Analytical Solution for a Periodic Boundary Random-value Problem Via Stochastic Fixed Points with PPF Dependence Technique

In this paper, some random common fixed point and coincidence point theorems are established with PPF dependence for generalized random contractions in a separable Banach space. Our results introduce stochastic versions and extensions of recent results as [3, 21, 25] and others. In addition, an application to establish PPF dependent solution of a periodic boundary random-valued problem is given to illustrate the usability of obtained results.


Introduction
Fixed point theorem with the PPF (past, present and future) dependence was initiated in the work of Bernfeld et al. [3], where they gave great results concerning with fixed point for the mappings which are different domains and ranges. Also, they introduced the notion of Banach type contraction and proved some important results under this contraction. Many authors worked in this direction and obtained common fixed point for pair or families of contractive mappings in metric and abstract space (see [2,5,6,8,9,10,14,18,19]) and others. These results are useful for proving the existence solutions for certain functional differential equations.
Nice work presented by Spacek [26] and Hans [12] for studying random fixed point theorems in abstract spaces, they initiated this idea to get stochastic generalizations of the deterministic fixed point theorems in separable Banach spaces. After that, Bharucha-Reid [4] introduced his paper which gave power for this theory, he attracted the attention of several mathematicians and expanded this theory. So several random fixed point theorems have been obtained in the literature. Random fixed point theorems are useful for proving the existence results for random solutions of nonlinear random equations in separable Banach spaces (see [17,22,23,24]) and others. A common assumption among all these operators in question to map an abstract space into itself, i.e. the domain and the range of the operators are the same.
Recently, Dhage [7] proved two basic random fixed point theorems for the operators in separable Banach spaces with PPF dependence and apply them to some nonlinear random differential equations for proving the existence and uniqueness of PPF dependent random solutions. After that, Hussian et al. [16] generalized and extended this results in the same space satisfying generalized random contractive conditions ofĆirić type.
for some c ∈ I. Using the above definition Parvaneh et al. [21] proved some fixed point results with PPF dependence for Hybrid rational and Suzuki-Edelstein contractions.
Continuing in this line, we prove some common random fixed point results with PPF dependence for mappings in a separable Banach space satisfying general random contractions without the continuity. Our results generalize some known results in the literature. Finally an application is given to certain nonlinear functional random differential equations for proving the existence results for random solutions with a PPF dependent.

Random fixed point results with PPF dependence
Let (Ω, X) be a measurable space and E be a separable Banach space with norm ∥.∥ E . We equip the Banach space E with a σ−algebra, χ E of Borel subset of E such that (E, χ E ) becomes a measurable space. ( where E 1 and E 2 are two Banach spaces and we denote a random operator T on for some c ∈ I and for all ω ∈ Ω. Let denote the class of all functions β : [0, +∞) → [0, 1) satisfy the following condition The random mapping T : Ω × E • → E is called a hybrid rational Geraghty random contractive if there exists β ∈ and c ∈ I such that for all ω ∈ Ω. Now, we present our first main results.

Theorem 2
Let (Ω, X) be a measurable space and let E be a separable Banach space. If the operator T : Ω × E • → E satisfy the condition of hybrid rational Geraghty random contractive (2) then the following statements hold in E: (a) If ℜ c is algebraically closed with respect to difference, then for given ξ • ∈ E • and c ∈ I, every sequence {ξ n (ω)} of measurable functions satisfying and for n ∈ N converges to a PPF dependent random fixed point of T.
(b) If ℜ c is topologically closed, then for a given ξ • ∈ E • , every sequence {ξ n (ω)} of iterates of T constructed as in (a), converges to a unique PPF dependent random fixed point ξ * (ω) of T.

This leads to
since β ∈ as in (1), then lim n→+∞ ∥ξ n−1 (ω) − ξ n (ω)∥ E• = 0, which is a contradiction, so q = 0 that is We claim that the measurable sequence {ξ n (ω)} is a Cauchy sequence in E • . Suppose the contrary, then we obtain lim m,n→+∞ By triangle inequality and since T is a hybrid rational Geraghty random contractive mapping, one can write Taking the limit as m, n → +∞ in the above inequality and applying (7), we observe that lim m,n→+∞ also lim m,n→+∞ Applying (9) and (10) in (8), it follows that which is a contradiction again. Therefore lim n,m→+∞

