Fixed Point Theory and Applications
Volume 2009 (2009), Article ID 584178, 7 pages
doi:10.1155/2009/584178
Research Article

Fixed Point Theorems for Random Lowersemi-continuous Mappings

Raúl Fierro,1,2 Carlos Martínez,1 and Claudio H. Morales3

1Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Cerro Barón, Valparaíso, Chile
2Laboratorio de Análisis Estocástico CIMFAV, Universidad de Valparaíso, Casilla 5030, Valparaíso, Chile
3Department of Mathematics, University of Alabama in Huntsville, Huntsville, AL 35899, USA

Received 31 January 2009; Accepted 1 July 2009

Academic Editor: Naseer Shahzad

Copyright © 2009 Raúl Fierro et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove a general principle in Random Fixed Point Theory by introducing a condition named ( 𝒫 ) 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 ( 𝑋 , 𝑑 ) be a metric space and 𝑆 a closed and nonempty subset of 𝑋 . Denote by 2 𝑋 (resp., 𝒞 ( 𝑋 ) ) the family of all nonempty (resp., nonempty and closed) subsets of 𝑋 . A mapping 𝑇 ∶ 𝑆 → 2 𝑋 is said to satisfy 𝑐 𝑜 𝑛 𝑑 𝑖 𝑡 𝑖 𝑜 𝑛 ( 𝒫 ) if, for every closed ball 𝐵 of 𝑆 with radius 𝑟 ≥ 0 and any sequence { 𝑥 𝑛 } in 𝑆 for which 𝑑 ( 𝑥 𝑛 , 𝐵 ) → 0 and 𝑑 ( 𝑥 𝑛 , 𝑇 ( 𝑥 𝑛 ) ) → 0 as 𝑛 → ∞ , there exists 𝑥 0 ∈ 𝐵 such that 𝑥 0 ∈ 𝑇 ( 𝑥 0 ) where 𝑑 ( 𝑥 , 𝐵 ) = i n f { 𝑑 ( 𝑥 , 𝑦 ) ∶ 𝑦 ∈ 𝐵 } . If Ω is any nonempty set, we say that the operator 𝑇 ∶ Ω × 𝑆 → 2 𝑋 satisfies 𝑐 𝑜 𝑛 𝑑 𝑖 𝑡 𝑖 𝑜 𝑛 ( 𝒫 ) if for each 𝜔 ∈ Ω , the mapping 𝑇 ( 𝜔 , ⋅ ) ∶ 𝑆 → 2 𝑋 satisfies 𝑐 𝑜 𝑛 𝑑 𝑖 𝑡 𝑖 𝑜 𝑛 ( 𝒫 ) . 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 𝑆 be a nonempty separable complete subset of a metric space 𝑋 and 𝑇 ∶ Ω × 𝐶 → 𝐶 ( 𝑋 ) a continuous random operator satisfying condition (A). Then 𝑇 has a deterministic fixed point if and only if 𝑇 has a random fixed point.

We shall show that the above result is still valid if the operator 𝑇 is only lower semi-continuous. In addition, the assumption that each value 𝑇 ( 𝑥 ) 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 ( Ω , 𝒜 ) be a measurable space and let ( 𝑋 , 𝑑 ) be a metric space. A mapping 𝐹 ∶ Ω → 2 𝑋 , is said to be measurable if 𝐹 − 1 ( 𝐺 ) = { 𝜔 ∈ Ω ∶ 𝐹 ( 𝜔 ) ∩ 𝐺 ≠ 𝜙 } is measurable for each open subset 𝐺 of 𝑋 . 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 𝑋 is separable and if, for each closed subset 𝐶 of 𝑋 , the set 𝐹 − 1 ( 𝐶 ) is measurable, then 𝐹 is measurable.

Let 𝐶 be a nonempty subset of 𝑋 and 𝐹 ∶ 𝐶 → 2 𝑋 , then we say that 𝐹 is lower (upper) semi-continuous if 𝐹 − 1 ( 𝐴 ) is open (closed) for all open (closed) subsets 𝐴 of 𝑋 . We say that 𝐹 is continuous if 𝐹 is lower and upper semi-continuous.

