Analysis of Stochastic Weighted Impulsive Neutral <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.52687pt" id="M1" height="13.4804pt" version="1.1" viewBox="-0.0657574 -8.95353 11.3962 13.4804" width="11.3962pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-245" d="M642 419L635 448C586 435 552 411 532 392C518 379 509 351 496 304C484 260 462 196 445 150C424 93 388 40 314 30L413 508L406 510L344 491L257 36C205 46 175 84 175 169C175 195 180 241 185 277C189 304 190 331 190 358C190 413 175 448 141 448C111 448 69 429 23 373L30 347C51 365 68 375 81 375C93 375 108 368 108 327C108 307 104 268 101 239C98 211 95 180 95 155C95 33 159 -8 248 -14L213 -186C206 -220 221 -254 230 -261L261 -230L306 -11C339 -4 366 6 389 20C431 46 473 87 503 155C533 224 557 286 571 325C586 367 606 393 642 419Z"/></g></svg>-</span>Hilfer Integro-Fractional Differential System with Delay

Ahmad, Manzoor; Zada, Akbar; Ahmad, Jihad; Abd El Salam, Mohamed A.

doi:https://doi.org/10.1155/2022/1490583

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2022 | Article ID 1490583 | https://doi.org/10.1155/2022/1490583

Analysis of Stochastic Weighted Impulsive Neutral -Hilfer Integro-Fractional Differential System with Delay

Manzoor Ahmad,¹Akbar Zada,¹Jihad Ahmad,²and Mohamed A. Abd El Salam³

Academic Editor: Ali Ahmadian

Received13 Nov 2021

Accepted13 Feb 2022

Published19 Mar 2022

Abstract

This paper is devoted to the study of stochastic fractional differential equations. Particularly, we study the existence and uniqueness of solutions of nonlinear stochastic weighted impulsive -Hilfer neutral integro-fractional differential system with infinite delay. We obtain the results with the help of fixed-point theorems and Banach contraction principle. Additionally, we investigate the Ulam–Hyers stability of the solutions using basic schemes of fractional calculus. For application of the theory, we add an example.

1. Introduction

Fractional calculus (FC) is the generalization of integer orders differentiations and integrations to fractional orders [1]. It is observed that many interdisciplinary applications can be modeled elegantly with the help of fractional differential operators. The fractional parameters allow a class of characterization of several eventual uncertainties presented in the dynamical models [2]. Fractional differential models are considered to be more accurate as compared to integer order differential models [3]. Many researchers have elaborated applications of fractional derivatives [4] in the time-dependent damping behavior of several viscoelastic behavior science. Furthermore, FC has also great applications in nonlinear oscillation of earthquake [5], fluid dynamics [6–12], traffic models [13], seepage flow in porous media [14] etc. Repudiating fractional differential equations (FDEs) is like saying that irrational numbers do not exist.

Researchers are investigating and analyzing FDEs with different approaches. Very recently Sousa and Olivera [15] extended the concept of Hilfer fractional derivative to -Hilfer fractional derivative and discussed its important properties and applications. The -Hilfer fractional derivative is quite different from other classical fractional derivatives because the kernel is in the form of function. Moreover, it has been proved that -Hilfer fractional derivative generalizes many existing fractional derivatives. FDEs in which the highest fractional derivatives of unknown term appear both with and without delays are known as neutral FDEs. In the last few years, the study of neutral FDEs has developed dramatically. This is due to the fact that the qualitative behavior of neutral FDEs is quite different from those of nonneutral FDEs. Neutral FDEs also play an important role and have many applications. For instance, it gives more better sketch of population fluctuations. Also, neutral FDEs with delay appear in models of electrical networks containing lossless transmission lines [16].

In the qualitative behavior of dynamical systems, the concept of stability is very important because every workable control system is designed to be stable. In physical applications, stability of solutions is very meaningful because deviations in mathematical models inevitably result from errors in measurement. A stable solution can become unstable due to such deviations. If the water flow in a sea is not stable, then jumping of an ant into the sea will probable cause flood. The investigations of stability properties of solutions have attracted many researchers through its potential applications. Especially, the Ulam–Hyers (UH) stability analysis and its applications have been studied by many researchers [17]. The definition of UH stability has applicable significance since it means that if one is investigating the UH stability, then one does not require to reach to the exact solution. It guarantees that there is always exact solution to every approximate solution [18]. This is quite helpful in many fields such as numerical analysis [19], economics [20], optimizations theory [21], and biology [22], where finding the exact solution is very difficult or time consuming, see [23].

Due to lack of investigations on the existence and UH stability to the solutions of weighted impulsive implicit neutral -Hilfer stochastic integro FDEs in the literature, it is of great interest to perform some investigations in this area. Y. Guo et al. [24] investigated the impulsive neutral functional stochastic differential equation for existence and UH stability given bywhere represent the Riemann–Liouville fractional derivative of order , . The functions are continuous and is Wiener process, and is a phase space. The notations represent appropriate functions, are Riemann–Liouville fractional integrals having orders and , respectively. and are defined by

The histories is defined by , .

