Some notes on a second-order random boundary value problem ∗

for all ω ∈ Ω, where f : Ω× [0, 1]×R→ R has certain regularities and Ω is a nonempty set. By a random solution of system (1), we mean a measurable mapping u : Ω → C([0, 1],R) satisfying (1), whereC([0, 1],R) denote the space of all continuous functions defined on [0, 1]. The interest for the random version of well-known ordinary differential equations is motivated by the necessity to model and understand certain nonspecific dynamic processes of natural phenomena arising in the applied sciences; see the books of Bharucha-Reid [2] and Skorohod [13]. For some interesting contributions to this problem,

By a random solution of system (1), we mean a measurable mapping u : Ω → C([0, 1], R) satisfying (1), where C([0, 1], R) denote the space of all continuous functions defined on [0, 1].The interest for the random version of well-known ordinary differential equations is motivated by the necessity to model and understand certain nonspecific dynamic processes of natural phenomena arising in the applied sciences; see the books of Bharucha-Reid [2] and Skorohod [13].For some interesting contributions to this problem, see Itoh [4], Li and Duan [5], Papageorgiou [9], Sinacer et al. [12], Tchier et al. [14].Clearly, in absence of ω, system (1) reduces to The approach developed in this paper uses a combination of classical tools based on Green's functions theory and fixed-point theorems for operators in Banach spaces.Indeed, we recall that the Green's function associated to (2) is given by The space C([0, 1], R), endowed with the metric x(t) − y(t) , is a complete metric space.In this setting, Samet et al. [11] investigated the solvability of system (2) by using a new concept of α-ψ-contractive type mapping, which generalizes the Banach contraction in [1] and many others fixed-point theorems in the literature (see, for example, Nieto and Rodríguez-López [7] and Ran and Reurings [10]).Motivated by [11], we propose a study of system (1).Precisely, we extend the original notion of α-admissible mapping to work with measurable mappings, then we give a random version of the main result in [11], finally, we establish the existence of at least one solution for system (1).The interesting feature of our work is that we do not impose contractive conditions to all points of the involved space, but just to the ones satisfying a specific inequality relation (defined by using a given function α; see Definition 2 below).This means that we enlarge the class of operators such that our results apply.Also, by appropriate choices of the function α, we are able to control the whole process (as shown by the proofs of the results).

Mathematical background
Here we give some concepts and notations from the existing literature.We denote the Borel σ-algebra on a metric space X by B(X).Let (Ω, Σ) be a measurable space so that by Σ ⊗ B(X) we mean the smallest σ-algebra on Ω × X containing all the sets M × B (with M ∈ Σ and B ∈ B(X)).
We recall a definition that we need in the statement of the main theorem.
Definition 1.Let (Ω, Σ) be a measurable space, X and Y be two metric spaces.A mapping Let (Ω, Σ) be a measurable space, X be a separable metric space, and Y be a metric space.A mapping h : Ω × X → Y is said to be superpositionally measurable (supmeasurable for short) if, for all Σ-measurable mapping u : Ω → X, the mapping ω → h(ω, u(ω)) is Σ-measurable from Ω into Y .
Moreover, a mapping f : Ω × X → X is called random operator whenever, for any , the space of all continuous functions from X into Y endowed with the compact-open topology).
Definition 2. Let (Ω, Σ) be a measurable space, (X, d) be a metric space, and T : Ω × X → X be a given mapping.We say that T is a random α-ψ-contractive mapping if there exist functions α : for all u, v ∈ X and ω ∈ Ω such that α(ω, u, v) 1. https://www.mii.vu.lt/NA Remark 1. From ( 4) we retrieve the random version of the Banach contraction condition whenever To show the role of function α : Ω × X × X → [0, +∞), we give the following example (see also Examples 2.1 and 2.2 in [11]).
Example 1.In both the following cases, T : Ω × X → X is a random α-admissible mapping.

Random fixed point results
In this section, we prove the existence of a random fixed point for a given mapping.Let (Ω, Σ) be a measurable space, (X, d) be a separable complete metric space, T : Ω × X → X and α : Ω × X × X → [0, +∞).The hypotheses are the following: (H1) T is a random α-admissible mapping; (H2) There exists a measurable mapping u Let Ω, X, T , and α be as in (ii) of Example 1.If u 0 : Ω → X is defined by u 0 (ω 1 ) = u 0 (ω 2 ) = 1, then hypothesis (H2) holds true for each Σ.
In the next theorem, we replace hypothesis (H3) (T is Carathéodory) by hypothesis (H5), which is a regularity condition on the metric space (X, d).
Proof.A similar reasoning as in the proof of Theorem 2 gives us that the sequence {u n (ω)} is Cauchy in the complete metric space (X, d) for all ω ∈ Ω.This means that there exists ξ : Ω → X such that u n (ω) → ξ(ω) as n → +∞ for all ω ∈ Ω.On the other hand, from (5) and hypothesis (H5), we have Now, using the triangle inequality, (4), and ( 6), we get Taking the limit as n → +∞ and since ψ ω is continuous at t = 0, we have The hypothesis that T is supmeasurable implies that u n is measurable for all n ∈ N and hence ξ is measurable.Thus, ξ is a random fixed point of T .
We have discussed the existence of a random fixed point for a given mapping under suitable hypotheses.Next, the question of uniqueness also applies to our study.So, to obtain a unique random fixed point, we consider the following hypothesis: (H6) For all u, v ∈ X and ω ∈ Ω, there exists z(ω) ∈ X such that α(ω, u, z(ω)) 1 and α(ω, v, z(ω)) 1.
https://www.mii.vu.lt/NATheorem 4. Adding hypothesis (H6) to the ones in the statement of Theorem 2 (resp. Theorem 3), we obtain uniqueness of the random fixed point of T .
An interesting feature of the function α is the fact that it is suitable to obtain easily the ordered counterparts of many fixed-point theorems without requiring any change to the proofs of previous theorems.We recall that the study of fixed points in partially ordered spaces has been considered in Ran and Reurings [10] and further investigated in a lot of papers (see, for example, Nieto and Rodríguez-López [7,8], Vetro [15], and references therein).The ordered approach is significant and largely motivated by nice applications to matrix equations (see [10]) and boundary value problems (see [7,8]).
So, our results can be immediately read in an ordered context by defining α : for all ω ∈ Ω, where denote an order relation on the set X (two elements u, v ∈ X are called comparable if u v or v u).In this case, hypotheses (H1)-(H6) reduce to (O1) T is a nondecreasing mapping w.r.t. ; (O2) There exists a measurable mapping u 0 : Ω → X such that u 0 (ω) T (ω, u 0 (ω)) for all ω ∈ Ω; (O3) T is a Carathéodory mapping; (O4) There exists function ψ ω ∈ Ψ , ω ∈ Ω such that d(T (ω, u), T (ω, v)) ψ ω (d(u, v)) for all u, v ∈ X such that u v; (O5) If {u n } is a sequence in X such that u n u n+1 for all n ∈ N ∪ {0} and u n → u ∈ X as n → +∞, then u n u for all n ∈ N ∪ {0}; (O6) For all u, v ∈ X, there exists z ∈ X such that u z and v z.