A mapping 𝐹 ∶ Ω × 𝑋 → 𝑌 is called a random operator if, for each 𝑥 ∈ 𝑋 , the mapping 𝐹 ( ⋅ , 𝑥 ) ∶ Ω → 𝑌 is measurable. Similarly a multivalued mapping 𝐹 ∶ Ω × 𝑋 → 2 𝑌 is also called a random operator if, for each 𝑥 ∈ 𝑋 , 𝐹 ( ⋅ , 𝑥 ) ∶ Ω → 2 𝑌 is measurable. A measurable mapping 𝜉 ∶ Ω → 𝑌 is called a measurable selection of the operator 𝐹 ∶ Ω → 2 𝑌 if 𝜉 ( 𝜔 ) ∈ 𝐹 ( 𝜔 ) for each 𝜔 ∈ Ω . A measurable mapping 𝜉 ∶ Ω → 𝑋 is called a random fixed point of the random operator 𝐹 ∶ Ω × 𝑋 → 𝑋 (or 𝐹 ∶ Ω × 𝑋 → 2 𝑋 ) if for every 𝜔 ∈ Ω , 𝜉 ( 𝜔 ) = 𝐹 ( 𝜔 , 𝜉 ( 𝜔 ) ) (or 𝜉 ( 𝜔 ) ∈ 𝐹 ( 𝜔 , 𝜉 ( 𝜔 ) ) ). For the sake of clarity, we mention that 𝐹 ( 𝜔 , 𝜉 ( 𝜔 ) ) = 𝐹 ( 𝜔 , ⋅ ) ( 𝜉 ( 𝜔 ) ) .

Let 𝐶 be a closed subset of the Banach space 𝑋 , and suppose that 𝐹 is a mapping from 𝐶 into the topological vector space 𝑌 . We say the 𝐹 is demiclosed at 𝑦 0 ∈ 𝑌 if, for any sequences { 𝑥 𝑛 } in 𝐶 and { 𝑦 𝑛 } in 𝑌 with 𝑦 𝑛 ∈ 𝐹 ( 𝑥 𝑛 ) , { 𝑥 𝑛 } converges weakly to 𝑥 0 and { 𝑦 𝑛 } converges strongly to 𝑦 0 , then it is the case that 𝑥 0 ∈ 𝐶 and 𝑦 0 ∈ 𝐹 ( 𝑥 0 ) . On the other hand, we say that 𝐹 is hemicompact if each sequence { 𝑥 𝑛 } in 𝐶 has a convergent subsequence, whenever 𝑑 ( 𝑥 𝑛 , 𝐹 ( 𝑥 𝑛 ) ) → 0 as 𝑛 → ∞ .

3. Main Results

Theorem 3.1. Let 𝐶 be a closed separable subset of a complete metric space 𝑋 , and let 𝑇 ∶ Ω × 𝐶 → 2 𝑋 be measurable in 𝜔 and enjoy 𝑐 𝑜 𝑛 𝑑 𝑖 𝑡 𝑖 𝑜 𝑛 ( 𝒫 ) . Suppose, for each 𝜔 ∈ Ω , that ℎ ( 𝜔 , 𝑥 ) = 𝑑 ( 𝑥 , 𝑇 ( 𝜔 , 𝑥 ) ) is upper semi-continuous and the set 𝐹 ( 𝜔 ) ∶ = { 𝑥 ∈ 𝐶 ∶ 𝑥 ∈ 𝑇 ( 𝜔 , 𝑥 ) } ≠ 𝜙 . ( 3 . 1 ) Then 𝑇 has a random fixed point.

