Random fixed point theorem in generalized Banach ...

Random Oper. Stoch. Equ. 2016; 24 (2):93–112

Research Article Moulay Larbi Sinacer, Juan Jose Nieto and Abdelghani Ouahab*

Random fixed point theorem in generalized Banach space and applications DOI: 10.1515/rose-2016-0007 Received July 1, 2015; revised September 14, 2015; accepted January 14, 2016

Abstract: In this paper, we prove some random fixed point theorems in generalized Banach spaces. We establish a random version of a Krasnoselskii-type fixed point theorem for the sum of a contraction random operator and a compact operator. The results are used to prove the existence of solution for random differential equations with initial and boundary conditions. Finally, some examples are given to illustrate the results. Keywords: Multivalued map, measurable selection, generalized metric space, fixed point, random operator, random differential equation MSC 2010: 47H10, 47H30, 54H25 || Communicated by: Enzo Orsingher

1 Introduction Probabilistic functional analysis is an important mathematical area of research due to its applications to probabilistic models in applied problems. Random operator theory is needed for the study of various classes of random equations. Indeed, in many cases the mathematical models or equations used to describe phenomena in biological, physical, engineering, and systems sciences contain certain parameters or coefficients which have specific interpretations, but whose values are unknown. Therefore, it is more realistic to consider such equations as random operator equations. These equations are much more difficult to handle mathematically than deterministic equations. Important contributions to the study of the mathematical aspects of such random equations have been undertaken in [8, 28, 40] among others. The problem of fixed points for random mappings was initialed by the Prague school of probabilities. The first results were studied ˘ cek and Han˘s in the context of Fredholm integral equations with random kernel. in 1955–1956 by Spa˘ In a separable metric space, random fixed point theorems for contraction mappings were proved by Han˘s ˘ cek [41], Han˘s and, Spa˘ ˘ cek [19] and Mukherjee [26, 27]. Then random fixed point theorems of [17, 18], Spa˘ Schauder or Krasnoselskii type were given by Mukherjea (cf. Bharucha-Reid [8, p. 110]), Prakasa Rao [33] and Bharucha-Reid [9]. Now it has become the full fledged research area and vast amount of mathematical activities have been carried out in this direction, see for examples [5, 7, 38, 39, 42]. In 1958, Krasnoselskii [23] established his famous fixed point theorem. The result combined the Banach contraction principle and Schauder’s fixed point theorem. The existence of fixed points for the sum of two deterministic operators has attracted tremendous interest, and their applications are frequent in nonlinear analysis. Many improveMoulay Larbi Sinacer: Laboratory of Mathematics, Sidi-Bel-Abbès University, Algeria, e-mail: [email protected] Juan Jose Nieto: Departamento de Análisis Matemático, Facultad de Matemáticas Universidad de Santiago de Compostela, Santiago de Compostela, 15782, Spain; and Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, 21589, Jeddah, Saudi Arabia, e-mail: [email protected] *Corresponding author: Abdelghani Ouahab: Laboratory of Mathematics, Sidi-Bel-Abbès University, Algeria, e-mail: [email protected]

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94 | M. L. Sinacer, J. J. Nieto and A. Ouahab, Random fixed point theorems

ments of Krasnoselskii’s theorem have been established in the literature in the course of time by modifying the above assumptions; see for example [13–15]. Rao [33] obtained a probabilistic version of Krasnoselskii’s theorem which is the sum of a contraction random operator and a compact random operator on a closed convex subset of a separable Banach space. Moreover, Itoh [21] extended Rao’s result to a sum of a nonexpansive random operator and a completely continuous random operator on a weakly compact convex subset of a separable uniformly convex Banach space. Very recently, in [6] Arunchai and Plubtieng obtained a random fixed point theorem for the sum of a weakly-strongly continuous random operator and a nonexpansive random operator which contains as a special Krasnoselskii type. Several well-known fixed point theorems of single-valued mappings such as Banach’s and Schauder’s have been extended in generalized Banach spaces; see [4, 16]. Very recently, in generalized Banach space a Krasnoselskii-type fixed point theorem of singleand multi-valued operators was studied by Petre [31], Petre and Petruşel [32] and Ouahab [37]. Our goal in this work is to give a random version of Pervo-type, Schauder-type and Krasnoselskii-type fixed point theorems in a generalized Banach space. The paper is organized as follows. In Sections 2 and 3 we introduce all the background material needed in this paper such as generalized metric space, some fixed point theorems and random variable. The aim of Section 4 is devoted to establishing a random version of the Pervo, Schauder and Krasnoselskii fixed point theorems in generalized metric and Banach spaces. In Section 5 and 6 we apply our results to a random Cauchy problem and second order boundary value problems.

