Abstract

This study examines the existence of mild solutions for nonlinear random impulsive integrodifferential inclusions with time-varying delays under sufficient conditions. Our study is based on the Martelli fixed point theorem, Pachpatte’s inequality, and the fixed point theorem due to Covitz and Nadler. Besides, we generalize, extend, and develop some well-known results in the existing literature.

1. Introduction

Differential equations (DEs) with impulses are applied to simulate processes that experience rapid changes at discrete moments that is why the dynamics of impulsive DEs have drawn the interest of many academicians in recent decades (see [1–3]). Additionally, generally speaking, impulsive effects emerge as prevalent occurrences in the realm of natural phenomena, induced by sudden disturbances that transpire at precise instances, like in the case of different threshold-based biological models, bursting explosive biological medicine models, and the optimal control model in economics (for more details, see [4, 5]).

Real-world systems and natural events are virtually always affected by stochastic disruptions. Mathematical representations cannot disregard stochastic aspects due to a mix of uncertainty and complexity. To encompass a variety of abrupt incidents, incorporating impulsive occurrences, partial DEs driven by stochastic processes or equations with random impulses offer a natural and efficient approach. Many researchers have thought about the study of integro-DEs with random impulses, including those in [6–10]. Dassios explored the singular system within a category of DEs with multiple delays in his work documented in [11]. Keten et al., as referenced in [12], examined the conditions for the existence and singularity of solutions for nonlinear DEs incorporating the Caputo-Fabrizio operator in the Banach spaces, leveraging the exponential decay principle. Sakar, in [13], enhanced the homotopy analysis method by incorporating an optimally determined auxiliary parameter based on the residual error function, applied to solve neutral functional DEs with proportional delays. Shafqat et al. [14] delved into the investigation of both local and global existence as well as the singularity of mild solutions for the Navier-Stokes equations involving the time fractional differential operator. Recently, many new results related to random impulsive DEs have been obtained. For example, Yin et al. in [15] investigated existence and multiplicity of mild solutions for first-order Hamilton random impulsive DEs with the Dirichlet boundary conditions, employing the generalized saddle point theorem. Another notable contribution comes from Guo et al., as documented in [16], where they delved into the optimal control problem associated with random impulsive DEs, utilizing fundamental analytical techniques and theories related to stochastic processes. The scholarly discourse also extends to the study by Liu et al. in [17], which focuses on the existence of positive solutions within a class of impulsive fractional DEs involving -Hilfer fractional derivatives. Moreover, Li et al., featured in [18], undertook an investigation into the existence of solutions for the Sturm-Liouville DEs featuring random impulses and boundary value problems, employing Dhage’s fixed point theorem as a pivotal tool in their analysis. The existence, singularity, and stability of the delay integro-DEs with random impulse were established by [6, 7].

The existence of semilinear functional differential inclusions accompanied by random impulses of a particular nature was examined by Vinodkumar in [19] of the type

The existence of integrodifferential inclusions with impulsive was examined by Kadam et al. in [9] of the type

Motivated by the above-mentioned works and to our knowledge, no notable work yet that conducts that research has been published on the nonlinear random impulsive integrodifferential inclusions (NRIIDIns) with time-varying delays under sufficient conditions; here, we consider NRIIDIns with time-varying delays of the type where is the infinitesimal generator of strongly continuous semigroup of linear operators with , the function is the family of all nonempty measurable subsets of , and are continuous functions with . The set of piecewise continuous functions is which maps into with some given , , and for . Here, is an arbitrary real number. Obviously, is a matrix-valued function for all according to their paths with the norm for all fulfilling are given numbers, is any given norm in , and is a function defined from ; here, the set is denoted by . Denote by the simple counting process generated by , that is, , and denote the -algebra generated by . Then, is a probability space.

The basic idea behind this work is to extend the work of Vinodkumar [19] and Kadam et al. [9]. At the same time, our approach is smooth and depends on the Martelli fixed point theorem, Pachpatte’s inequality, and the fixed point theorem due to Covitz and Nadler. Moreover, we are achieving the same results with fewer hypotheses just by using Pachpatte’s inequality to establish boundedness condition.

The structure of this article is as follows: Section 2 introduces fundamental concepts and preliminary information. In Section 3, we employ Martelli’s fixed point theorem for condensing maps to explore the existence of NRIIDIns with time-varying delays, specifically focusing on the convex case of the multivalued function (MF). Our investigation delves into the problems of existence while incorporating the Wintner growth condition. In Section 4, we investigate the existence of RIFDIns in the non-convex case of the MF, employing the fixed point theorem attributed to Covitz and Nadler. Section 5 gives an illustrative example that effectively applies the insights garnered from Section 3. Finally, Section 6 provides the concluding remarks, along with acknowledgments of the study.

2. Preliminaries

Consider a real separable Hilbert space and a nonempty set . The random variable is defined from to for , where . Moreover, suppose that follow the Erlang distribution, where , and let and be independent of each other as for . For simplification, we denote . Let denote the Banach space of each -measurable square integrable random variables in . Suppose that is any fixed time and denotes the Banach space , which is the family of all -measurable, -valued random variables with the norm

The family of all -measurable, -valued random variable is denoted by .