Proof. Let 𝑍 = { 𝑧 𝑛 } be a countable dense subset of 𝐶 . Define 𝐹 ∶ Ω → 2 𝐶 by 𝐹 ( 𝜔 ) = { 𝑥 ∈ 𝐶 ∶ 𝑥 ∈ 𝑇 ( 𝜔 , 𝑥 ) } . Firstly, we show that 𝐹 is measurable. To this end, let 𝐵 0 be an arbitrary closed ball of 𝐶 , and set 𝐿 𝐵 0 = ∞ 𝑘 = 1 0 𝑥 0 2 0 0 𝑑 𝑧 ∈ 𝑍 𝑘 1 𝜔 ∈ Ω ∶ 𝑑 ( 𝑧 , 𝑇 ( 𝜔 , 𝑧 ) ) < 𝑘 , ( 3 . 2 ) where 𝑍 𝑘 = 𝐵 𝑘 ∩ 𝑍 and 𝐵 𝑘 = { 𝑥 ∈ 𝐶 ∶ 𝑑 ( 𝑥 , 𝐵 0 ) < 1 / 𝑘 } . We claim that 𝐹 − 1 ( 𝐵 0 ) = 𝐿 ( 𝐵 0 ) . To see this, let 𝜔 ∈ 𝐹 − 1 ( 𝐵 0 ) . Then there exists 𝑥 ∈ 𝐵 0 such that 𝑥 ∈ 𝑇 ( 𝜔 , 𝑥 ) . Since ℎ ( 𝜔 , ⋅ ) is upper semi-continuous, for each 𝑘 ∈ ℕ , there exists 𝑧 𝑛 𝑘 ∈ 𝑍 𝑘 such that 𝑑 ( 𝑧 𝑛 𝑘 , 𝑇 ( 𝜔 , 𝑧 𝑛 𝑘 ) ) < 1 / 𝑘 . Therefore 𝜔 ∈ 𝐿 ( 𝐵 0 ) . On the other hand, if 𝜔 ∈ 𝐿 ( 𝐵 0 ) , then there exists a subsequence { 𝑧 𝑛 𝑘 } of { 𝑧 𝑛 } such that 𝑑 𝑧 𝑛 𝑘 , 𝐵 0 < 1 𝑘 𝑧 , 𝑑 𝑛 𝑘 , 𝑇 𝜔 , 𝑧 𝑛 𝑘 < 1 𝑘 ( 3 . 3 ) for all 𝑘 ∈ ℕ . This means that 𝑑 ( 𝑧 𝑛 𝑘 , 𝐵 0 ) → 0 and 𝑑 ( 𝑧 𝑛 𝑘 , 𝑇 ( 𝜔 , 𝑧 𝑛 𝑘 ) ) → 0 as 𝑛 → ∞ . Consequently, by 𝑐 𝑜 𝑛 𝑑 𝑖 𝑡 𝑖 𝑜 𝑛 ( 𝒫 ) , there exists 𝑥 0 ∈ 𝐵 0 such that 𝑥 0 ∈ 𝑇 ( 𝜔 , 𝑥 0 ) . Hence 𝜔 ∈ 𝐹 − 1 ( 𝐵 0 ) . Then we conclude that 𝐹 − 1 ( 𝐵 0 ) = 𝐿 ( 𝐵 0 ) , and thus 𝐹 − 1 ( 𝐵 0 ) is measurable. To complete the proof, let 𝐺 be an arbitrary open subset of 𝐶 . Then by the separability of 𝐶 , 𝐺 = ∞ 𝑛 = 1 𝐵 𝑛 w h e r e e a c h 𝐵 𝑛 i s a c l o s e d b a l l o f 𝐶 . ( 3 . 4 ) Since 𝐹 − 1 ⋃ ( 𝐺 ) = ∞ 𝑛 = 1 𝐹 − 1 ( 𝐵 𝑛 ) , we conclude that 𝐹 is measurable. Additionally, we show that 𝐹 ( 𝜔 ) is closed for each 𝜔 ∈ Ω . To see this, let 𝑥 𝑛 ∈ 𝐹 ( 𝜔 ) such that 𝑥 𝑛 → 𝑥 ∈ 𝐶 . Then, let 𝐵 0 = { 𝑥 } be a degenerated ball centered at 𝑥 and radius 𝑟 = 0 , and since 𝑑 ( 𝑥 𝑛 , 𝑇 ( 𝜔 , 𝑥 𝑛 ) ) = 0 , 𝑐 𝑜 𝑛 𝑑 𝑖 𝑡 𝑖 𝑜 𝑛 ( 𝒫 ) implies that 𝑥 ∈ 𝑇 ( 𝜔 , 𝑥 ) . Hence 𝑥 ∈ 𝐹 ( 𝜔 ) and thus by the Kuratowski and Ryll-Nardzewski Theorem [8], 𝐹 has a measurable selection 𝜉 ∶ Ω → 𝐶 such that 𝜉 ( 𝜔 ) ∈ 𝑇 ( 𝜔 , 𝜉 ( 𝜔 ) ) for each 𝜔 ∈ Ω .