Sathiyaraj et al. [25] studied Ulam–Hyers stability results to the solutions of the following fractional stochastic differential system involving Hilfer fractional derivative:where represent the Hilfer fractional derivative of order and type , , and is a matrix of dimension . The nonlinear functions and are continuous.

There exists a vast literature over existence and uniqueness (EU) of solutions related to FDEs involving Caputo or Riemann–Liouville fractional derivatives, etc., while in setting of nonlinear weighted impulsive implicit neutral -Hilfer stochastic integro FDEs, as far as we know, existence, uniqueness, and stability have not been discussed.

Motivated by the work discussed above, in this study, we study the EU and Ulam–Hyers stability of the following nonlinear weighted impulsive neutral -Hilfer stochastic integro-FDEs’ system:where represent the -Hilfer fractional derivative of order and type , is the Riemann–Liouville fractional left-sided integral of order , and is the increasing function with , , ,

The functions , and are appropriate functions, is a phase space of mappings from into Hilbert space , , with , for and norm . Furthermore, is a Wiener process, . The histories is defined by , .

The rest of this study is arranged into the following way. In Section 2, we recall some helpful definitions and results relating to both fractional derivatives and fractional integrals. The EU of solutions to the considered system (4) is discussed in Section 3. Ulam–Hyers stability result is established in Section 4. A particular example is given in Section 5.

2. Preliminaries

This portion is dedicated to introduce some notions, definitions, and preliminary results.

Throughout this study, let denote the complete filtered probability space such that contains all null sets of . Also, is supposed to be -Wiener process on the probability space with the covariance operator satisfying . Consider and as two separable Hilbert spaces and as the space of all bounded linear operators from into . The notation denotes the same norms in , , and ; we also denote the inner product of and by (.,.). Furthermore, we assume that there exists a complete orthonormal system in and a bounded sequence of nonnegative real numbers such that and a sequence of independent Brownian motion satisfying

Let and , for .

Consider the space is continuous everywhere except some points at which and do exist and , . We also introduce the spaces:andwith norms

where is mathematical expectation.

Before coming to fractional order neutral differential equations, we define the abstract phase space . Let be a continuous function satisfying . The Hilbert space induced by is given byendowed with norm,

Define the space:where is the restriction of into and .

We use the notation for the seminorm in the space defined bywhere

Definition 1. (see [15]). Let and be increasing, positive monotone function on such that is continuous on ; then, the -Riemann–Liouville fractional integral of a function with respect to another function is defined as

Definition 2. (see [15]). The -Hilfer fractional derivative of a function of order and type is given by

Lemma 1 (see [15]). Let . Then,Ifthen

Lemma 2 (see [15]). If , and , then

Lemma 3 (see [15]). Let and be continuous; then, for any and a function defined bywhich is the solution of -Hilfer fractional differential equation, .

Lemma 4 (see [26]. Banach fixed-point theorem). Let be a closed subset of a Banach space ; then, any contraction mapping has a unique fixed point.

3. Main Results

In this portion, we demonstrate and exhibit the existence and uniqueness for the solution of the considered system on the interval under Banach contraction principle and Mönch’s fixed-point theorem. We also discuss the UH stability for the solution of considered problem (4). Before coming to the main results, we assume some hypothesis as follows.

: let the function satisfy the following assumptions:(i)The function , for all , is measurable and continuous, for . .(ii)There exists constant , , and positive integrable function such thatfor all while satisfies(iii)There exists constant and functions such that, for any bounded subsets ,for , where , , and , and is Hausdorff measure of noncompactness.(iv)We can find constants all greater than zero, such that, for all and ,: let the function satisfy the following conditions.(v)The function is continuous, and there exists a positive constant such that(vi)There exists constant , and a function such that, for any bounded subset ,(vii)There exists a constant , such that, for all and ,: let the function satisfy the following conditions.(viii) is continuous for a.e. and is measurable for all .(ix)The function is continuous, and we can find a positive constants and such that(x)There exists constant , and a function such that, for any bounded subset ,: is continuous and satisfies the conditions as follows.(xi)Let there exist constants , , and such that(xii)There exists a constant such that, for all ,:

Before studying the qualitative behavior of solutions to (4), we consider a simplified form of (4) given bywhere represent the -Hilfer fractional derivative of order and type , is the -Riemann–Liouville fractional left-sided integral of order .

Theorem 1. Let , , and be continuous, then any solution ofhas the formand

Proof. For any , we considerApplying the integral on both sides of (39) and using Lemma 2, we havewhich implies thatNow, for andunder the conditionFrom (41) and Lemma 3, one hasFrom (41), we haveUsing (45) in (44), we obtainContinuing in the same manner, for , we haveNext on the basis of Theorem 1, we suppose the solution of equation (4).

