Multiplicity Solutions of Fractional Impulsive <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 10.3455 13.4804" width="10.3455pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g></svg>-</span>Laplacian Systems: New Result

Guefaifia, Rafik; Abdalla, Mohamed; Bouali, Tahar; Kamache, Fares; Cherif, Bahri Belkacem; Mekawy, Ibrahim

doi:https://doi.org/10.1155/2021/9957952

Journal of Function Spaces

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Fractional Problems with Variable-Order or Variable Exponents

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Volume 2021 | Article ID 9957952 | https://doi.org/10.1155/2021/9957952

Multiplicity Solutions of Fractional Impulsive -Laplacian Systems: New Result

Rafik Guefaifia,¹Mohamed Abdalla,^2,3Tahar Bouali,^1,4Fares Kamache,¹Bahri Belkacem Cherif,^5,6and Ibrahim Mekawy⁶

Academic Editor: Simone Secchi

Received25 Mar 2021

Revised18 Apr 2021

Accepted26 May 2021

Published09 Jun 2021

Abstract

In this paper, the existence of multiplicity distinct weak solutions is proved for differentiable functionals for perturbed systems of impulsive nonlinear fractional differential equations. Further, examples are given to show the feasibility and efficacy of the key findings. This work is an extension of the previous works to Banach space.

1. Introduction

This paper explores the perturbed impulsive fractional differential system where , , for , and are the left and right Riemann–Liouville fractional derivatives of order , respectively, , , , , , , , and are -Caratheodory functions, and they satisfy in the following standard summability condition: for any with and , and is a -Lipschitz continuous function with the Lipschitz constant , i.e., for every , satisfying for . The operator is defined as and where and is a -Lipschitz continuous function with the Lipschitz constant , i.e.,

Here, and are the partial derivatives of and with respect to for , respectively.

In science and engineering, fractional differential equations (FDEs) have recently proved to be useful methods for modeling a broad variety of phenomena. In viscoelasticity, electrochemistry, power, porous media, and electromagnetism, for instance, see [1–33] and the references therein. Many articles have recently investigated the existence of solutions to boundary value problems for FDEs, and we refer the reader to one of them [2, 18–20, 34–46] and the references therein. For example, Kamache et al. [40] investigated the existence of three solutions for a class of fractional -Laplacian systems using a variational structure and critical point theory.

In [36], we investigated the existence of solutions of the periodic boundary value problem for a nonlinear impulsive fractional differential equation with periodic boundary conditions: where is the standard Riemann–Liouville fractional derivative, is the sequential Riemann–Liouville fractional derivative presented by Miller and Ross on p. 209 of [14], , , and is continuous at every point . By using the method of upper and lower solutions and its associated monotone iterative method, the author studies the existence and uniqueness of the solution of the periodic boundary value problem for the nonlinear impulsive fractional differential equation (7).

Upon using variational methods and critical point theory, the presence of one weak solution for the system was also demonstrated in [19] with and for and .

Impulsive effects are a common phenomenon triggered by short-term perturbations that are negligible in relation to the original operation’s total duration. Such perturbations can be approximated fairly well as instantaneous changes of state or in the form of impulses. Such phenomena governing equations can be interpreted as impulsive differential equations. There has been a surge in interest in the study of impulsive differential equations in recent years, as these equations provide a natural framework for mathematical modeling of many real-world phenomena, especially in control theory, physics, chemistry, population dynamics, biotechnology, economics, and medical fields. Under such boundary conditions, the presence of solutions for impulsive differential equations with variational structures is determined by variational methods. See, for example, [36] as well as the references therein. Many scholars have recently studied fractional differential equations with impulses using variational methods, fixed point theorems, and critical point theory, due to the rapid growth in the theory of fractional calculus and impulsive differential equations, as well as their broad applications in a variety of fields (see, for example, [35, 44] and the references therein for a thorough discussion, as well as the sources therein for more details). For example, Gao et al. provided sufficient conditions for the existence and uniqueness of solutions for a class of impulsive integrodifferential equations with nonlocal conditions involving the Caputo fractional derivative using the Schaefer fixed point theorems (see [45]).

The existence of infinitely many solutions for the system (1) was discussed in [46] using variational methods. Some new parameters to guarantee that the system (1), in the case , has at least two nontrivial and nonnegative solutions were obtained in [30] under appropriate hypotheses and using variational methods.

Recently, in Reference [27], perturbed systems of impulsive nonlinear fractional differential equations were studied, including continuous nonlinear Lipschitz terminology where at least three distinct weak solutions were demonstrated based on the modern critical point theory of differentiable functions, but here, we will prove the existence of three distinct weak solutions for differentiable functionals for perturbed systems of impulsive nonlinear fractional differential equations.