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

Theorem 3.2. Let 𝐶 be a closed separable subset of a complete metric space 𝑋 , and let 𝑇 ∶ Ω × 𝐶 → 2 𝑋 be a lower semi-continuous random operator, which enjoys 𝑐 𝑜 𝑛 𝑑 𝑖 𝑡 𝑖 𝑜 𝑛 ( 𝒫 ) . Suppose, for each 𝜔 ∈ Ω , that the set 𝐹 ( 𝜔 ) ∶ = { 𝑥 ∈ 𝐶 ∶ 𝑥 ∈ 𝑇 ( 𝜔 , 𝑥 ) } ≠ 𝜙 . ( 3 . 5 ) Then 𝑇 has a random fixed point.

Proof. Due to Theorem 3.1, it is enough to show that ℎ ( 𝜔 , ⋅ ) is upper semi-continuous. To see this, we will prove that 𝐴 = { 𝑥 ∈ 𝐶 ∶ 𝑑 ( 𝑥 , 𝑇 ( 𝜔 , 𝑥 ) ) < 𝛼 } is open in 𝐶 for 𝛼 > 0 . Let 𝑎 ∈ 𝐴 and select 𝜖 = 𝛼 − 𝑑 ( 𝑎 , 𝑇 ( 𝜔 , 𝑎 ) ) . Then there exists 𝑦 ∈ 𝑇 ( 𝜔 , 𝑎 ) so that 𝑑 ( 𝑎 , 𝑦 ) < 𝜖 / 3 + 𝑑 ( 𝑎 , 𝑇 ( 𝜔 , 𝑎 ) ) . Since 𝑇 ( 𝜔 , ⋅ ) is lower semi-continuous, there exists a positive number 𝑟 < 𝜖 / 3 such that 𝑇 ( 𝜔 , 𝑢 ) ∩ 𝐵 ( 𝑦 ; 𝜖 / 3 ) ≠ ∅ for all 𝑢 ∈ 𝐵 ( 𝑎 ; 𝑟 ) . Hence, we may choose 𝑧 𝑢 ∈ 𝑇 ( 𝜔 , 𝑢 ) ∩ 𝐵 ( 𝑦 ; 𝜖 / 3 ) for which, 𝑑 𝑢 , 𝑧 𝑢 ≤ 𝑑 ( 𝑢 , 𝑎 ) + 𝑑 ( 𝑎 , 𝑦 ) + 𝑑 𝑦 , 𝑧 𝑢 < 𝛼 , ( 3 . 6 ) and consequently, 𝑑 ( 𝑢 , 𝑇 ( 𝜔 , 𝑢 ) ) < 𝛼 . Therefore, 𝐴 is open, and proof is complete.

We observe that if the mapping ℎ ( 𝑥 ) = 𝑑 ( 𝑥 , 𝑇 ( 𝑥 ) ) is upper semi-continuous, then not necessarily the mapping 𝑇 is lower semi-continuous. Consider the following example.

Let 𝑇 ∶ ℝ → 2 ℝ be defined by [ ] 𝑇 ( 𝑥 ) = 1 , 𝑥 ≠ 0 2 , 3 , 𝑥 = 0 . ( 3 . 7 ) Then ℎ ( 𝑥 ) = | 𝑥 − 1 | for 𝑥 ≠ 0 while ℎ ( 0 ) = 2 , which is upper semi-continuous. On the other hand, 𝑇 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 𝐶 be a weakly compact separable subset of a Banach space 𝑋 , and let 𝑇 ∶ Ω × 𝐶 → 2 𝑋 be a lower semi-continuous random operator. Suppose, for each 𝜔 ∈ Ω , that 𝐼 − 𝑇 ( 𝜔 , ⋅ ) is demiclosed at 0 and the set 𝐹 ( 𝜔 ) ∶ = { 𝑥 ∈ 𝐶 ∶ 𝑥 ∈ 𝑇 ( 𝜔 , 𝑥 ) } ≠ 𝜙 . ( 3 . 8 ) Then 𝑇 has a random fixed point.