We use the following notations: , , , and . In a Hilbert space , a multivalued map is a convex (closed) valued, if is convex (closed) is bounded on bounded sets if is bounded in that is

is called upper semicontinuous (u.s.c.) on , if . The set is nonempty, closed subset of , and if open set of containing an open neighbourhood of s.t is called completely continuous if is relatively compact,

Assuming the multivalued mapping denoted as exhibits complete continuity and has a nonempty compact value, it can be deduced that is u.s.c. if its graph is closed. A mapping is classified as condensing, being u.s.c., when applied to any bounded subset s.t the measure of noncompactness, denoted by , satisfying , where is the Kuratowski measure of noncompactness as defined in [20].

Remark 1 (see [21]). An utterly continuous multivalued function stands as the simplest instance of a condensing map. A fixed point for a function emerges when an element exists within such that resides in the range of . The collection of fixed points for the multivalued operator shall be referred to as the set fix .

Define the function defined by where and The function is called a Hausdorff metric on .

The multivalued map is called measurable if for all the function defined by

Definition 2. A multivalued operator is said to be (a)contraction such that(b) has a fixed point if s.t For additional information about multivalued maps, consult references [21–23]. Our results regarding existence are rooted in the fixed point theorems of Martelli [24] and the work of Covitz and Nadler [25].

Theorem 3 (see [24]). Let be a Hilbert space and a u.s.c. and condensing map. If the set then has a fixed point.

Theorem 4 (see [25]). Let be a Banach space. If is a contraction, then fix .

Definition 5. A map and are given functions. Moreover, , are continuous functions with . We assume the following. (1) is measurable for each (2)Each is continuous for almost all (3) ( is positive integer), s.tfor . define the set of selections of by

Lemma 6 (see [26]). Consider a compact interval denoted as and let represent a Hilbert space. Let be an Lp-Carathéodory multivalued map with and be a linear continuous mapping from , which is a closed graph operator in .

Definition 7 (see [8]). A semigroup is said to be uniformly bounded if ( is constant) s.t

The following Pachpatte’s inequalities play the crucial role in our analysis.

Lemma 8 (see [27], p. 33). Let , , and be nonnegative continuous functions defined on , for which the inequality holds, where is nonnegative constant. Then,

Definition 9. For a given , a stochastic process is said to be a solution to equation (3) in , if (1) is -adapted for (2)where and is the index function

3. Existence Result: Convex Case

Here, we list the following assumptions for our convenience.

A1. There exists a continuous function such that

, where is a continuous increasing function satisfying .

A2. There exists a continuous function such that

A3. , are continuous functions with .

A4. is uniformly bounded if there is a constant such that

A5. is the infinitesimal generator of strongly continuous .

A6. is a compact convex Lp-Carathéodory multivalued function.

Theorem 10. Under the assumptions A1-A6, problem (3) has at least one mild solution defined on provided that the following estimate is fulfilled: where , , and .

Proof. Let be an arbitrary number fulfilling (19). We transform the system (3) into a fixed point problem. We consider the operator defined as We will demonstrate that the operator satisfies all the conditions outlined in Theorem 3. The proof is presented through the following sequence of steps.
Step 1. To prove is convex, . Since has convex values, it follows that is convex, so that ; then, , which implies clearly that is convex.
Step 2. To prove is bounded on .
Let , be a bounded subset of . We prove that is a bounded subset of . , let . Then, s.t , we get Then, Hence, , we get Then, , we get .
Step 3. sends bounded sets into equicontinuous sets of .
Let be a bounded set in and . If , a function s.t , we get Thus, Then, where where , Equations (29) and (30) are independent of . This demonstrates that as approaches , the right-hand side of equation (26) approaches zero. The compact nature of for implies continuity within the uniform operator topology.
Step 4. maps bounded sets into relatively compact sets in .
Let be fixed and a real number fulfilling . For . We define a function by Since is a compact operator, the set is relatively compact in , . Additionally, , we get Using A1–A6, we get Hence, there exist precompact sets that can be brought arbitrarily close to the collection . The collection is itself precompact within the space . By combining the outcomes of steps 1 to 4 and invoking the Ascoli-Arzela theorem, it follows that qualifies as a compact multivalued map, consequently establishing its character as a condensing map.
Step 5. has a closed graph.
Let and with . We shall show that . There exists , such that Now, we prove that , s.t where
Take into account the linear continuous operator defined by Then, we get From Lemma 6, it becomes evident that the composite operator possesses a closed graph. This observation aligns with the definition of , which yields the following relationship: As and , , s.t Hence, , which follows that the graph is closed.
Step 6. A priori bounds.
Here, it remains to show that the set Let ; then for , and there exists s.t where Then, by A1-A6, we have Observing that the right-hand side’s final term in the inequality above increases in and selecting , we arrive at the conclusion that Then, From the above inequality, the last term of the right side increases in , we would obtain Let us define the function as follows: From the above inequality (47), the right side is denoted by , we get This implies where the last inequality is obtained by (19). From (57) and applying the mean value theorem, it can be deduced that there exists a constant such that and here . Since holds for every , we have , where depends only on and the functions and . is bounded, as evidenced by this. Theorem 3 leads us to conclude that has a fixed point defined on , which is a solution of (3).