Approximate Endpoints for Set-Valued Contractions in Metric Spaces

Abstract

The existence of approximate fixed points and approximate endpoints of the multivalued almost -contractions is established. We also develop quantitative estimates of the sets of approximate fixed points and approximate endpoints for multivalued almost -contractions. The proved results unify and improve recent results of Amini-Harandi (2010), M. Berinde and V. Berinde (2007), Ćirić (2009), M. Păcurar and R. V. Păcurar (2007) and many others.

1. Introduction and Preliminaries

In fixed point theory, one of the main directions of investigation concerns the study of the fixed point property in topological spaces. Recall that a topological space is said to have the fixed point property if every continuous mapping has a fixed point. The major contribution to this subject has been provided by Tychonoff. Every compact convex subset of a locally convex space has the fixed point property. Another important branch of fixed point theory is the study of the approximate fixed point property. Recently, the interest in approximate fixed point results arise in the study of some problems in economics and game theory, including, for example, the Nash equilibrium approximation in games; see [1–3] and references therein.

We establish some existence results concerning approximate fixed points, endpoints, and approximate endpoints of multivalued contractions. We also develop quantitative estimates of the sets of approximate fixed points and approximate endpoints for set-valued almost -contractions. The results presented in this paper extend and improve the recent results of [4–10] and many others.

Now, we give some notions and definitions.

Let be a metric space and let and denote the families of all nonempty subsets and nonempty closed subsets of , respectively. Let and be two Hausdorff topological spaces and a multivalued mapping with nonempty values. Then is said to be

(1)upper semicontinuous (u.s.c.) if, for each closed set , is closed in ;

(2)lower semicontinuous (l.s.c.) if, for each open set , is open in ;

(3)continuous if it is both u.s.c. and l.s.c.;

(4)closed if its graph is closed;

(5)compact if is a compact subset of .

For any subsets , of a metric space , we consider the following notions:

: the distance between the sets and ;

: the diameter of the sets and ;

: the diameter of the set ;

: the Hausdorff metric on induced by the metric .

Let be a multivalued mapping. An element such that is called a fixed point of . We denote by the set of all fixed points of , that is,

A mapping is called

a multivalued contraction (or multivalued-contraction) if there exists a number such that

(1.1)

a multivalued almost contraction [6] or a multivalued-almost contraction if there exist two constants and such that

(1.2)

a generalized multivalued almost contraction [6] if there exists a function satisfying for every such that

(1.3)

It is important to note that any mapping satisfying Banach, Kannan, Chatterjea, Zamfirescu, or Ćirić (with the constant in ) type conditions is a single-valued almost contraction; see [5, 6, 8, 11].

2. Approximate Fixed Points of Multivalued Contractions

Definition 2.1.

A multivalued mapping is said to have the approximate fixed point property [2] provided

(2.1)

or, equivalently, for any , there exists such that

(2.2)

or, equivalently, for any , there exists such that

(2.3)

where denotes a closed ball of radius centered at .

We first prove that every generalized multivalued almost contraction has the approximate fixed point property.

Lemma 2.2.

Every generalized multivalued almost contraction has the approximate fixed point property.

Proof.

Let be an arbitrary metric space and a generalized multivalued almost contraction. Let and be such that

(2.4)

By passing to the subsequences, if necessary, we may assume that the sequence is convergent. Then we have

(2.5)

Since , we get . This completes the proof.

Corollary 2.3 (see [5, Theorem ], [10, Theorem ]).

Let be a metric space and a single-valued almost contraction. Then has the approximate fixed point property.

The authors in [5, 10] obtained the following quantitative estimate of the diameter of the set, , of approximate fixed points of single-valued almost contraction .

Theorem 2.4 (see [5, Theorem ], [10, Theorem ]).

Let be a metric space. If is a single-valued almost contraction with , then

(2.6)

The following simple example shows that the conclusion of Theorem 2.4 is not valid for set-valued almost contractions.

Example 2.5.

Let be defined by . Then and so is multivalued almost contraction with . Further, and so . This shows that conclusion of Theorem 2.4 is not true whenever is multivalued almost contraction.