Proof. In order to apply Theorem 3.2, we just need to prove that 𝑇 enjoys 𝑐 𝑜 𝑛 𝑑 𝑖 𝑡 𝑖 𝑜 𝑛 ( 𝒫 ) . To this end, let 𝜔 be fixed in Ω . Suppose that 𝐵 0 is a closed ball of 𝐶 with radius 𝑟 ≥ 0 where { 𝑥 𝑛 } is a sequence in 𝐶 such that 𝑑 ( 𝑥 𝑛 , 𝐵 0 ) → 0 and 𝑑 ( 𝑥 𝑛 , 𝑇 ( 𝜔 , 𝑥 𝑛 ) ) → 0 as 𝑛 → ∞ . Since 𝐶 is separable, the weak topology on 𝐶 is metrizable, and thus there exists a weakly convergent subsequence { 𝑥 𝑛 𝑘 } of { 𝑥 𝑛 } , so that 𝑥 𝑛 𝑘 → 𝑥 weakly, while 𝑑 ( 𝑥 𝑛 𝑘 , 𝑇 ( 𝜔 , 𝑥 𝑛 𝑘 ) ) → 0 as 𝑘 → ∞ . Consequently, for each 𝑘 ∈ ℕ , there exists 𝑧 𝑘 ∈ 𝑇 ( 𝜔 , 𝑥 𝑛 𝑘 ) such that ‖ 𝑥 𝑛 𝑘 − 𝑧 𝑘 ‖ ⟶ 0 a s 𝑘 ⟶ ∞ . ( 3 . 9 ) Hence, the demiclosedness of 𝐼 − 𝑇 ( 𝜔 , ⋅ ) implies that 𝑥 ∈ 𝑇 ( 𝜔 , 𝑥 ) , and thus 𝑇 ( 𝜔 , ⋅ ) enjoys 𝑐 𝑜 𝑛 𝑑 𝑖 𝑡 𝑖 𝑜 𝑛 ( 𝒫 ) .
Before we give an extension of the main result of [4], we observe that 𝑐 𝑜 𝑛 𝑑 𝑖 𝑡 𝑖 𝑜 𝑛 ( 𝒫 ) is basically applied to those closed balls directly used to prove the measurability of the mapping 𝐹 , as will be seen in the proof of the next result.

Theorem 3.4. Let 𝐶 be a closed separable subset of a complete metric space 𝑋 , and let 𝑇 ∶ Ω × 𝐶 → 𝒞 ( X ) be a continuous hemicompact random operator. If, for each 𝜔 ∈ Ω , the set 𝐹 ( 𝜔 ) ∶ = { 𝑥 ∈ 𝐶 ∶ 𝑥 ∈ 𝑇 ( 𝜔 , 𝑥 ) } ≠ 𝜙 , ( 3 . 1 0 ) then 𝑇 has a random fixed point.

