Fixed Point Theorems for Random Lowersemi-continuous Mappings

Abstract

We prove a general principle in Random Fixed Point Theory by introducing a condition named () which was inspired by some of Petryshyn's work, and then we apply our result to prove some random fixed points theorems, including generalizations of some Bharucha-Reid theorems.

1. Introduction

Let be a metric space and a closed and nonempty subset of . Denote by (resp., ) the family of all nonempty (resp., nonempty and closed) subsets of . A mapping is said to satisfy if, for every closed ball of with radius and any sequence in for which and as , there exists such that where . If is any nonempty set, we say that the operator satisfies if for each , the mapping satisfies . We should observe that this latter condition is related to a condition that was originally introduced by Petryshyn [1] for single-valued operators, in order to prove existence of fixed points. However, in our case, the condition is used to prove the measurability of a certain operator. On the other hand, in the year 2001, Shahzad (cf. [2]) using an idea of Itoh (cf. [3]), see also ([4]), proved that under a somewhat more restrictive condition, named condition (A), the following result.

Theorem 1 S.

Let be a nonempty separable complete subset of a metric space and a continuous random operator satisfying condition (A). Then has a deterministic fixed point if and only if has a random fixed point.

We shall show that the above result is still valid if the operator is only lower semi-continuous. In addition, the assumption that each value is closed has been relaxed without an extra assumption. Furthermore we state a new condition which generalizes condition (A) and allow us to generalize several known results, such as, Bharucha-Reid [5, Theorem  7], Domínguez Benavides et al. [6, Theorem  3.1] and Shahzad [2, Theorem  2.1].

2. Preliminaries

Let be a measurable space and let be a metric space. A mapping , is said to be measurable if is measurable for each open subset of . This type of measurability is usually called weakly (cf. [7]), but since this is the only type of measurability we use in this paper, we omit the term "weakly". Notice that if is separable and if, for each closed subset of , the set is measurable, then is measurable.

Let be a nonempty subset of and , then we say that is lower (upper) semi-continuous if is open (closed) for all open (closed) subsets of . We say that is continuous if is lower and upper semi-continuous.

A mapping is called a random operator if, for each , the mapping is measurable. Similarly a multivalued mapping is also called a random operator if, for each , is measurable. A measurable mapping is called a measurable selection of the operator if for each . A measurable mapping is called a random fixed point of the random operator (or ) if for every (or ). For the sake of clarity, we mention that

Let be a closed subset of the Banach space , and suppose that is a mapping from into the topological vector space . We say the is demiclosed at if, for any sequences in and in with , converges weakly to and converges strongly to , then it is the case that and . On the other hand, we say that is hemicompact if each sequence in has a convergent subsequence, whenever as .

3. Main Results

Theorem 3.1.

Let be a closed separable subset of a complete metric space , and let be measurable in and enjoy . Suppose, for each , that is upper semi-continuous and the set

(3.1)

Then has a random fixed point.

Proof.

Let be a countable dense subset of . Define by . Firstly, we show that is measurable. To this end, let be an arbitrary closed ball of , and set

(3.2)

where and . We claim that . To see this, let . Then there exists such that . Since is upper semi-continuous, for each , there exists such that . Therefore . On the other hand, if , then there exists a subsequence of such that

(3.3)

for all . This means that and as . Consequently, by , there exists such that . Hence . Then we conclude that , and thus is measurable. To complete the proof, let be an arbitrary open subset of . Then by the separability of ,

(3.4)

Since , we conclude that is measurable. Additionally, we show that is closed for each . To see this, let such that . Then, let be a degenerated ball centered at and radius , and since , implies that . Hence and thus by the Kuratowski and Ryll-Nardzewski Theorem [8], has a measurable selection such that for each .

As a consequence of Theorem 3.1, we derive a new result for a lower semi-continuous random operator.

Theorem 3.2.

Let be a closed separable subset of a complete metric space , and let be a lower semi-continuous random operator, which enjoys . Suppose, for each , that the set

(3.5)

Then has a random fixed point.

Proof.

Due to Theorem 3.1, it is enough to show that is upper semi-continuous. To see this, we will prove that is open in for . Let and select . Then there exists so that . Since is lower semi-continuous, there exists a positive number such that for all . Hence, we may choose for which,

(3.6)

and consequently, . Therefore, is open, and proof is complete.

We observe that if the mapping is upper semi-continuous, then not necessarily the mapping is lower semi-continuous. Consider the following example.

Let be defined by

(3.7)

Then for while , which is upper semi-continuous. On the other hand, is not lower semi-continuous.

Now, we derive several consequences of Theorem 3.2. We first obtain an extension of one of the main results of [6].

Theorem 3.3.

Let be a weakly compact separable subset of a Banach space , and let be a lower semi-continuous random operator. Suppose, for each , that is demiclosed at and the set

(3.8)

Then has a random fixed point.