2 Preliminaries In this section we recall from the literature some notations, definitions, and auxiliary results which will be used throughout this paper. If x, y ∈ ℝn with x = (x1 , . . . , x n ) and y = (y1 , . . . , y n ), then by x ≤ y we mean x i ≤ y i for all i = 1, . . . , n. Also we set |x| = (|x1 |, . . . , |x n |), max(x, y) = (max(x1 , y1 ), . . . , max(x n , y n )) and ℝ+n = {x ∈ ℝn : x i > 0}. If c ∈ ℝ, then x ≤ c means x i ≤ c for each i = 1, . . . , n. Definition 2.1. Let X be a nonempty set. By a vector-valued metric on X we mean a map d : X × X → ℝn with the following properties: (i) d(u, v) ≥ 0 for all u, v ∈ X; if d(u, v) = 0, then u = v. (ii) d(u, v) = d(v, u) for all u, v ∈ X. (iii) d(u, v) ≤ d(u, w) + d(w, v) for all u, v, w ∈ X. We call the pair (X, d) a generalized metric space with d1 (x, y) .. d(x, y) := ( ). . d n (x, y) Notice that d is a generalized metric space on X if and only if d i , i = 1, . . . , n, are metrics on X. For r = (r1 , . . . , r n ) ∈ ℝ+n , we will denote by B(x0 , r) = {x ∈ X : d(x0 , x) < r} = {x ∈ X : d i (x0 , x) < r i , i = 1, . . . , n} the open ball centered in x0 with radius r and B(x0 , r) = {x ∈ X : d(x0 , x) ≤ r} = {x ∈ X : d i (x0 , x) ≤ r i , i = 1, . . . , n} the closed ball centered in x0 with radius r. We mention that for a generalized metric space, the notions of open subset, closed set, convergence, Cauchy sequence and completeness are similar to those in usual metric spaces. Let (X, d) be a generalized metric space we define the following metric spaces: Let X i = X, i = 1, . . . , n. Consider ∏n X i with d:̄ i=1

n

̄ 1 , . . . , x n ), (y1 , . . . , y n )) = ∑ d i (x i , y i ). d((x i=1


M. L. Sinacer, J. J. Nieto and A. Ouahab, Random fixed point theorems

|

95

The diagonal space of ∏ni=1 X i is defined by n

̃ = {(x, . . . , x) ∈ ∏ X i : x ∈ X, i = 1, . . . , n}. X i=1

Thus it is a metric space with the following distance: n

d∗ ((x, . . . , x), (y, . . . , y)) = ∑ d i (x, y) for each x, y ∈ X. i=1

∏ni=1

̃ is closed set in It is clear that X Xi . ̃ Intuitively, X and X are equivalent. This is shown in the following result. ̃ Lemma 2.2. Let (X, d) be a generalized metric space. Then there exists a homeomorphism map h : X → X. ̃ defined by Proof. Consider h : X → X h(x) = (x, . . . , x) for all x ∈ X. Obviously, h is bijective. To prove that h is a continuous map, let x, y ∈ X. Thus n

d∗ (h(x), h(y)) ≤ ∑ d i (x, y). i=1

For ϵ > 0 we take δ = x ∈ B(x0 , δ) we have

( nϵ ,

Now we show that

h−1

...,

ϵ n ),

let x0 ∈ X be fixed and B(x0 , δ) = {x ∈ X : d(x0 , x) < δ}. Then for every d∗ (h(x0 ), h(x)) ≤ ϵ.