Proof. Due to Theorem 3.2, it would be enough to show that 𝑇 ( 𝜔 , ⋅ ) enjoys 𝑐 𝑜 𝑛 𝑑 𝑖 𝑡 𝑖 𝑜 𝑛 ( 𝒫 ) for every 𝜔 ∈ Ω . To see this, let 𝐵 0 be a closed ball of 𝐶 , and let { 𝑥 𝑛 } be a sequence in 𝐶 such that 𝑑 ( 𝑥 𝑛 , 𝐵 0 ) → 0 and 𝑑 ( 𝑥 𝑛 , 𝑇 ( 𝜔 , 𝑥 𝑛 ) ) → 0 as 𝑛 → ∞ . Then by the hemicompactness of 𝑇 , there exists a convergent subsequence { 𝑥 𝑛 𝑘 } of { 𝑥 𝑛 } , so that 𝑥 𝑛 𝑘 → 𝑥 ∈ 𝐵 0 . Hence 𝑑 ( 𝑥 𝑛 𝑘 , 𝑇 ( 𝜔 , 𝑥 𝑛 𝑘 ) ) → 0 as 𝑘 → ∞ . This means that, for each 𝑘 ∈ ℕ , there exists 𝑧 𝑘 ∈ 𝑇 ( 𝜔 , 𝑥 𝑛 𝑘 ) such that 𝑑 𝑥 𝑛 𝑘 , 𝑧 𝑘 ⟶ 0 a s 𝑘 ⟶ ∞ . ( 3 . 1 1 ) Consequently, 𝑧 𝑘 → 𝑥 . On the other hand, since 𝑇 is upper semi-continuous at 𝑥 , for every 𝜖 > 0 there exist 𝑘 0 ∈ ℕ such that 𝑇 𝜔 , 𝑥 𝑛 𝑘 ⊂ 𝐵 ( 𝑇 ( 𝜔 , 𝑥 ) ; 𝜖 ) f o r a l l 𝑘 ≥ 𝑘 0 . ( 3 . 1 2 ) Hence, 𝑥 ∈ 𝐵 ( 𝑇 ( 𝜔 , 𝑥 ) ; 𝜖 ) . Since 𝜖 is arbitrary and 𝑇 ( 𝜔 , 𝑥 ) is closed, we derive that 𝑥 ∈ 𝑇 ( 𝜔 , 𝑥 ) , and thus 𝑇 satisfies 𝑐 𝑜 𝑛 𝑑 𝑖 𝑡 𝑖 𝑜 𝑛 ( 𝒫 ) .

Corollary 3.5. Let 𝐶 be a locally compact separable subset of a complete metric space 𝑋 , and let 𝑇 ∶ Ω × 𝐶 → 𝒞 ( X ) be a continuous random operator. Suppose, for each 𝜔 ∈ Ω , that the set 𝐹 ( 𝜔 ) ∶ = { 𝑥 ∈ 𝐶 ∶ 𝑥 ∈ 𝑇 ( 𝜔 , 𝑥 ) } ≠ 𝜙 . ( 3 . 1 3 ) Then 𝑇 has a random fixed point.

Proof. Let 𝐺 be an arbitrary open subset of 𝐶 , and let 𝑥 ∈ 𝐺 . Since 𝐶 is locally compact, there exists a compact ball 𝐵 centered at 𝑥 such that 𝐵 ⊂ 𝐺 . Now, we prove that 𝑐 𝑜 𝑛 𝑑 𝑖 𝑡 𝑖 𝑜 𝑛 ( 𝒫 ) holds with respect to 𝐵 . To see this, let 𝜔 ∈ Ω , and let { 𝑥 𝑛 } be a sequence in 𝑋 such that 𝑑 ( 𝑥 𝑛 , 𝐵 ) → 0 and 𝑑 ( 𝑥 𝑛 , 𝑇 ( 𝜔 , 𝑥 𝑛 ) ) → 0 as 𝑛 → ∞ . Then there exists a sequence { 𝑦 𝑛 } in 𝐵 so that 𝑑 ( 𝑥 𝑛 , 𝑦 𝑛 ) → 0 as 𝑛 → ∞ . Since 𝐵 is compact, there exists a convergent subsequence { 𝑦 𝑛 𝑘 } of { 𝑦 𝑛 } such that 𝑦 𝑛 𝑘 → 𝑥 , and consequently 𝑥 𝑛 𝑘 → 𝑥 with 𝑥 ∈ 𝐵 as well as 𝑑 ( 𝑥 𝑛 𝑘 , 𝑇 ( 𝜔 , 𝑥 𝑛 𝑘 ) ) → 0 as 𝑘 → ∞ . Since 𝑇 is upper semi-continuous, we derive, as in the proof of Theorem 3.4, that 𝑥 ∈ 𝑇 ( 𝑥 ) . In addition, since 𝑇 is lower semi-continuous, we may follow the proof of Theorem 3.1, to conclude that 𝐹 − 1 ( 𝐵 ) is measurable. Hence, the separability of 𝐶 implies that we can select countably many compact balls 𝐵 𝑖 centered at corresponding points 𝑥 𝑖 ∈ 𝐺 such that 𝐹 − 1 ( 𝐺 ) = 𝑖 ∈ ℕ 𝐹 − 1 𝐵 𝑖 . ( 3 . 1 4 ) Therefore, 𝐹 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 𝐶 be a compact and convex subset of a Fréchet space 𝑋 , and let 𝑇 ∶ Ω × 𝐶 → 𝐶 be a continuous random operator. Then 𝑇 has a random fixed point.