ANALYTICAL SOLUTION FOR A PERIODIC BOUNDARY RANDOM-VALUE PROBLEM
This shows that {ξ n (ω)} is a Cauchy sequence of measurable function on Ω to E • . Since E • is a separable Banach space hence it complete, there is a measurable function ξ * : Ω → E • such that lim n→+∞ ξ n (ω) = ξ * (ω) for all ω ∈ Ω. Now, we prove that ξ * (ω) is a random fixed point with PPF dependence of the random operator T on E • . From (2), we get Taking the limit as n → +∞ in the above inequality, yields and Applying (12) and (13) in (11), we deduce By bart (a) above, the sequence {ξ n (ω)} of measurable functions as constructed in (a) converges to a random fixed point ξ * (ω) with PPF dependence. Since ℜ c is topologically closed, then ξ * (ω) ∈ ℜ c . Consider η * (ω) ̸ = ξ * (ω) be another PPF dependent random fixed point of T for all ω ∈ Ω. Then which is a contradiction, therefore ξ * (ω) = η * (ω) for all ω ∈ Ω, this mean that the random mapping T has a unique PPF dependent fixed point in On taking β(t) = r(ω) and γ(t) = λ(ω), where r(ω) and λ(ω) are measurable functions satisfying 0 ≤ r(ω) < 1 and λ(ω) ≥ 0 in Theorem ??, we obtain the following result: Suppose that ℜ c is topologically closed and algebraically closed with respect to the difference, then T has a unique PPF dependent random fixed point ξ * (ω) ∈ ℜ c .
If we take λ(ω) = 0 in Corollary 1, we get the following corollary: for all ξ, η ∈ E • , c ∈ I and ω ∈ Ω. If ℜ c is topologically and algebraically closed with respect to the difference, then T has a unique PPF dependent random fixed point ξ * (ω) ∈ ℜ c .
Then from (14), we have where r (ω) = a (ω) + b (ω) , hence by Corollary 2, we obtain the required. Now, we prove the existence of PPF dependent random fixed point for random mapping satisfying Suzuki-Edelstein type theorem for nonlinear random contractions in Razumikhin class.

Theorem 3
Let (Ω, X) be a measurable space, E be a separable Banach space and T : Ω × E • → E is random operator. Suppose that there exists ψ ∈ Ψ such that for all ξ, η ∈ E • and for some c ∈ I. Assume that ℜ c is topologically and algebraically closed with respect to the difference, then T has a unique PPF dependent random fixed point ξ * (ω) ∈ ℜ c . Moreover for a fixed ξ • (ω) ∈ E • , if the measurable sequence {ξ n (ω)} of iterates of T be defined by T (ω, ξ n−1 (ω)) = ξ n (c, ω) for all n ∈ N and ω ∈ Ω. Therefore {ξ n (ω)} converges to a PPF dependent random fixed point of T .
If we take ψ(t) = r(ω) (where r(ω) as in Corollary 1 in Theorem 3, we obtain the following result:

Random coincidence point with PPF dependence
We begin this section with the following definitions: for some c ∈ I and ω ∈ Ω. The random operators T : Ω × E • → E and S : Ω × E • → E • are said to satisfy a condition of hybrid rational Geraghty random contraction if there exists β ∈ and c ∈ I such that for all ω ∈ Ω. Now, according to Theorem 2, we state and prove the following PPF dependent random coincidence theorem.

Theorem 4
Let (Ω, X) be a measurable space, E be a separable Banach space and T : Ω × E • → E and S : Ω × E • → E • satisfy a condition of hybrid rational Geraghty random contraction (22) such that S(ℜ c ) ⊆ ℜ c . Consider S(ℜ c ) is topologically and algebraically closed with respect to the difference. Then T and S have a PPF dependent random coincidence point in ℜ c .

Proof
The idea of the proof be defined as a new random mapping satisfy all conditions of Theorem 2 as follows. Since ζ(ω)), for all ζ ∈ C • and ω ∈ Ω. Since S | C• is one to one, then F welldefined.
Hence φ(ω) is a PPF dependent random coincidence point of S and T , this complete the proof.
According to the previous corollaries and Theorem 4, we can obtain the following results: for all ξ, η ∈ E • and ω ∈ Ω, (r (ω) , λ (ω) as in Corollary 1 where Let S(ℜ c ) is topologically and algebraically closed with respect to the difference. Then T and S have a PPF dependent random coincidence point.

