PERIODIC BOUNDARY VALUE PROBLEMS OF SECOND ORDER RANDOM DIFFERENTIAL EQUATIONS

In this paper, an existence and the existence of extremal random solutions are proved for a periodic boundary value problem of second order ordinary random differential equations. Our investigations have been placed in the space of real-valued functions defined and continuous on closed and bounded intervals of real line together with the applications of the random version of a nonlinear alternative of Leray-Schauder type and an algebraic random fixed point theorem of Dhage [5]. An example is also indicated for demonstrating the realizations of the abstract theory developed in this paper.


Statement of the Problem
Let R denote the real line and let J = [0, 2π] be a closed and bounded interval in R. Let C 1 (J, R) denote the class of real-valued functions defined and continuously differentiable on J. Given a measurable space (Ω, A) and for a given measurable function x : Ω → C 1 (J, R), consider a second order periodic boundary value problem of ordinary random differential equations (in short PBVP) −x (t, ω) = f (t, x(t, ω), ω) a.e.t ∈ J x(0, ω) = x(2π, ω), x (0, ω) = x (2π, ω) for all ω ∈ Ω, where f : By a random solution of the random PBVP (1) we mean a measurable function x : Ω → AC 1 (J, R) that satisfies the equations in (1), where AC 1 (J, R) is the space of EJQTDE, 2009 No. 21, p. 1 continuous real-valued functions whose first derivative exists and is absolutely continuous on J.
The random PBVP ( 1) is new to the theory of periodic boundary value problems of ordinary differential equations.When the random parameter ω is absent, the random PBVP (1) reduces to the classical PBVP of second order ordinary differential equations, where, f : The study of PBVP (2) has been discussed in several papers by many authors for different aspects of the solutions.See for example, Lakshmikantham and Leela [12], Leela [13], Nieto [14,15], Yao [16], and the references therein.In this paper, we discuss the random PBVP (1) for existence as well as for existence of extremal solutions under suitable conditions of the nonlinearity f which thereby generalize several existence results of the PBVP (2) proved in the above mentioned paper.Our analysis rely on the random versions of nonlinear alternative of Leray-Schauder type (see Dhage [5,6]) and an algebraic random fixed point theorem of Dhage [5].
The paper is organized as follows: In Section 2 we give some preliminaries and definitions needed in the sequel.The main existence result is given in Section 3, while the result on extremal solutions is given in Section 4. Finally, in Section 5, an example is presented to illustrate the abstract result developed in Section 3.

