for each .
Then, by (b), the sequence
is Cauchy in
. Hence, it is convergent to an element
. Since
is an iterative sequence of the Picard type for
F starting from
, and
F has a closed graph, we obtain that
. Moreover, by the relation
letting
, we obtain the following a priori approximation for the fixed point:
Taking
in the above relation, we obtain the following retraction–displacement condition:
The proof is complete. □
We now present some stability concepts for the fixed point inclusion in the setting of a vector-valued metric space.
The concept of the Ulam–Hyers stability is now introduced; see also [
22].
Definition 5. Let be a vector-valued metric space and be a multi-valued operator. The fixed point inclusion is called Ulam–Hyers stable if there exists a matrix , such that for every (with for each ) and for each ε-fixed point of F (i.e., , there exists , such that The well-posedness of the fixed point inclusion
in a vector-valued metric space is defined, as follows. The concept is inspired by the single-valued case; see the papers of Reich and Zaslavski [
23,
24].
Definition 6. Let be a vector-valued metric space. Let be a multi-valued operator such that and let be a set retraction. Then, the fixed point inclusion is called well-posed in the sense of Reich and Zaslavski if for each and for any sequence , such that converges to zero as ; we have that The data dependence property is given in our next definition.
Definition 7. Let be a vector-valued metric space and be a multi-valued operator. Let be a multi-valued operator satisfying the following conditions:
- (i)
;
- (ii)
There exists (with for each ), such that , for all .
Then, the fixed point inclusion has the data dependence property if for each there exists , such that The notion of the Ostrowski stability property for a fixed point inclusion in the vector-valued metric space is now presented; see also [
12].
Definition 8. Let be a vector-valued metric space. Let be a multi-valued operator such that and let be a set retraction. Then, the fixed point inclusion is said to have the Ostrowski stability property if for each and for any sequence , such that converges to zero as ; we have that The following retraction–displacement condition will be important for our main results.
Definition 9. Let be a vector-valued metric space and let be a multi-valued operator such that . Then, we say that F satisfies the strong vectorial retraction–displacement condition if there exist a matrix and a set retraction , such that An abstract result concerning some stability properties of a multi-valued operator is given in our next result.
Theorem 8. Let be a vector-valued metric space and let be a multi-valued operator satisfying the strong vectorial retraction–displacement condition, such that . Then, the fixed point inclusion has the Ulam–Hyers stability property; it is well-posed and satisfies the data dependence property.
Proof. Suppose that there exists a matrix
and a set retraction
, such that
In order to prove the Ulam–Hyers stability property, let us consider
(with
for each
) and
such that
. Then, by the strong vectorial retraction–displacement condition, we have
Thus, the fixed point inclusion
is Ulam–Hyers stable.
For the well-posedness property of the fixed point inclusion, let us consider the sequence
, such that the sequence
converges to zero as
. Then, for each
, we have
and, again by the strong vectorial retraction–displacement condition, we conclude that
Let us now prove the data dependence of the fixed point set. Let us consider a multi-valued operator to have the properties:
(i) ;
(ii) There exists
(with
for each
), such that
Take any
and denote
. Then, by the strong vectorial retraction–displacement condition, we have that
The proof is now complete.
□
The following result shows that any multi-valued vectorial contraction of the Feng–Liu type has a strong vectorial retraction–displacement condition.
Theorem 9. Let be a complete vector-valued metric space and be a multi-valued vectorial contraction of the Feng–Liu type. Suppose that F has a closed graph. Then, F satisfies the strong vectorial retraction–displacement condition.
Proof. By Theorem 7, we know that
and, for every
, there exists an iterative sequence
of the Picard type for
F starting from the arbitrary
, which converges to a fixed point
of
F. Moreover, the following relation holds
Thus, we can define the set-retraction
,
with the property
Hence, the strong vectorial retraction–displacement condition from Definition 9 is satisfied. □
By combining the above two theorems, we obtain the following stability properties for the multi-valued vectorial contraction of the Feng–Liu type.
Theorem 10. Let be a complete vector-valued metric space and be a multi-valued vectorial contraction of the Feng–Liu type. Suppose that F has a closed graph. Then, the fixed point inclusion is well-posed in the sense of Reich and Zaslavsi, has the Ulam–Hyers stability property and satisfies the data dependence property.
Proof. By Theorem 7, we have that , while Theorem 9 implies that F has the strong vectorial retraction–displacement property. The conclusions follow by Theorem 8.
□
Remark 3. It is an open question to prove the Ostrowski stability property for a multi-valued vectorial contraction of the Feng–Liu type.
4. An Application to a System of Operatorial Inclusions
In this section, we will present an existence result for a system of operatorial inclusion in complete metric spaces. The approach is based on the vectorial technique for multi-valued Feng–Liu operators.
Let
and
be two complete metric spaces and let
and
be two multi-valued operators with a closed graph. We consider the following system of operatorial inclusions
Denote by
and define on
Z the vectorial metric
given by
Let
and define the following nonempty sets:
and
where
and
are the distances from a point to a set with respect to
and
, respectively.
Denote also
We suppose that for every
there exist
and
, such that
and
where
is a matrix with nonnegative elements. We also suppose that the matrix
is convergent to zero.
Under the above assumptions, we have the following existence and approximation result.
Theorem 11. Let us consider the system of operatorial inclusions (6). Under the above assumptions, the system (6) has at least one solution . Moreover, for each , there exist two sequences and with the following properties: (A) and , for each ;
(B) converges to and converges to as ;
(C)
(D) .
Proof. We denote
and, for
, consider the multi-valued operator
given by
. Notice that the fixed points
of
G are solutions for the operatorial inclusion (
6).
Let
and
, such that
and
We consider the vectorial gap function
We denote
. Then, by our assumptions, the set
is nonempty for each
. Moreover, by (
8) and (
9), we obtain that for each
, there exists
, such that
where
Thus,
G satisfies all the assumptions of the Theorem 7. As a consequence, for each
there exists an iterative sequence of the Picard type
, which converges to a fixed point
of
G, and the following relations hold:
- (I)
;
- (II)
.
Thus, the proof is complete. □
Remark 4. In the above mentioned conditions, some stability results (well-posedness, Ulam–Hyers stability and data dependence property) for the system of operatorial inclusions (6) can be established by applying the abstract results proved in Section 3. In particular, an existence and approximation result for the multi-valued altering points problem can be obtained. We notice that, if
and
are two metric spaces and
and
are two multi-valued operators, then the following system of operatorial inclusions is called an altering points problem for multi-valued operators:
The above problem has important applications in the theory of generalized/multivalued variational inequalities, see, e.g., [
25].
Theorem 12. Let and be two complete metric spaces and let and be two multi-valued operators with a closed graph. Let and the setsandWe suppose that for every , there exists , such thatandwhereis a matrix with nonnegative elements. We also suppose that the matrixis convergent to zero, whereThen, the altering points problem (12) has at least one solution in . Moreover, for each , there exist two sequences and with the following properties: (A) and , for each ;
(B) converges to and converges to as ;
(C)
(D) .
Example. Let
and
, both endowed with the absolute value metric. Let
and
be given by
and
The above operators satisfy the multi-valued vectorial contraction condition of the Feng–Liu type, having as solutions of the altering point problem the pairs
,
and
.
The above results generalize some altering points theorems, as given for the single-valued case in [
26].