Theorem 2.6.

Let be a metric space. If is a generalized multivalued almost contraction, then has a fixed point provided either is compact and the function is lower semicontinuous or is closed and compact.

Proof.

By Lemma 2.2, we have . The lower semicontinuity of the function and the compactness of imply that the infimum is attained. Thus there exists an such that and so .

Suppose that is closed and compact. According to Lemma 2.2, has the approximate fixed point property. Therefore, for any , there exist and such that

(2.7)

Now, since is compact, we may assume that converges to a point as . Consequently, also converges to as . Since is closed, then This completes the proof.

Let be a single-valued mapping and a multivalued mapping. Then is called a multivalued almost-contraction [6, 8] if there exist constants and such that

(2.8)

We say that is a generalized multivalued almost-contraction if there exists a function satisfying for every such that

(2.9)

The mappings and are said to have an approximate coincidence point property provided

(2.10)

or, equivalently, for any , there exists such that

(2.11)

A point is called a coincidence (common fixed) point of and if ().

Theorem 2.7.

Every generalized multivalued almost -contraction in a metric space has the approximate coincidence point property provided each is -invariant. Further, if is compact and the function is lower semicontinuous, then and have a coincidence point.

Proof.

Let be a generalized multivalued almost -contraction and let and be such that

(2.12)

By passing to the subsequences, if necessary, we may assume that the sequence is convergent. Then we have

(2.13)

since each is -invariant, that is, for each , we have . Since

(2.14)

we get .

Further, the lower semi-continuity of the function and the compactness of imply that the infimum is attained. Thus there exists such that and so as required. This completes the proof.

Corollary 2.8.

Every multivalued almost -contraction in a metric space has the approximate coincidence point property provided each is -invariant. Further, if is compact and the function is lower semicontinuous, then and have a coincidence point.

Recently, Ćirić [7] has introduced multivalued contractions and obtained some interesting results which are proper generalizations of the recent results of Klim and Wardowski [9], Feng and Liu [12], and many others. In the results to follow, we obtain approximate fixed point property for these multivalued contractions.

Theorem 2.9.

Let be a metric space and a multivalued mapping from into Suppose that there exist a function such that

(2.15)

and and satisfying the following two conditions:

(2.16)

where . Then has the approximate fixed point property. Further, has a fixed point provided either is compact and the function is lower semicontinuous or is closed and compact.

Proof.

Let and be the sequences that satisfy (2.16). By passing to subsequences, if necessary, we may assume that both of the sequences and are convergent (note that is bounded since ). Then we have

(2.17)

Since , we get .

Further, the lower semi-continuity of the function and the compactness of imply that the infimum is attained. Thus there exists such that and so .

The second assertion follows as in the proof of Theorem 2.6. This completes the proof.

Theorem 2.10.

Let be a metric space and a multivalued mapping from into Suppose that there exist a function such that

(2.18)

and and satisfying the following two conditions:

(2.19)

where . Then has the approximate fixed point property. Further, has a fixed point provided either is compact and the function is lower semicontinuous or is closed and compact.

Proof.

Let and satisfy (2.19). By passing to subsequences, if necessary, we may assume that the sequence is convergent. Then we have

(2.20)

Since , we get .

Further, the lower semi-continuity of the function and the compactness of imply that the infimum is attained. Thus there exists such that and so .

The second assertion follows as in the proof of Theorem 2.6. This completes the proof.

3. Endpoints of Multivalued Nonlinear Contractions

Let be a multivalued mapping. An element is said to be a endpoint (or stationary point) [13] of if . We say that a multivalued mapping has the approximate endpoint property [4] if

(3.1)

Let be a single-valued mapping and a multivalued contraction. We say that the mappings and have an approximate endpoint property provided

(3.2)

A point is called an endpoint of and if .

For each , set

(3.3)

Lemma 3.1.

Let be a metric space. Let be a single-valued mapping such that for all , where is a constant. If is a multivalued almost -contraction with , then

(3.4)

Proof.

For any , we have

(3.5)

and so

(3.6)

Since , from (3.6), we have

(3.7)

The following simple example shows that under the assumptions of Lemma 3.1, may be empty.