̃ → X defined by :X h−1 (x, . . . , x) = x

̃ (x, . . . , x) ∈ X

̃ Then is a continuous map. Let (x, . . . , x), (y, . . . , y) ∈ X. d(h−1 (x, . . . , x), h−1 (y, . . . , y)) = d(x, y). Let ϵ = (ϵ1 , . . . , ϵ n ) > 0; we take δ =

min1≤i≤n ϵ i n

̃ Set and we fix (x0 , . . . , x0 ) ∈ X.

̃ : d∗ ((x0 , . . . , x0 ), (x, . . . , x)) < δ}. B((x0 , . . . , x0 ), δ) = {(x, . . . , x) ∈ X For (x, . . . , x) ∈ B((x0 , . . . , x0 ), δ) we have n

d∗ ((x0 , . . . , x0 ), (x, . . . , x)) < δ 󳨐⇒ ∑ d i (x0 , x) < i=1

Then d i (x0 , x)
0 there is a measurable and bounded function h r ∈ L1 ([0, b], ℝ) such that |f(t, x, ω)| ≤ h r (t, ω) for a.e. t ∈ [0, b] for all ω ∈ Ω and x ∈ ℝ with |x| ≤ r. Theorem 5.3 (Carathéodory). Let X be a separable metric space and let G : Ω × X → X be a mapping such that G( ⋅ , x) is measurable for all x ∈ X and G(ω, ⋅ ) is continuous for all ω ∈ Ω. Then the map (ω, x) → G(ω, x) is jointly measurable. As a consequence of the above theorem we can easily prove the following result. Lemma 5.4. Let X be a separable generalized metric space and let G : Ω × X → X be a mapping such that G( ⋅ , x) is measurable for all x ∈ X and G(ω, ⋅ ) is continuous for all ω ∈ Ω. Then the map (ω, x) → G(ω, x) is jointly measurable. Proof. Let C be a closed subset of X. Then G−1 (C) = {(ω, x) ∈ Ω × X : G(ω, x) ∈ C} = {(ω, x) ∈ Ω × X : d(G(ω, x), C) = 0}, where d1 (x, y) .. d(x, y) = ( ), . d n (x, y)

(x, y) ∈ X × X.

Since X is a separable generalized metric space, there exists a countable dense subset D of X. Let C m = {x ∈ X : d(x, C) < ϵ m }, where

1 m

ϵ m := ( ... ) . 1 m

Hence G(ω, x) ∈ C if and only if for every m ≥ 1 there exists an a ∈ D such that d(x, a) < ϵ m and G(ω, x) ∈ C m , therefore G−1 (C) = ⋂ ⋃ {ω ∈ Ω : G(ω, a) ∈ C m } × {x ∈ X : d(x, a) < ϵ m } ∈ A × B(X). m≥1 a∈D

This implies that G( ⋅ , ⋅ ) is joint measurable.


104 | M. L. Sinacer, J. J. Nieto and A. Ouahab, Random fixed point theorems Theorem 5.5. Let f, g : [0, b] × ℝ × ℝ × Ω → ℝ be two Carathédory functions. Assume that the following condition holds: (L1 ) There exist random variables p1 , p2 , p3 , p4 : Ω → ℝ such that |f(t, x, y, ω) − f(t, x̃ , ̃y , ω)| ≤ p1 (ω)|x − x̃| + p2 (ω)|y − ̃y| and |g(t, x, y, ω) − g(t, x̃ , ̃y , ω)| ≤ p3 (ω)|x − x̃| + p4 (ω)|y − ̃y|, where M(ω) = (

bp1 (ω) bp3 (ω)

bp2 (ω) ). bp4 (ω)