Corollary 6
Let T : Ω × E • → E and S : Ω × E • → E • be two random mappings, there exists c ∈ I such that S(ℜ c ) ⊆ ℜ c and is topologically and algebraically closed with respect to the difference. Then T and S have a PPF dependent random coincidence point.

Corollary 7
for all ξ, η ∈ E • and a(ω), b(ω) as in Corollary 3. Let S(ℜ c ) is topologically and algebraically closed with respect to the difference. Then T and S have a PPF dependent random coincidence point.

Corollary 8
Let (Ω, X) be a measurable space, E be a separable Banach space, T : Ω × E • → E and S : Ω × E • → E • be two random mappings such that S(ℜ c ) ⊆ ℜ c and for all ξ, η ∈ E • , where ψ ∈ Ψ. Let S(ℜ c ) is topologically and algebraically closed with respect to the difference. Then T and S have a PPF dependent random coincidence point.

Corollary 9
Let T : Ω × E • → E and S : Ω × E • → E • be two random mappings such that S(ℜ c ) ⊆ ℜ c and for all ξ, η ∈ E • , where 0 ≤ r(ω) < 1. Let S(ℜ c ) is topologically and algebraically closed with respect to the difference. Then T and S have a PPF dependent random coincidence point.
Note that the random operator T in Theorems 2-3 and Corollaries 1-4 and the random pair (S, T ) in Theorems 4-8 and Corollaries 5-9 are not required to satisfy any continuity condition on the domains of their definition.

Application to random differential and integral equations
Fixed point theorems have many applications in many mathematical disciplines especially in differential and integral equations (see [1,13,15,20]. In this section, we shall prove the existence of PPF dependence solution to a periodic boundary random-valued problem type (in short PBRVP). Let It's clear that ℵ is a Banach space with this norm. For t ∈ I define a function t → x t ∈ ℵ by where the argument s represents the delay in the argument solution. Let (Ω, X) be a measurable space. Define a mapping x : Ω → C(J, R), we denote a function x(t, ω), which is continuous in the variable t for each ω ∈ Ω, we also write x(t, ω) = x(ω)(t). Given the measurable functions φ : Ω → ℵ and x : Ω → C(I, R), consider the first-order periodic boundary random-valued problem (PBRVP) for all t ∈ I and ω ∈ Ω, where f : I × R × ℵ → R. By a random solution x of PBRVP (23) we mean a measurable function x : Ω → C(J, R) that satisfies the equation (23)  In this fashion, we will prove the existence of random solutions with PPF dependence for the PBRVP (23) defined on J with the condition of Theorem 3. We consider the following hypotheses: (H 1 ) The function ω → f (t, x, ω) is measurable for each t ∈ I and x ∈ ℵ and the function (t, x) → f (t, x, ω) is jointly continuous for each ω ∈ Ω. (H 2 ) Assume that there exists λ > 0 such that for each ∼ x, ∼ y : Ω → C(J, R) and ξ, η ∈ ℵ with

Theorem 5
Suppose the conditions (H 1 ) and (H 2 ) hold, then PBRVP (23) has a unique PPF dependent random solution defined on J.

Proof
Suppose that E = C(J, R) which is a separable Banach space. Given a function x : Ω → C(J, R), define a mapping We define a norm on where t ∈ I. Using variation of parameters formula, we get which yields Since x(0, ω) = x(T, ω), we have Applying (25) in (24), we can write where First, we show that S is a random operator on Ω × ∼ E. Since hypothesis (H 1 ) holds, by Caratheodory theorem, the function ω → f (t, x, ω) is measurable for all t ∈ I and x ∈ ℵ. As integral is the limit of the finite sum of measurable function, the map Secondly, we claim that the random operator S is continuous on x * (t, ω)) = ∼ (x * (t, ω)) t∈I . This yields the functional random integral equation (26) has a random solution with PPF dependence defined on J which implies that the PBRVP (23) has a PPF dependent random solution. Moreover, here the Razumikhin class ℜ 0 , 0 ∈ [−r, T ] is C([0, T ], R) which is topologically and algebraically closed with respect to the difference, so this solution is unique.

Conclusion
In this paper we introduced some random fixed point theorems with PPF (past, present and future) in a separable Banach space under hybrid rational and Suzuki Edelstein random contractions type without using the continuity condition of the mappings. The new approach improves, extends and generalizes the existing results in the literature of the fixed point theory. As an application, we give the existence of random solution to certain nonlinear functional random differentiable equations with a PPF dependent.