Definition 3. Let be the solution of equation (4); then, satisfies the following -Hilfer fractional integral equation:which is a mild solution of equation (4).

Lemma 1 (see [27]). Let ; then, , satisfyingwhere and .
Next, we consider some properties and definitions of noncompactness.
The Hausdorff measure of noncompactness defined on each bounded set , where is a Banach space which is given by

Lemma 2 (see [28]). In the real Banach space , if are bounded, then the following properties are satisfied. (1) Monotone: if, for all bounded subsets of , , then ; (2) nonsingular: , for each and every nonempty subset ; (3) regular: the subset is precompact if and only if ; (4) for , ; (5) ; (6) ; (7) if the subset is both bounded and equicontinuous, then is continuous on and(8) If is a sequence of Bochner integrable functions from the interval into the Banach space satisfying , for almost and each , then the function and(9) If is bounded, then, for every , there is a sequence such thatEstablishing our main results, the above lemma about Hausdorff measurement of noncompactness will be used.

Lemma 3 (see [29]). Let be a closed and convex subset of a Banach space , . If a map is continuous, which satisfies Mönchs condition, i.e., , implies is compact. Then, has a fixed point in .

Lemma 4 (see [30]). If and is the standard Wiener process, thenwhere

Theorem 2. Let the conditions to be true; then, system (4) possess at least one solution on the interval .

Proof. Transform the considered system (4) into equivalent fixed-point problem. Define an operator byThe operator has fixed point in the interval if and only if (4) has a solution. For , we defineThen, . Let , ; then, it can be easily observed that , for , where is given byif and only if for andSetting the space , induced by , aswith the following norm:Consider a close ball . Then, for every , is a closed, bounded, and convex subset of . For any , we haveDefine an operator byWe will complete the proof in several steps.Step 1: we prove that there exists some positive real number for which . If it is not true, then, for each , there is some such that .Now, it follows from the assumptions thatNow, we determine to from the conditions and :Hence, we haveDividing each term by and taking , then, from the above,yieldsThis contradicts . Thus, there exists some for which .Step 2: let such that in as .Now, for any , we haveAlso, for all , where , we haveThus,so that is continuous.Step 3: the operator is equicontinuous, for all .For , the class of functions is equicontinuous. For any with , , , and , we haveThe right side of the last inequality is independent of and approaches to zero as , since and as . Therefore, as . Hence, the operator is equicontinuous, for all .Step 4: it remains to show that the Mönchs condition holds.LetwhereAssume that is countable and . We show that , where is the Hausdorff measure of noncompactness. Without loss of generality, we may take . Since is equicontinuous on the interval , therefore, is equicontiuous on as well.
Using Lemmas 2 and 4 along with conditions , we haveThus, we haveAs both and are continuous, it follows from Lemma 2 that .
Thus, from Mönchs condition, we haveSince , we get , from which it follows that is compact. Thus, has a fixed point in . Thanks to Lemma 4, this completes the proof.

Theorem 3. Let the assumptions hold, provided that , whereThen, the operator has a unique fixed point on .

Proof. Define the operator byFor , the proof is trivial. For any and , we considerAlso, for all , where , we haveFrom the condition , the mapping is contractional; therefore, has a unique fixed point.

4. UH Stability Analysis

In this section, we are establishing UH stability result.

For any , consider the following inequality:

Definition 4. The considered system (4) is UH stable if we can find constant such that, for every , there exists a solution which satisfies inequality (82) and has the same initial value as , where is the solution of (4). Then, satisfies the relation

Theorem 4. Let conditions , , , and be satisfied; then, solution to (4) is UH stable.

Proof. Let inequality (82) is true, and consider the following system:Using the method of fundamental solution to (84), we obtainIn the interval , it is obvious that the solution to (84) is UH stable. Now, for and , we haveThus,whereIf we consider the interval , thenSince is continuous and for , the conclusion implies that there exists some positive constant say , such thatTherefore,Thus,whereContinuing the same process, we can prove that, for any , there exists a constant such thatBy Definition 1, system (4) is UH stable.

5. Illustrative Example

In this section, we illustrate the above results by an example.

Example 1. Consider the following impulsive neutral -Hilfer stochastic integro-fractional differential systemSet the functions asFor any , we defineThe space is a Banach space having the properties given below.
: if is continuous on , , then is also continuous on and .
: is a Banach space.
:Additionally, let , , , , and . Thus, assumptions to are satisfied.
Now,As , thus, system (96) has a unique solution. Also, ; therefore, system (96) is UH stable on .

6. Conclusion

The fractional stochastic neutral impulsive differential equations have applications in various fields such as viscoelasticity, automatic control, and electrochemistry. Based on the some well-known fixed-point theorems of fractional calculus, technique of stochastic analysis existence results for the considered system is given. Likewise under specific assumptions and conditions, we have found the UH stability result for the solution of (4).

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final form of the manuscript.

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