Proof. As we know, every Fréchet space is a complete metric space, and since 𝐶 is compact, 𝐶 itself is a complete separable metric space. In addition, for each 𝜔 ∈ Ω , there exists 𝑥 ∈ 𝐶 such that 𝑇 ( 𝜔 , 𝑥 ) = 𝑥 . This means that the set 𝐹 ( 𝜔 ) , defined in Theorem 3.1, is nonempty. Since 𝐶 is compact, any sequence in 𝐶 contains a convergent subsequence, which means that 𝑇 is trivially a hemicompact operator. Consequently, by Theorem 3.4, 𝑇 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 𝑋 be a metric space, and let Ω be a measurable space. A random operator 𝑇 ∶ Ω × 𝑋 → 𝑋 is said to be a random contraction if there exists a mapping 𝑘 ∶ Ω → [ 0 , 1 ) such that 𝑑 ( 𝑇 ( 𝜔 , 𝑥 ) , 𝑇 ( 𝜔 , 𝑦 ) ≤ 𝑘 ( 𝜔 ) 𝑑 ( 𝑥 , 𝑦 ) f o r a l l 𝑥 , 𝑦 ∈ 𝑋 . ( 3 . 1 5 ) Theorem 3.7. Let 𝑋 be a complete separable metric space, and let 𝑇 ∶ Ω × 𝑋 → 𝑋 be a continuous random operator such that 𝑇 2 is a contraction with constant 𝑘 ( 𝜔 ) for each 𝜔 ∈ Ω . Then 𝑇 has a unique random fixed point.

Proof. For each 𝜔 ∈ Ω , the mapping 𝑇 2 has a unique fixed point, 𝜉 ( 𝜔 ) , which is also the unique fixed point of 𝑇 . It remains to show that the mapping 𝜉 ∶ Ω → 𝑋 defined by 𝑇 ( 𝜔 , 𝜉 ( 𝜔 ) ) = 𝜉 ( 𝜔 ) is measurable. To see this, let 𝑓 0 ∶ Ω → 𝑋 be an arbitrary measurable function. Then, we claim that 𝑇 ( 𝜔 , 𝑓 0 ( 𝜔 ) ) is measurable. To this end, let 𝑍 = { 𝑧 𝑛 } be a countable dense set of 𝑋 . Let 𝜔 ∈ Ω and let 𝑘 ∈ ℕ . Define ℎ 𝑘 ∶ Ω ⟶ 𝑋 b y ℎ 𝑘 ( 𝜔 ) = 𝑧 𝑚 , ( 3 . 1 6 ) where 𝑚 is the smallest natural number for which 𝑑 ( 𝑧 𝑚 , 𝑓 0 ( 𝜔 ) ) < 1 / 𝑘 . Since 𝑓 0 is measurable, so are the sets 𝐸 𝑚 = { 𝜔 ∈ Ω ∶ 𝑑 ( 𝑧 𝑚 , 𝑓 0 ( 𝜔 ) ) < 1 / 𝑘 } , which, as a matter of fact, conform a disjoint covering of Ω . Consequently, { ℎ 𝑘 } is a sequence of measurable functions that converges pointwise to 𝑓 0 . On the other hand, the range of each ℎ 𝑘 is a subset of 𝑍 , and hence constant on each set 𝐸 𝑚 . Since the mapping 𝑇 is measurable in 𝜔 , then, for each 𝑘 ∈ ℕ , 𝑇 ( 𝜔 , ℎ 𝑘 ( 𝜔 ) ) is also measurable. Therefore the continuity of 𝑇 on the second variable implies that 𝑇 𝜔 , ℎ 𝑘 ( 𝜔 ) ⟶ 𝑇 𝜔 , 𝑓 0 ( 𝜔 ) a s 𝑘 ⟶ ∞ , ( 3 . 1 7 ) for each 𝜔 ∈ Ω . Hence 𝑇 ( 𝜔 , 𝑓 0 ( 𝜔 ) ) is measurable. Define the sequence 𝑓 𝑛 ( 𝜔 ) = 𝑇 𝜔 , 𝑓 𝑛 − 1 ( 𝜔 ) , 𝑛 ∈ ℕ . ( 3 . 1 8 ) Then { 𝑓 𝑛 } is a sequence of measurable functions. Since 𝑓 𝑛 ( 𝜔 ) = 𝑇 𝑛 ( 𝜔 , 𝑓 0 ( 𝜔 ) ) , the fact that 𝑇 2 is a contraction implies that 𝑓 𝑛 ( 𝜔 ) → 𝜉 ( 𝜔 ) . Therefore, the mapping 𝜉 is measurable, which completes the proof.
As a direct consequence of Theorem 3.7, we derive the extension mentioned earlier where the space 𝑋 is more general, and the randomness on the mapping 𝑘 has been removed.

