Generalized Random α-ψ-Contractive Mappings with Applications to Stochastic Differential Equation

The main purpose in this paper is to define the modification form of random α-admissible and random α-ψ-contractive maps. We establish new random fixed point theorems in complete separable metric spaces. The interpretation of our results provide the main theorems of Tchier and Vetro (2017) as directed corollaries. In addition, some applications to second order random differential equations are presenred here to interpret the usability of the obtained results.


Introduction
It is well known that random fixed point theorems are stochastic generalizations of a classical (or deterministic) fixed point theorem.In a separable complete metric space (Polish space), the existence of a random fixed point for contraction mapping has been considered recently by Špaček [1] and Hanš [2,3].In atomic probability measure spaces, Mukherjea [4] introduced the random fixed point theorem versions of Schauder's fixed point theorems.Random ordinary differential equations are ordinary differential equations that include a stochastic process in their vector field.The recenly interest for the random version of some ordinary differential equations can see by Tchier and Vetro in [5].It is more realistic to consider such equations as random operator equations, which are much more difficult to handle mathematically than deterministic equations.In 1976, Bharucha-Reid [6] presented the random fixed point results to verify the unique and measurable solutions of random operator equations.In 1977, the generalized random fixed point theorems of [1] for multivalued contraction in Polish spaces and their applications for solving some random differential equation results in Banach spaces were introduced by Itoh [7].In 1984, Sehgal and Waters [8] proved the random fixed point theorem versions of the well-known Rothe's fixed point theorem.Since then, many improvements of random fixed point theorems have been established in the literature in several ways; see for example [9][10][11][12][13][14][15].It is the most widely-applied random fixed point result in different areas of mathematics, statistics, engineering, and physics, among other.Recently, we received an enormous number of applications with considerable attention in various areas such as probability theory, nonlinear analysis, and for the study of random integral and random differential equations arising in various sciences (see, [16][17][18][19][20][21][22]).
In 2012, Samet et al. [23] investigated a new concept of α-admissible and α-ψ-contractive maps and also demonstrated some fixed point results in complete metric spaces.In the same way, Karapinar and Samet [24] initiated the definition of the concepts of generalized α-ψ-contractive-type mappings.In 2013, Salimi et al. [25] generalized these notions of α-admissible and α-ψ-contractive mappings and obtained certain fixed point results.Our results are proper extensions of the recent results in [26,27].
Our goal in this work is to prove some random fixed point theorems for obtaining the generalization of random α-ψ-contractive maps in Polish spaces.By using our main results, we can assure the existence of random solutions of a second order random differential equation.

Preliminaries
We denote the Borel σ-algebra on a metric space M by B(M).Let (Ω, Σ) be a measurable space with Σ a σ-algebra of subsets of Ω.By Σ × B(M), we mean the smallest σ-algebra on Ω × M containing all the sets A × B (with A ∈ Σ and B ∈ B(M)).
Theorem 1. [28].Let (Ω, Σ) be a measurable space, M be a separable metric space, and N be a metric space.
Corollary 1. [28].Let (Ω, Σ) be a measurable space, M be a separable metric space, and N be a metric space.If f : Ω × M → N is a Carathéodory mapping and u : Definition 2. [28].Let (Ω, Σ) be a measurable space, M be a separable metric space, and N be a metric space.A function f : Ω × M → N is said to be superpositionally measurable (for short; sup-measurable), if for all u : Remark 1. [28].Corollary 1 says that a Carathéodory function is sup-measurable.Furthermore, every Σ × B(M)-measurable function f : Ω × M → N is sup-measurable.
Lemma 1. [28].Given that M and N are two locally-compact metric spaces, a mapping f : the space of all continuous functions from M into N endowed with the compact-open topology).

Main Results
Firstly, we will start this section by introducing the concept of mapping as the following definitions.
It is easy to see that if we take η(ω, u, v) = 1 in the above definition, then it can be reduced to Definition 4. Definition 7. Suppose that (Ω, Σ) is a measurable space, (M, d) is a separable space, and T : Ω × M → M is a given mapping.The mapping T is called a generalized random α-ψ-contractive map if there exists a function where: for all u, v ∈ M and ω ∈ Ω.
The following are our main results.
The argument implies that the sequence {u n (ω)} is Cauchy.Since (M, d) is a complete space, there is ζ : Ω → M such that u n (ω) → ζ(ω) as n → +∞ for all ω ∈ Ω.Since T is a Carathéodory mapping (Hypothesis (H3)), then, for all n ∈ N, u n is measurable, and also, u n+1 (ω) = T(ω, u n (ω)) → T(ω, ζ(ω)) as n → +∞ for all ω ∈ Ω.By the uniqueness of the limit, we have is a random fixed point of T. Note that ζ is measurable since it is a limit of a sequence of measurable space.
Then, T has a random fixed point; this means that ζ : Proof.A similar reason comes from the proof of the above theorem.Then, the sequence {u n (ω)} is a Cauchy sequence for all ω ∈ Ω.This means that there exists ζ : Ω → M such that u n (ω) → ζ(ω) as n → +∞ for all ω ∈ Ω.On the other hand, from (2) and Hypothesis (G5), we have: Now, using the triangle inequality (3) and (G4), we get: where: Since O(ω, (u n−1 (ω), u n (ω))) > 0, then: By taking the limit as n → ∞ in the above inequality, we have: for all ω ∈ Ω.By the hypothesis that T is sup-measurable, we see that u n for all n ∈ N is measurable and also ζ is measurable.Thus, ζ is a random fixed point of T.
The boundary value problem (4) can be written as the random integral equation: for all t ∈ [0, 1] and ω ∈ Ω. Define the random integral operator for all u ∈ C([0, 1], R) and ω ∈ Ω.Then, Problem ( 4) is equivalent to finding a random fixed point of F.
, is Carathéodory.By Lemma 1, the integral in ( 6) is the limit of a finite sum of measurable functions.Therefore, the mapping ω → F(ω, u) is measurable, and hence, F is a random operator.
Now, we prove a random fixed point of random integral operator F.
By an application of Theorem 4, we deduce that Problem (4) admits a random solution.
Hypotheses (ii) and (iii) of Theorem 4 hold true by defining θ(ω, u, v) = 1, for all ω ∈ Ω and u, v ∈ C([0, 1], R).Consequently, Hypothesis (i) is satisfied with ψ ω (t) = t 7e .Therefore, by Theorem 4, the integral operator F has a random fixed point, and the two-point boundary value problem (7) has at least one random solution.  .Therefore, by Theorem 4, the two-point boundary value problem (9) has at least one random solution.

Conclusions
We present the random version of α-ψ-contractive mappings with respect to η, previously known in complete separable metric spaces.We proved random fixed point theorems in complete separable metric spaces and proved random solutions of second order random differential equations.The presented theorems extend and improve the corresponding results given in the literature such as Tchier and Vetro, in [5].
Author Contributions: All authors read and approved the final manuscript.

Funding:
Petchra Pra Jom Klao was supported by the Doctoral Scholarship for the Ph.D. program of King Mongkut's University of Technology Thonburi (KMUTT).