Most precisely, in this work, we extend the last work [38] to Banach space, where we show that there are at least three weak solutions for the system (1), which involves two parameters and . Furthermore, we do not need any asymptotic conditions of the nonlinear term at infinity in our new findings. The proof is based on a three-critical point theorem proved by Bonanno and Candito in [32], which we will revisit in the following section (Theorem 1). Theorem 10 is our most important finding. As a result, Theorem 11 can be deduced. Theorem 11 is shown in Example 1. When it comes to a scalar situation (), we obtain Theorems 14 and 15 as special cases of Theorems 10 and 11. Theorem 15 is shown in Example 2. Under appropriate conditions on the nonlinear term at zero and at infinity, we obtain the presence of at least two positive solutions in Theorem 16.

The present paper is organized as follows. In Section 2, we recall some basic definitions and preliminary results, while Section 3 is devoted to the existence of multiple weak solutions for the eigenvalue system (1).

2. Preliminaries

Let be a nonempty set and be two functions. For all , , we define

Theorem 1 ([32], Theorem 3.3). Let be a reflexive real Banach space; let be a coercive and continuously Gateaux differentiable and sequentially weakly lower semicontinuous functional whose Gateaux derivative admits a continuous inverse on , where is the dual space of , and let be a continuously Gateaux differentiable functional whose Gateaux derivative is compact, such that
: is convex, and .
: for every such that and , one has Assume that there are three positive constants , , and with , such that Then, for each , the functional admits three distinct critical points , , and such that , , and .
Now, we introduce some important fractional calculus concepts and properties that will be used in this paper.
Let be the set of all functions with and the norm Denote the norm of the space for by The following lemma yields the boundedness of the Riemann–Liouville fractional integral operators from the space to the space , where .

Definition 2 [35]. The left and right Riemann–Liouville fractional derivatives of order for the function are defined in the following forms, respectively, where is a function defined on and for , and is the standard gamma function given by

Definition 3 (see [40]). Let for and . (i)If and , then the left and right Caputo fractional derivatives of order for function denoted by and , respectively, exist almost everywhere on and for , where and are represented by respectively.(ii)If and , then and are represented by and

Lemma 4. Let for , , and . Then

Proposition 5 (see [40]). From fractional integration, we have provided that , , and or .

Definition 6 (see [40]). Let for . The fractional derivative space (denoted by for short) is defined by the closure , that is, with respect to the weighted norm for every and for .

Remark 7. It is obvious that the fractional derivative space is the space of functions having an order Riemann–Liouville fractional derivative and for . From [12] (Proposition 3.1), we know that for , the space is a reflexive and separable Banach space.

Lemma 8 (see [40]). Let for , and . For any , we have Moreover, if , then where . Upon using (23), we observe that for , which is equivalent to (15). Then, we have and if , then with Now, we let be the Cartesian product of Sobolev spaces , i.e., , which is a reflexive Banach space endowed with the norm Obviously, is compactly embedded in .

Definition 9. We mean by a (weak) solution of the system (1) any function such that for every .
Put for .
We need the following conditions:
: for .
: , and there exists a constant such that : , where and .
Put

3. Main Results

In this section, we present our key findings regarding the existence of at least three weak system solutions (1). For any , we denote by the set . For positive constants and , set

For the rest of this article, positive constants will be used ( and ), and let and be the vectors in defined by respectively.

Set for , and

Fixing four positive constants , , , and , put for .

Theorem 10. Let be nonnegative. Assume that there exist positive constants , , , , and with and such that
: Then, for every and every nonnegative function satisfying , there exists given by (39) such that, for each , the system (1) has at least three solutions , , and such that , , and .

Proof. Our aim is to apply Theorem 1 to the system (1). We take and introduce the functionals and for , as follows: and we put Clearly, and are continuously Gateaux differentiable functionals whose Gateaux derivatives at the point are given by for every . Clearly, , and we easily observe that
We can show by (42) that is sequentially weakly lower semicontinuous. Indeed, taking the sequentially weakly lower semicontinuity property of the norm into account and since is continuous for , it is enough to prove that is weakly continuous in . In fact, for , if converges to in , then there exists such that . Therefore, we have where . So, we have ; thus, is weakly continuous. Hence, is sequentially weakly lower semicontinuous in . We show what is required. Since , one has for ; from (43) and the condition , we see that and bearing the condition in mind, it follows ; namely, is coercive and convex.
For , define by for . Clearly, and for . A direct calculation shows that for . Furthermore, for . Thus, , and for . By using (50) and (52), we have Choose , , and . From the conditions , , and , we achieve and . From the definition of and considering Equations (24), (27), and (50), one has Hence, since is nonnegative, one has In a similar way, we have Therefore, since and , one has On the other hand, for each , one has Since , one has This means Furthermore, This means Then, In a similar way, we have Hence, from (57)–(67), we get Therefore, and of Theorem 1 are verified.
Now, we show that the functional satisfies the assumption of Theorem 1. Let and be two local minima for . Then, and are critical points for ; they are weak solutions for the system (1). Since we assumed is nonnegative and since is nonnegative, for fixed and , we have , and consequently, for all . Hence, Theorem 1 implies that for every and , the functional has three critical points , , in such that , , and , that is, , , and . Then, taking into account the fact that the weak solutions of the system (1) are exactly critical points of the functional , we have the desired conclusion.☐

