Existence of Mild Solutions for Multi-Term Time-Fractional Random Integro-Differential Equations with Random Carathéodory Conditions

In this paper, we investigate the existence of mild solutions to a multi-term fractional integro-differential equation with random effects. Our results are mainly relied upon stochastic analysis, Mönch’s fixed point theorem combined with a random fixed point theorem with stochastic domain, measure of noncompactness and resolvent family theory. Under the condition that the nonlinear term is of Carathéodory type and satisfies some weakly compactness condition, we establish the existence of random mild solutions. A nontrivial example illustrating our main result is also given.


Introduction
For more than three decades, fractional calculus has played an important role in the study of linear and nonlinear fractional integro-differential equations that arise from the modeling of nonlinear phenomena, optimal control of complex systems and other scientific research (see, e.g., [1,2]). Multi-term time-fractional differential systems have also attracted a great interest in recent years, see for instance [3][4][5][6] and references cited therein. As inherently deterministic extensions, random fractional differential equations exist in many applications and have been studied by many authors and more details from historical points of view and recent developments of such equations are reported to the monographs [7,8], papers [9][10][11], and the references cited therein. To be more precise, the existence results and qualitative properties for fractional differential equations with random effects are examined in [12,13] and references cited therein. Very recently, considerable attention has been given to multi-term time-fractional differential systems. For instance, Pardo and Lizama [6] studied the existence of mild solutions under Carathéodory type conditions by using measure of noncompactness techniques, Singh and Pandey [14] have established the existence and uniqueness of mild solutions for multi-term time-fractional differential systems with non-instantaneous impulses and finite delay by using Banach fixed point theorem whereas Chang and Ponce [15], with the help of the theory of fractional resolvent families, established the existence of mild solutions to a multi-term fractional differential equation. It may be noted here that the mentioned works are confined to deterministic systems. Inspired by the aforementioned papers [6,[12][13][14], this work focuses on the existence of mild solution of problem (1) with multi-term time-fractional integrodifferential equations with random effects in the form B(t, s)ϑ(s, ω)ds, ω , ϑ(0, ω) = ϑ 0 (ω), ϑ (0, ω) = ϑ 1 (ω), (1) for 0 < t ≤ b < ∞ and ω ∈ Ω, where the state ϑ(·, ·) takes values in a separable Banach space X with norm · , (Ω, F , P) is a complete probability space, C D u stand for the Caputo fractional derive of order u > 0, 0 < β ≤ γ n ≤ · · · ≤ γ 1 ≤ 1 and α k ≥ 0, k = 1, 2, · · · n be given and A : D(A) ⊂ X → X is the infinitesimal generator of a bounded and strongly continuous cosine family. F : [0, b] × X × X × Ω → X is a random nonlinear function to be specified later. The operator B(·, ·) : ∆ → R + is a continuous operator satisfying where ∆ = {(t, s) ∈ R 2 : 0 ≤ s ≤ t < b} and ϑ 0 (·) and ϑ 1 (·) are given random functions.
To the best of our knowledge, the study of the existence of mild solutions of multi-term time-fractional integro-differential equations with random effects by the abstract form (1) has not yet been treat in the literature. The main contributions of this paper are: Firstly, the study of existence of random multi-term time-fractional integro-differential equations of the form (1) via measure of noncompactness is an untreated topic in the literature. Secondly, the nonlinear term satisfies a weak compactness condition that does not require the compactness of the resolvent family and sufficient conditions for the existence of mild solutions where the solution operators are only equicontinuous, are established by means of Mönch fixed point theorem and a random fixed point theorem with stochastic domain via the noncompactness measure. At last, our theorems guarantee the effectiveness of existence results under some weakly compactness condition and the work can considered as a supplemented for the case that the corresponding (β, γ k )-resolvent operator is compact and deterministic one. The results are established using of the (β, γ k )-resolvent operators developed in [6].
This paper is organized as follows. Section 2 contains preliminary details. In Section 3, we show the existence of random mild solutions by Mönch's fixed point theorem combined with a fixed point theorem with stochastic domain and (β, γ k )-resolvent family. A nontrivial example illustrating our main result (Theorem 2; see below) is also given.