If M(ω) converges to 0, then problem (5.1) has a unique random solution. Proof. Consider the operator N : C([0, b], ℝ) × C([0, b], ℝ) × Ω → C([0, b], ℝ) × C([0, b], ℝ) given by (x, y) 󳨃→ (A1 (t, x, y, ω), A2 (t, x, y, ω)), where

t

A1 (x, y, ω) = ∫ f(s, x(s, ω), y(s, ω), ω) ds + x0 (ω) 0

and

t

A2 (x, y, ω) = ∫ g(s, x(s, ω), y(s, ω), ω) ds + y0 (ω). 0

First we show that N is a random operator on C([0, b], ℝ) × C([0, b], ℝ) × Ω. Since f and g are Carathédory functions, it follows that ω → f(t, x, y, ω) and ω → g(t, x, y, ω) are measurable maps in view of Lemma 5.4. Further, the integral is a limit of a finite sum of measurable functions, therefore the following two maps are measurable: ω 󳨃→ A1 (x(t, ω), y(t, ω), ω), ω 󳨃→ A2 (x(t, ω), y(t, ω), ω). As a result, N is a random operator on C([0, b], ℝ) × C([0, b], ℝ) × Ω into C([0, b], ℝ) × C([0, b], ℝ). We show that the random operator N satisfies the conditions of Theorem 4.1 on C([0, b], ℝ)×C([0, b], ℝ). Let (x, y), (x̃ , ̃y) ∈ C([0, b], ℝ) × C([0, b], ℝ). Then t 󵄨󵄨 t 󵄨󵄨 󵄨 󵄨 |A1 (t, x(t), y(t), ω) − A1 (t, x̃(t), ̃y(t), ω)| = 󵄨󵄨󵄨󵄨 ∫ f(s, x(s, ω), y(s, ω), ω) ds − ∫ f(s, x̃(s, ω), ̃y(s, ω), ω) ds󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 0 0 t

≤ ∫ |f(s, x(s, x), y(s, ω), ω) − f(s, x̃(s, ω), ̃y(s, ω), ω)| ds 0 t

t

≤ ∫ p1 (ω)|x(s, ω) − x̃(s, ω)| ds + ∫ p2 (ω)|y(s, ω) − ̃y(s, ω)| ds, 0

0

and so ‖A1 ( ⋅ , x, y, ω) − A1 (t, x̃ , ̃y , ω))‖∞ ≤ ‖x − x̃‖∞ p1 (ω)b + ‖y − ̃y‖∞ p2 (ω)b. Similarly, we obtain ‖A2 (x, y, ω) − A2 (x̃ , ̃y , ω)‖∞ ≤ ‖x − x̃‖∞ p3 (ω)b + ‖y − ̃y‖∞ p4 (ω)b. Hence d(N(x, y, ω), N(x̃ , ̃y , ω)) ≤ ̃ M(ω)d((x, y), (x̃ , ̃y)), where d(x, y) = (

‖x − y‖∞ ) ‖x − y‖∞



|

105

and bp1 (ω) ̃ M(ω) = ( bp3 (ω)

bp2 (ω) ) = bM(ω). bp4 (ω)

From Theorem 4.2 there exists unique random solution of problem (5.1). The following result is known as the Gronwall–Bihari theorem. Lemma 5.6 ([10]). Let u, ḡ : [a, b] → ℝ be positive real continuous functions. Assume there exist c > 0 and a continuous nondecreasing function ϕ : ℝ → (0, +∞) such that t

̄ ds u(t) ≤ c + ∫ g(s)ϕ(u(s))

for all t ∈ J.

a

Then

t

̄ ds) for all t ∈ J u(t) ≤ H −1 ( ∫ g(s) a

provided +∞

b

c

a

dy ̄ ds. > ∫ g(s) ∫ ϕ(y)

Here H −1 refers to the inverse of the function u

H(u) = ∫ c

dy ϕ(y)

for u ≥ c.

We consider the following set of hypotheses in what follows: (H1 ) The functions f and g are random Carathéodory on [0, b] × ℝ × Ω. (H2 ) There exist measurable and bounded functions γ1 , γ2 : Ω → L1 ([0, b], ℝ) and a continuous and nondecreasing function ψ1 , ψ2 : ℝ+ → (0, ∞) such that |f(t, x, y)| ≤ γ1 (t, ω)ψ(|x| + |y|),

|g(t, x, y)| ≤ γ2 (t, ω)ψ2 (|x| + |y|)

a.e. t ∈ [0, b]

for all ω ∈ Ω and x, y ∈ ℝ. Now, we give a prove of the existence result of problem (5.1) by using a Leray–Schauder-type random fixed point theorem in a generalized Banach space. Theorem 5.7. Assume that hypotheses (H1 ) and (H2 ) hold. If ∞