Corollary 3.8. Let 𝑋 be a complete separable metric space, and let 𝑇 ∶ Ω × 𝑋 → 𝑋 be a random contraction operator with constant 𝑘 ( 𝜔 ) for each 𝜔 ∈ Ω . Then 𝑇 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 𝑋 be a complete separable metric space, and let 𝑇 ∶ Ω × 𝑋 → 𝑋 be a continuous random operator. Suppose, for each 𝜔 ∈ Ω , that there exists 𝑛 ∈ ℕ such that 𝑇 𝑛 is a contraction with constant 𝑘 ( 𝜔 ) . Then 𝑇 has a unique random fixed point.

Proof. As in the proof of the theorem, the mapping 𝑇 has a unique fixed point for each 𝜔 ∈ Ω . The rest of the proof follows the proof of the theorem with the appropriate changes of the second power of 𝑇 by the 𝑛 t h power of 𝑇 .

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

Open Question
Suppose that 𝑋 is a complete separable metric space, and let 𝑇 ∶ Ω × 𝑋 → 𝒞 𝐵 ( X ) be a continuous random operator such that 𝑇 2 is a contraction with constant 𝑘 ( 𝜔 ) for each 𝜔 ∈ Ω . Then does 𝑇 have a unique random fixed point?

Acknowledgments

This work was partially supported by Dirección de Investigación e Innovación de la Pontificia Universidad Católica de Valparaíso under grant 124.719/2009. In addition, the first author was supported by Laboratory of Stochastic Analysis PBCT-ACT 13.

References

  1. W. V. Petryshyn, “Fixed point theorems for various classes of 1-set-contractive and 1-ball-contractive mappings in Banach spaces,” Transactions of the American Mathematical Society, vol. 182, pp. 323–352, 1973.
  2. N. Shahzad, “Random fixed points of set-valued maps,” Nonlinear Analysis: Theory, Methods & Applications, vol. 45, no. 6, pp. 689–692, 2001.
  3. S. Itoh, “Random fixed-point theorems with an application to random differential equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 261–273, 1979.
  4. K.-K. Tan and X.-Z. Yuan, “Random fixed-point theorems and approximation in cones,” Journal of Mathematical Analysis and Applications, vol. 185, no. 2, pp. 378–390, 1994.
  5. A. T. Bharucha-Reid, “Fixed point theorems in probabilistic analysis,” Bulletin of the American Mathematical Society, vol. 82, no. 5, pp. 641–657, 1976.
  6. T. Domínguez Benavides, G. López Acedo, and H. K. Xu, “Random fixed points of set-valued operators,” Proceedings of the American Mathematical Society, vol. 124, no. 3, pp. 831–838, 1996.
  7. C. J. Himmelberg, “Measurable relations,” Fundamenta Mathematicae, vol. 87, pp. 53–72, 1975.
  8. K. Kuratowski and C. Ryll-Nardzewski, “A general theorem on selectors,” Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 13, pp. 397–403, 1965.
  9. O. Hanš, “Random operator equations,” in Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, pp. 185–202, University California Press, Berkeley, Calif, USA, 1961.