Preliminaries
In this section, we recall some basic concepts, notations, definitions, lemmas, and preliminary facts, which are used throughout this article. We set (Ω, F , P) be a complete probability space. Let X be a separable Banach space and denote C( is measurable. First, we recall some basics definitions and properties related to random operators which are used in this paper. Definition 1. A mapping F : I × X × X × Ω → X is said to be random Carathéodory if the following hold: (a) The mapping (t, ω) → F(t, , x, y, ω) is jointly measurable for all x ∈ X and for all y ∈ X; (b) The mapping (x, y) → F(t, , x, y, ω) is jointly continuous for almost each t ∈ [0, b] and for all ω ∈ Ω; Definition 2 (see [16]). Let X be a separable Banach space with Borel σ-algebra B. A mapping Υ : Ω × X → X is called a random operator if Υ(., y) is measurable for each y ∈ X.
It is generally expressed as Υ(ω, y) = Υ(ω)y. We will use these two expressions interchangeably in this paper.
(ii) We say that Υ is continuous if every Υ(ω) is continuous.
Definition 4 (see [16]). For a random operator Υ, a mapping y : Ω → X is called a random (stochastic) fixed point of Υ if for P-almost all ω ∈ Ω, we have for every open set O ⊆ X (i.e., y is measurable).
then Υ has a stochastic fixed point.
In this work, the existence of a mild solution to problem (1) is related to the existence of resolvent family introduce by Pardo and Lizama [6].

Resolvent Family
Now, we recall some definitions and basic results on fractional calculus. Let Γ(·) denote the gamma function and define g x for x > 0 by It is known that g x satisfies the following properties: (i) for any a, b > 0, (g a g b )(t) = g a+b (t); (ii) for a, λ > 0 and Re(λ) > 0, g a (λ) = 1/λ a , where (·) and (· ·)(·) denote the Laplace transformation and convolution, respectively.
The most frequently encountered tools in the theory of fractional calculus are provided by the Riemann-Liouville and Caputo fractional differential operators.
Definition 5. The Riemann-Liouvulle fractional integral of a function f ∈ L 1 loc ([0, ∞), X) of order η > 0 with lower limit zero is defined as follows

Definition 6.
Let η > 0 be given and denote m = η . The Caputo fractional derivative of order η > 0 of a function f ∈ C m ([0, ∞), X) with lower limit zero is given by For more progress and important properties about fractional calculus and its applications, we refer the reader to [1,2] and references therein. The following definition was introduced by Pardo and Lizama [6] and provides a suitable representation of a mild solution for Problem (1) in terms of a specific family of bounded and linear operators.
Definition 7 (see [6]). Let A be a closed linear operator on a Banach space X with domain D(A) and let β > 0, γ k , α k , k = 1, 2, · · · n be real positive numbers. Then A is called the generator of a (β, γ k )-resolvent family if there exists κ ≥ 0 and a strongly continuous function R β,γ k : R + → L(X) such that where Re(λ) > κ and ϑ ∈ X.
Motivated by Pardo and Lizama [6], we introduce the concept of random mild solution for Equation (1) as follows.

Measures of Noncompactness
We recall some fundamental definitions and lemmas related to the measure of noncompactness. We introduce first the definition for Hausdorff's measure of noncompactness and its properties.
Definition 9 (see [17]). The Hausdorff measure of noncompactness χ(·) defined on bounded set E of Banach space X is χ(E) = inf{ > 0 : E can be covered by finite number of balls of radii smaller then }.
More details on the Hausdorff's measure of noncompacness can be found in Goebel [17] and Deimling [18].
The notations χ(·) and χ C (·) stand for the Hausdorff measure of noncompactness on the bounded set of X and C( The next results play an important role in demonstrating our main result. Lemma 3 (see [17]). Let V ⊂ C([0, b], X) be bounded, then Lemma 4 (see [19]). Let {Z n : n ∈ N} be a sequence of Bochner integrable functions from Definition 10 (see [20]) Then the following statements hold: To prove our existence results, we shall use the following Mönch's fixed point theorem combined with the stochastic fixed point theorem (i.e., Lemma 2). Lemma 6 (see [21]). Let V be a closed and convex subset of X and 0 ∈ V. Then a continuous mapping Q : V → X which satisies Mönch condition (i.e., W ⊆ V is countable and W ⊆ con({0} ∪ Q(W)) =⇒ W is compact) has a fixed point in V.