b

∫ |γ1 (ω)| + |γ2 (ω)|
0 such that |f(t, x, y, ω)| ≤ C1 |x| + C2 |y| + C3 , |g(t, x, y, ω)| ≤ C̄ 1 |x| + C̄ 2 |y| + C̄ 3

a.e. t ∈ [0, b]

for all ω ∈ Ω and x, y ∈ ℝ. Let 1 − C1 − K1 M̄ = ( −C̄ 1

−C2 ). 1 − C̄ 2 − K2

If det M̄ > 0, then problem (6.1) has at least one random solution. Proof. Let N : C([0, 1], ℝ) × C([0, 1], ℝ) × Ω → C([0, 1], ℝ) × C([0, 1], ℝ) be an operator defined by N(x, y, ω) = F(x, y, ω) + B(x, y, ω),

(x, y, ω) ∈ C([0, 1], ℝ) × C([0, 1], ℝ) × Ω,

where F(x, y, ω) = (F1 (x, y, ω), F2 (x, y, ω)),

B(x, y, ω) = (B1 (x, y, ω), B2 (x, y, ω))

with 1

1

̄ s)f(s, x(s, ω), y(s, ω), ω) ds, F1 (t, x, y, ω) = ∫ K(t,

B1 (x, y, ω) = L1 ( ∫ x(s, ω)dα(s))t,

0

0 1

1

̄ s)g(s, x(s, ω), y(s, ω), ω) ds, , F2 (t, x, y, ω) = ∫ K(t,

B2 (x, y, ω) = L2 ( ∫ y(s, ω)dβ(s))t.

0

0

Since K1 , K2 ∈ [0, 1), it follows that

K1 M̄ = ( 0

0 ) K2

converges to zero. This implies that B is a contraction operator. By the same method used in Theorem 4.2 we can prove that F is a completely continuous operator. Now, we show that the set 󵄨󵄨 y x , , ω) = (x, y)} M = {(x, y) : Ω → C([0, 1], ℝ) × C([0, 1], ℝ) is measurable 󵄨󵄨󵄨󵄨 λ(ω)F(x, y, ω) + B( λ(ω) λ(ω) 󵄨 is bounded for some measurable mapping λ : Ω → ℝ with 0 < λ(ω)) < 1 on Ω. Let (x, y) ∈ M. Then 1

1

̄ s)f(s, x(s, ω), y(s, ω), ω) ds + λ(ω)L1 ( ∫ x(t, ω) = λ(ω) ∫ K(t, and

0

0

1

1

̄ s)g(s, x(s, ω), y(s, ω), ω) ds + λ(ω)L2 ( ∫ y(t, ω) = λ(ω) ∫ K(t, 0

0

x(s, ω) dα(s))t λ(ω)

y(s, ω) dβ(s))t. λ(ω)

Thus |x(t, ω)| ≤ C1 |x(t, ω)| + C2 |y(t, ω)| + C3 + K1 |x(t, ω)| + |L1 (0)| and |y(t, ω)| ≤ C̄ 1 |x(t, ω)| + C̄ 2 |y(t, ω)| + C̄ 3 + K2 |y(t, ω)| + |L2 (0)|.


110 | M. L. Sinacer, J. J. Nieto and A. Ouahab, Random fixed point theorems

This implies that (

1 − C1 − K1 −C̄ 1

−C2 x(t, ω) C3 + |L1 (0)| )( )≤( ̄ ). 1 − C̄ 2 − K2 y(t, ω) C3 + |L2 (0)|

Therefore x(t, ω) C3 + |L1 (0)| M̄ ( )≤( ̄ ). y(t, ω) C3 + |L2 (0)|

(6.2)

̄ −1 is order preserving. We apply (M) ̄ −1 to Since M̄ satisfies the hypotheses of Lemma 2.8, it follows that (M) both sides of inequality 6.2 to obtain (

x(t, ω) ̄ −1 (C3 + |L1 (0)|) . ) ≤ (M) y(t, ω) C̄ 3 + |L2 (0)|

Hence, from Lemma 4.9, we conclude that the operator N has at least one random fixed point which is solution of problem (6.1)