Example 3.2.

Let be a multivalued mapping defined by for each and the identity mapping. Then and so is a multivalued almost -contraction with . However, for each .

Lemma 3.3.

Let be a metric space. Let be a continuous single-valued mapping and a lower semicontinuous multivalued mapping. Then, for each , is closed.

Proof.

Let be such that with as . Let . Since is lower semicontinuous, then there exists such that . Since , then and so . Since is continuous, . Therefore, , that is, . This completes the proof.

Theorem 3.4.

Let be a complete metric space. Let be a continuous single-valued mapping such that , where is a constant. Let be a lower semicontinuous multivalued almost -contraction. Then and have a unique endpoint if and only if and have the approximate endpoint property.

Proof.

It is clear that, if and have an endpoint, then and have the approximate endpoint property. Conversely, suppose that and have the approximate endpoint property. Then

(3.8)

Also it is clear that, for each , . By Lemma 3.3, is closed for each . Since and have the approximate endpoint property, then for each . Now, we show that . To show this, let . Then, from Lemma 3.1,

(3.9)

and so . It follows from the Cantor intersection theorem that

(3.10)

Thus is the unique endpoint of and .

If is the identity mapping on , then the above result reduces to the following.

Corollary 3.5.

Let be a metric space. If is a multivalued almost contraction with , then

(3.11)

where .

Corollary 3.6.

Let be a complete metric space. Let be a lower semicontinuous multivalued almost contraction with . Then has a unique endpoint if and only if has the approximate endpoint property.

Corollary 3.7 (see [4, Corollary ]).

Let be a complete metric space. Let be a multivalued -contraction. Then has a unique endpoint if and only if has the approximate endpoint property.

Theorem 3.8.

Let be a complete metric space and a multivalued mapping from into . Suppose that there exist a function such that

(3.12)

and and satisfying the two following conditions:

(3.13)

where . Then has the approximate endpoint property. Further, has an endpoint provided is compact and the function is lower semicontinuous.

Proof.

We first prove that has the approximate endpoint property. Let and that satisfy (3.13). By passing to subsequences, if necessary, we may assume that the sequence is convergent. Then we have

(3.14)

Since , we get

(3.15)

Thus has the approximate endpoint property. The lower semi-continuity of the function and the compactness of imply that the infimum is attained. Thus there exists such that . Therefore, . This completes the proof.

The following theorem extends and improves Theorem in [4].

Theorem 3.9.

Let be a complete metric space. Let be a continuous single-valued mapping such that , where is a constant. Let be a multivalued mapping satisfying

(3.16)

where is a function such that and for each . Then and have a unique endpoint if and only if and have the approximate endpoint property.

Proof.

It is clear that, if and have an endpoint, then and have the approximate endpoint property. Conversely, suppose that and have the approximate endpoint property. Then

(3.17)

Also it is clear that, for each , . Since the mapping is continuous (note that and are continuous), we have that is closed. Now we show that . On the contrary, assume that . Since , then (note that the sequences and are nonincreasing and bounded below and then they have the limits). Let be such that . Given , from (3.16) and triangle inequality, we have

(3.18)

Therefore, we have

(3.19)

From (3.19), we have for each and so we get

(3.20)

Hence we have

(3.21)

From (3.21), we obtain

(3.22)

Since , from (3.22), we get . Thus

(3.23)

which is a contradiction and so . It follows from the Cantor intersection theorem that

(3.24)

Thus and hence . To prove the uniqueness of the endpoints of and , let be an arbitrary endpoint of and . Then =0 and so . Thus . This completes the proof.

From Theorem 3.9, we obtain the following improved version of the main result of [4].

Corollary 3.10.

Let be a complete metric space. Let be a multivalued mapping satisfying

(3.25)

where is a function such that and for each . Then has a unique endpoint if and only if has the approximate endpoint property.

Example 3.11.

Let with the usual metric . Let be a multivalued mapping defined by and be a function defined by

(3.26)

Then

(3.27)

Then and satisfy the conditions of Corollary 3.10, but the conditions of Theorem in [4] are not satisfied (note that ).