Some Existence Results
In this section, we investigate the existence of random mild solution for Equation (1). The following conditions will be used in our main theorem.
(H4) For a constant p 2 ∈ (0, 1) and bounded subsets V 1 , (H5) There exists a random function r : Ω → R\{0} such that The following existence theorem is one of the main results of this paper.

Theorem 2.
Assume that the assumptions (H1)-(H5) are valid, then the multi-term time-fractional integro-differential problem (1) has at least one mild random solution on [0, b] provided that Proof. It is noted that Consider the random operator N : We divide the proof into a sequence of steps.
Step 1. We show that the mapping N is a random operator with stochastic domain.
hence N is continuous.
To achieve this, we going to demontrate that the Mönch condition holds. Let ω ∈ Ω be arbitrary fixed. First, let show that N maps bounded sets into equicontinuous sets of D(ω). Let t 1 , t 2 ∈ [0, b] with t 2 > t 1 and ϑ ∈ D(ω). By the equicontinuouty of R β,γ k (t), we have By the equicontinuity of R β,γ k and Lebesgue dominated convergence theorem, we conclude that the right side of the above inequality tends to zero (independently of ϑ) as Now, let us assume that V = {ϑ (m) } ∞ m=1 be a countable subset of D(ω) and V ⊆ con({0} ∪ N (ω)(V)). Since N (ω)(V) is is bounded and equicontinuous, we have V = {ϑ (m) } ∞ m=1 is bounded and equicontinuous and therefore by Lemma 3, the function t → χ(V(t)) is continuous on [0, b]. By Lemma 4, we get By Lemma 3, we obtain From Mönch condition, we get From inequality (2), we deduce that χ(V) = 0. As a consequence of Theorem 6, we show that N has a fixed point ϑ(ω) ∈ D(ω). Since ∩ ω∈Ω D(ω) = ∅, the hypothesis that a measurable selector of int(D) holds. By Lemma 2, the random operator N has a stochastic fixed point ϑ (ω), which is a mild solution of (1). This complete the proof.
The proof can be complete similarly to Theorem 2.

Remark 1.
Consider the measure of noncompactness χ C and ν defined in C([0, b], X) by for all bounded subsets K of C([0, b], X), where ∆(K) stand for the collection of all countable subsets of K, O(t) = {x(t) , x ∈ O, t ∈ [0, b]} and L is an appropriate constant to be defined later. mod C is the modulus of equicontinuity of the function set O defined by From [20], we know that there exists a O which achieves the maximun in (5). Furthermore, the measure χ C is nonsingular, monotone and regular.
Applying the abaove regular measure of noncompactness, we obtain the following result. Proof. As in the proof of Theorem 2, we show that the operator N : D(ω) → D(ω) has a stochastic fixed point. We know that N is a random operator with stochastic domain which is continuous and maps bounded sets into equicontinuous sets of D(ω). So, in order to finish the proof it is sufficient to show that N satisfies the Mönch's condition.

Remark 2.
In comparison to Theorem 2, the result obtain in Theorem 3 is more general and interesting. Due to the choice of the measure of noncompactness, we can notice that inequality (2) in Theorem 2 is not necessary in Theorem 3.
In Theorem 3, when we replace the condition (H4) by (H3 ), we obtain the following result where the condition (H5) is released.
Proof. As in the proof of Theorem 3, there exists a random function r : Ω → R − {0} such that the operator N : D(ω) → D(ω) is a random operator with stochastic domaine D(ω) = {ϑ ∈ C([0, b], X) : ϑ ≤ r(ω)}. Furthermore, we know that N is a random operator with stochastic domain which is continuous and maps bounded sets into equicontinuous sets of D(ω). So, we only need to check that N satisfies the Mönch's condition. Following a similar argument as in the proof of Theorem 3, one can verify the conclusion. Remark 3. The random differential equation with delay is a special type of random functional differential equations. The random functional differential equations with state-dependent delay have many important applications in mathematical models of real phenomena. By applying the ideas and techniques as in this article and making some appropriate conditions, one can obtain the existence results for a class of multi-term time-fractional random integro-differential equations with state-dependent delay.

Conclusions
Random fractional integro-differential equations are one of the most important research topics in the past thirty years. In this paper, we investigate the existence of mild solutions to a multi-term fractional integro-differential Equation (1) with random effects (see . A nontrivial example illustrating Theorem 2 is also given.