7 An example Let Ω = ℝ be equipped with the usual σ-algebra consisting of Lebesgue measurable subsets of (−∞, 0), and let J := [0, 1]. Consider the following random differential equation system: tω2 x2 (t, ω) 󸀠 { { x (t, ω) = , { { (1 + ω2 )(1 + x2 (t, ω) + y2 (t, ω)) { { { { { tω2 y2 (t, ω) { 󸀠 , y (t, ω) = { (1 + ω2 )(1 + x2 (t, ω) + y2 (t, ω)) { { { { { x(0, ω) = sin ω, { { { { { y(0, ω) = cos ω, Here f(t, x, y, ω) =

2(1 +

tω2 x2 , + x2 + y2 )

ω2 )(1

g(t, x, y, ω) =

ω ∈ Ω.

2(1 +

(7.1)

tω2 y2 . + x2 + y2 )

ω2 )(1

Clearly, the map (t, ω) 󳨃→ f(t, x, y, ω) is jointly continuous for all x, y ∈ [1, ∞); the same for the map g. Also the maps x 󳨃→ f(t, x, y, ω) and y 󳨃→ f(t, x, y, ω) are continuous for all t ∈ J and ω ∈ Ω; similarly for the maps corresponding to the function g. Thus the functions f and g are Carathéodory on J × [1, ∞) × [1, ∞) × Ω. Firstly, we show that f and g are Lipschitz functions. Indeed, let x, y ∈ ℝ. Then 󵄨󵄨 󵄨󵄨 tω2 x2 tω2 x2 󵄨 󵄨󵄨 󵄨󵄨 |f(t, x, y, ω) − f(t, x̃ , ̃y , ω)| = 󵄨󵄨󵄨󵄨 − 2 2 2 2 2 2 󵄨󵄨 ̃ ̃ 2(1 + ω )(1 + x + y ) 2(1 + ω )(1 + x + y ) 󵄨󵄨 󵄨 󵄨 󵄨󵄨 2 󵄨󵄨 tω [(1 + x̃2 + ̃y2 )x2 − (1 + x2 + y2 )x̃2 󵄨󵄨󵄨 󵄨 = 󵄨󵄨󵄨 2 2 2 2 2 󵄨󵄨 󵄨󵄨 2(1 + ω )(1 + x + y )(1 + x̃ + ̃y ) 󵄨󵄨

tω2 |x2 + ̃y2 x2 − x̃2 − y2 x̃2 | 2(1 + + + y2 )(1 + x̃2 + ̃y2 ) ω2 ω2 ̃ ≤ |x − x | + |y − ̃y|. 2(1 + ω2 ) 2(1 + ω2 )

=

ω2 )(1

Then |f(t, x, y, ω) − f(t, x̃ , ̃y , ω)| ≤

x2

ω2 ω2 |x − x̃| + |y − ̃y|. 2 2(1 + ω ) 2(1 + ω2 )

Analogously, for the function g we get |g(t, x, y, ω) − g(t, x̃ , ̃y , ω)| ≤

ω2 ω2 ̃ |x − x | + |y − ̃y|. 2(1 + ω2 ) 2(1 + ω2 )



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111

We take p1 (ω) = p2 (ω) = p3 (ω) = p4 (ω) =

ω2 2(1 + ω2 )

and M(ω) =

ω2 2(1+ω2 ) ( ω2 2(1+ω2 )

ω2 2(1+ω2 ) ω2 ) . 2(1+ω2 )

We remark that |ρ(M(ω))| =

ω2 < 1. 2(1 + ω2 )

Then M(ω) converges to 0. Therefore, all the conditions of Theorem 4.2 are satisfied. Hence problem (7.1) has a unique random solution. Acknowledgment: This paper was completed while M. L. Sinacer and A. Ouahab visited the Department of MÁthematical AnÁlysis of the University of Santiago de Compostela. They would like to thank the department for its hospitality and support. Funding: The research has partially been supported by Ministerio de Economía y Competitividad (Spain), project MTM2010-15314, and co-financed by the European Community fund FEDER.

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