Proof.

In order to apply Theorem 3.2, we just need to prove that enjoys . To this end, let be fixed in . Suppose that is a closed ball of with radius where is a sequence in such that and as . Since is separable, the weak topology on is metrizable, and thus there exists a weakly convergent subsequence of , so that weakly, while as . Consequently, for each , there exists such that

(3.9)

Hence, the demiclosedness of implies that , and thus enjoys .

Before we give an extension of the main result of [4], we observe that is basically applied to those closed balls directly used to prove the measurability of the mapping , as will be seen in the proof of the next result.

Theorem 3.4.

Let be a closed separable subset of a complete metric space , and let be a continuous hemicompact random operator. If, for each , the set

(3.10)

then has a random fixed point.

Proof.

Due to Theorem 3.2, it would be enough to show that enjoys for every . To see this, let be a closed ball of , and let be a sequence in such that and as . Then by the hemicompactness of , there exists a convergent subsequence of , so that . Hence as . This means that, for each , there exists such that

(3.11)

Consequently, . On the other hand, since is upper semi-continuous at , for every there exist such that

(3.12)

Hence, . Since is arbitrary and is closed, we derive that , and thus satisfies .

Corollary 3.5.

Let be a locally compact separable subset of a complete metric space , and let be a continuous random operator. Suppose, for each , that the set

(3.13)

Then has a random fixed point.

Proof.

Let be an arbitrary open subset of , and let . Since is locally compact, there exists a compact ball centered at such that . Now, we prove that holds with respect to . To see this, let , and let be a sequence in such that and as . Then there exists a sequence in so that as . Since is compact, there exists a convergent subsequence of such that , and consequently with as well as as . Since is upper semi-continuous, we derive, as in the proof of Theorem 3.4, that . In addition, since is lower semi-continuous, we may follow the proof of Theorem 3.1, to conclude that is measurable. Hence, the separability of implies that we can select countably many compact balls centered at corresponding points such that

(3.14)

Therefore, is measurable.

Next, we get a stochastic version of Schauder's Theorem, which is also an extension of a Theorem of Bharucha-Reid (see [5, Theorem  10]). We also observe that our proof is much easier and quite short.

Corollary 3.6.

Let be a compact and convex subset of a Fréchet space , and let be a continuous random operator. Then has a random fixed point.

Proof.

As we know, every Fréchet space is a complete metric space, and since is compact, itself is a complete separable metric space. In addition, for each , there exists such that . This means that the set , defined in Theorem 3.1, is nonempty. Since is compact, any sequence in contains a convergent subsequence, which means that is trivially a hemicompact operator. Consequently, by Theorem 3.4, has a random fixed point.

Before obtaining an extension of Bharucha-Reid [5, Theorem 3.7], we define a contraction mapping for metric spaces. Let be a metric space, and let be a measurable space. A random operator is said to be a random contraction if there exists a mapping such that

(3.15)

Theorem 3.7.

Let be a complete separable metric space, and let be a continuous random operator such that is a contraction with constant for each . Then has a unique random fixed point.

Proof.

For each , the mapping has a unique fixed point, , which is also the unique fixed point of . It remains to show that the mapping defined by is measurable. To see this, let be an arbitrary measurable function. Then, we claim that is measurable. To this end, let be a countable dense set of . Let and let . Define

(3.16)

where is the smallest natural number for which . Since is measurable, so are the sets , which, as a matter of fact, conform a disjoint covering of . Consequently, is a sequence of measurable functions that converges pointwise to . On the other hand, the range of each is a subset of , and hence constant on each set . Since the mapping is measurable in , then, for each , is also measurable. Therefore the continuity of on the second variable implies that

(3.17)

for each . Hence is measurable. Define the sequence

(3.18)

Then is a sequence of measurable functions. Since , the fact that is a contraction implies that . Therefore, the mapping is measurable, which completes the proof.

As a direct consequence of Theorem 3.7, we derive the extension mentioned earlier where the space is more general, and the randomness on the mapping has been removed.

Corollary 3.8.

Let be a complete separable metric space, and let be a random contraction operator with constant for each . Then has a unique random fixed point.

Next, one can derive a corollary of the proof of Theorem 3.7, which is a theorem of Hans [9].

Corollary 3.9.

Let be a complete separable metric space, and let be a continuous random operator. Suppose, for each , that there exists such that is a contraction with constant . Then has a unique random fixed point.

Proof.

As in the proof of the theorem, the mapping has a unique fixed point for each . The rest of the proof follows the proof of the theorem with the appropriate changes of the second power of by the power of .

Notice that Theorem 3.7 holds for single-valued operators. The following question is formulated for multivalued operators taking closed and bounded values in .

Open Question

Suppose that is a complete separable metric space, and let be a continuous random operator such that is a contraction with constant for each . Then does have a unique random fixed point?