Auxiliary Results
Let E denote a Banach space with the norm • and let is totally bounded subset of E for any bounded subset B of E. Q is called completely continuous if is continuous and totally bounded on E. Note that every compact operator is totally bounded, but the converse may not be true.However, both the notions coincide on bounded sets in the Banach space E.
We further assume that the Banach space E is separable, i.e., E has a countable dense subset and let β E be the σ-algebra of Borel subsets of E. We say a mapping where A ×β E is the direct product of the σalgebras A and β E those defined in Ω and E respectively.The details of the measurablity of the functions appears in Himmelberg EJQTDE, 2009 No. 21, p. 2 [9].Note that a continuous map f from a Banach space E into itself is measurable, but the converse may not be true.
Let Q : Ω × E → E be a mapping.Then Q is called a random operator if Q(ω, x) is measurable in ω for all x ∈ E and it is expressed as Q(ω)x = Q(ω, x).In this case we also say that Q(ω) is a random operator on E. A random operator Q(ω) on E is called continuous (resp.compact, totally bounded and completely continuous) if Q(ω, x) is continuous (resp.compact, totally bounded and completely continuous) in x for all ω ∈ Ω.The details of completely continuous random operators in Banach spaces and their properties appear in Itoh [10].The study of random equations and their solutions have been discussed in Bharucha-Reid [1] and Hans [7] which is further applied to different types of random equations such as random differential and random integral equations etc. See Itoh [10], Bharucha-Reid [2] and the references therein.In this paper, we employ the following random nonlinear alternative in proving the main result of this paper.
Theorem 2.1 (Dhage [5,6]) Let U be a non-empty, open and bounded subset of the separable Banach space E such that 0 ∈ U and let Q : Ω × U → E be a compact and continuous random operator.Further suppose that there does not exists an u ∈ ∂U such that Q(ω)x = αx for all ω ∈ Ω, where α > 1 and ∂U is the boundary of U in E.. Then the random equation Q(ω)x = x has a random solution, i.e., there is a measurable function ξ : Ω → E such that Q(ω)ξ(ω) = ξ(ω) for all ω ∈ Ω.
An immediate corollary to above theorem in applicable form is Corollary 2.1 Let B r (0) and B r (0) be the open and closed balls centered at origin of radius r in the separable Banach space E and let Q : Ω × B r (0) → E be a compact and continuous random operator.Further suppose that there does not exists an u ∈ E with u = r such that Q(ω)u = αu for all ω ∈ Ω, where α > 1.Then the random equation Q(ω)x = x has a random solution, i.e., there is a measurable function The following theorem is often used in the study of nonlinear discontinuous random differential equations.We also need this result in the subsequent part of this paper.
The following lemma appears in Nieto [15] and is useful in the study of second order periodic boundary value problems of ordinary differential equations.EJQTDE, 2009 No. 21, p. 3 Lemma 2.1 For any real number m > 0 and σ ∈ L 1 (J, R), x is a solution to the differential equation if and only if it is a solution of the integral equation where, Notice that the Green's function G m is continuous and nonnegative on J × J and the numbers and exist for all positive real number m.
We need the following definitions in the sequel.
for each real number r > 0 there is a measurable and bounded function q r : Ω → We seek the random solutions of PBVP (1) in the Banach space C(J, R) of continuous real-valued functions defined on J.We equip the space C(J, R) with the supremum norm • defined by It is known that the Banach space C(J, R) is separable.By L 1 (J, R) we denote the space of Lebesgue measurable real-valued functions defined on R + .By • L 1 we denote the usual norm in L 1 (J, R) defined by For a given real number m > 0, consider the random PBVP, for all ω ∈ Ω, where the function Remark 3.1 We remark that the random PBVP (1) is equivalent to the random PBVP (6) and therefore, a random solution to the PBVP (6) implies the random solution to the PBVP (1) and vice versa.
The random PBVP ( 6) is equivalent to the random integral equation, for all t ∈ J and ω ∈ Ω, where the function G m (t, s) is defined by (5).
We consider the following set of hypotheses in what follows: for all ω ∈ Ω.Then the random PBVP (1) has a random solution defined on J.
Proof.Set E = C(J, R) and define a mapping for all t ∈ J and ω ∈ Ω.
Since the map t → G m (t, s) is continuous on J, Q(ω) defines a mapping Q : Ω × E → E. Define a closed ball B r (0) in E centered at origin 0 of radius r, where the real number r satisfies the inequality (8).We show that Q satisfies all the conditions of Corollary 2.1 on B r (0).
First we show that Q is a random operator on B r (0).Since f m (t, x, ω) is random Carathéodory, the map ω → f m (t, x, ω) is measurable in view of Theorem 2.2.Similarly, the product G m (t, s)f m (s, x(s, ω), ω) of a continuous and a measurable function is again measurable.Further, the integral is a limit of a finite sum of measurable functions, therefore, the map Next we show that the random operator Q(ω) is continuous on B r (0).Let {x n } be a sequence of points in B r (0) converging to the point x in B r (0).Then it is enough to prove that lim n→∞ Q(ω)x n (t) = Q(ω)x(t) for all t ∈ J and ω ∈ Ω.By dominated convergence theorem, we obtain: for all t ∈ J and ω ∈ Ω.This shows that Q(ω) is a continuous random operator on B r (0).Now, we show that Q(ω) is a compact random operator on B r (0).To finish, it is enough to prove that Q(ω)(B r (0)) is uniformly bounded and equi-continuous set in E EJQTDE, 2009 No. 21, p. 6 for each ω ∈ Ω.Since the map ω → γ(t, ω) is bounded and L 2 (J, R) ⊂ L 1 (J, R), by hypothesis (H 2 ), there is a constant c such that γ(ω) L 1 ≤ c for all ω ∈ Ω.Let ω ∈ Ω be fixed.Then for any x : Ω → B r (0), one has ) is an equi-continuous set in E. Let x ∈ B r (0) be arbitrary.Then for any t 1 , t 2 ∈ J, one has Hence for all t 1 , t 2 ∈ J, uniformly for all x ∈ B r (0).Therefore, Q(ω)(B r (0)) is an equi-continuous set in E.

Extremal Random Solutions
A closed set K of the Banach space E is called a cone if where θ is the zero element of E. We introduce an order relation ≤ in E with the help of the cone K in E as follows.Let x, y ∈ E, then we define The details of different types of cones and their properties appear in Deimling [3] and Heikkilä and Lakshmikantham.
We introduce an order relation ≤ in C(J, R) with the help of a cone K in it defined by Thus, we have x ≤ y ⇐⇒ x(t) ≤ y(t) for all t ∈ J.
for all ω ∈ Ω and for all x, y ∈ E for which x ≤ y.
We use the following random fixed point theorem of Dhage [4,5] in what follows.We need the following definitions in the sequel.
for all ω ∈ Ω.Similarly, a measurable function b : Ω → C(J, R) is called an upper random solution for the PBVP (1) if for all ω ∈ Ω.
EJQTDE, 2009 No. 21, p. 9 Note that a random solution for the random PBVP ( 1) is lower as well as upper random solution for the random PBVP (1) defined on J.
Remark 4.1 We remark that lower and upper random solutions to the PBVP (1) implies respectively the lower and upper random solutions to the PBVP (6) on J and vice versa.
Definition 4.3 A random solution r M for the random PBVP ( 1) is called maximal if for all random solutions of the random PBVP (1), one has x(t, ω) ≤ r M (t, ω) for all t ∈ J and ω ∈ Ω.Similarly, a minimal random solution to the PBVP (1) on J is defined.
Remark 4.2 We remark that maximal and minimal random solutions to the PBVP (1) implies respectively the maximal and minimal random solutions to the PBVP (6) on J and vice versa.
) is continuous and nondecreasing for all t ∈ J and ω ∈ Ω.
Definition 4.5 A function f (t, x, ω) is called random L 1 -Chandrabhan if for each real number r > 0 there exists a measurable function q r : Ω → L 1 (J, R) such that for all ω ∈ Ω |f (t, x, ω)| ≤ q r (t, ω)a.e.t ∈ J for all x ∈ R with |x| ≤ r.
We consider the following set of assumptions: (H 4 ) The PBVP (1) has a lower random solution a and upper random solution b with a ≤ b on J.
(H 5 ) The function q : J × Ω → R + defined by   Next, since (H 5 ) holds, the hypothesis (H 2 ) is satisfied with γ(t, ω) = q(t, ω) for all (t, ω) ∈ J × Ω and ψ(r) = 1 for all real number r ≥ 0. Now it can be shown as in the proof of Theorem 3.1 that the random operator Q(ω) is completely continuous on [a, b] into itself.Thus, the random operator Q(ω) satisfies all the conditions of Theorem 4.1 and so the random operator equation Q(ω)x = x(ω) has a least and a greatest random solution in [a, b].Consequently, the PBVP (1) has a minimal and a maximal random solution defined on J.This completes the proof.To see this, let hypothesis (H 6 ) hold.Since the cone K in C(J, R) is normal, the random order interval [a, b] is norm-bounded.Hence there is a real number r > 0 such that x ≤ r for all x ∈ [a, b].Now f m is L 1 -Chandrabhan, so there is a measurable function q r : Ω → C(J, R) such that |f m (t, x, ω)| ≤ q r (t, ω)a.e.t ∈ J for all x ∈ R with |x| ≤ r and for all ω ∈ Ω.Hence, hypotheses (H 3 ) and (H 5 ) hold with q(t, ω) = q r (t, ω) for all t ∈ J and ω ∈ Ω. EJQTDE, 2009 No. 21, p. 12

Theorem 4 . 1 (
Dhage [4]) Let (Ω, A) be a measurable space and let [a, b] be a random order interval in the separable Banach space E. Let Q : Ω×[a, b] → [a, b] be a completely continuous and nondecreasing random operator.Then Q has a least fixed point x * and a greatest random fixed point y * in [a, b].Moreover, the sequences {Q(ω)x n } with x 0 = a and {Q(ω)y n } with y 0 = a converge to x * and y * respectively.

2π 0 G
m (t, ω)f m (s, x n (s, ω), ω) ds, n ≥ 0 with x 0 = a, andy n+1 (t, ω) = 2π 0 G m (t, ω)f m (s, y n (s, ω), ω) ds, n ≥ 0 with y 0 = b for all t ∈ J and ω ∈ Ω. Proof.Set E = C(J, R) and define an operator Q : Ω × [a, b] → E by(9).We show that Q satisfies all the conditions of Theorem 4.1 on [a, b].It can be shown as in the proof of Theorem 3.1 that Q is a random operator on Ω × [a, b].We show that it is L 1 -Chandrabhan.First we show that Q(ω) is nondecreasing on [a, b].Let x, y : Ω → [a, b] be arbitrary such that x ≤ y on Ω.Then, Q(ω)x(t) ≤

Remark 4 . 5
The conclusion of the Theorem 4.2 also remains true if we replace the hypotheses (H 3 ) and (H 5 ) with the following one.(H6 ) The function f m is random L 1 -Chandrabhan on J × R × Ω.

)
EJQTDE, 2009No.21, p. 8 A cone K in the Banach space E is called normal, if the norm • is semi-monotone on K i.e., if x, y ∈ K, then x + y ≤ x + y .Again a cone K is called regular, if every monotone order bounded sequence in E converges in norm.Similarly, a cone K is called fully regular, if every monotone norm bounded sequence converges in E.