For positive constants , , and , set where

Theorem 11. Let satisfy the condition for all . Assume that there exist positive constants , , , and with and such that
: Then, for every and every nonnegative function satisfying , there exists given by (70) such that, for each , the system (1) has at least three solutions , , and such that , , and .☐

Proof. Choose and . So, by using , one has Moreover, taking into account that , by using , we have Hence, from , (73), and (74), it is easy to see that the assumption of Theorem 10 is satisfied, and since the critical points of the functional are the weak solutions of the system (1), we have the conclusion.☐

Now, we present the following example in which the hypotheses of Theorem 11 are satisfied.

Example 1. Consider the following system: where and for every , , , and for every and for . By expressions of and , we have and for every . Choosing , , , and , we clearly observe that all assumptions of Theorem 11 are satisfied. Hence, for every and every nonnegative function satisfying , there exists such that, for each , the system (74) has at least three solutions , , and such that , , and .☐

Remark 12. When does not depend on , in Theorem 10, the assumption can be written as as well as In this case, in Theorem 11, the assumption follows the form as well as

Remark 13. We observe that, in our results, no asymptotic conditions on and are needed and only algebraic conditions on are imposed to guarantee the existence of solutions. Moreover, in the conclusions of the above results, one of the three solutions may be trivial since the values of and for every , , are not determined.
As an application of Theorem 10, we consider the following problem: where , , , , and denote the left and right Riemann–Liouville fractional derivatives of order , respectively, , are -Caratheodory functions, and is a -Lipschitz continuous function with the Lipschitz constant , i.e., for every , satisfying , for , such that , , and is a -Lipschitz continuous function with the Lipschitz constant such that for any , satisfying , for . Put and for every .
Set for . We suppose that where .
For positive constants and , set Obviously, if changes sign on , then clearly .☐

Now, we give the following straightforward consequences of Theorems 10 and 11, respectively.

Theorem 14. Let be a nonnegative -Caratheodory function. Assume that there exist positive constants , , , , and with , , and such that Then, for every and every nonnegative -Caratheodory function , there exists given by such that, for each , the system (83) has at least three solutions , , and such that , , and .☐

Proof. By a similar argument as given in the proof of Theorem 10, we ensure the existence of the weak solutions , , and such that , , and . Now, we show that the weak solutions , , and are nonnegative. To this end, let be a nontrivial weak solution of the problem (83). Arguing by a contradiction, assume that the set is nonempty and of positive measure. Put for all . Clearly, , and one has Thus, from our sign assumptions on the data, we have Hence, since , in and we arrive at a contradiction.☐

Theorem 15. Let be a nonnegative -Caratheodory function. Assume that there exist positive constants , , , and with and such that Then, for every and every nonnegative function satisfying , there exists given by such that, for each , the system (83) has at least three solutions , , and such that , , and .☐

Here, in order to illustrate Theorem 15, we present the following example.

Example 2. where , for every , , , and , for every and for . By expression, we have Taking , , , and , we clearly observe that all assumptions of Theorem 15 are satisfied.
Then, for each and every nonnegative continuous function , there exists such that, for each , the system (83) has at least three nonnegative weak solutions , , and such that , , and .☐

Now, we list some consequences of Theorem 15 as follows.

Theorem 16. Let be a nonnegative continuous and nonzero function such that for every , where Then, there exists where , , and are positive constants with , such that for each , the problem where is a nonnegative continuous and nonzero function, has at least two distinct positive weak solutions.☐

Proof. Fix , put for all , and let such that and From (102), there is such that and and such that and . Therefore, Theorem 11 ensures the conclusion.☐

Finally, by way of example, we point out the following simple consequence of Theorem 16 when .

Theorem 17. Let be a continuous function such that for all and Then, for every , where the problem (105), in the case , has at least four distinct nontrivial weak solutions.

Proof. Setting and applying Theorem 16 to and , we have the result.☐

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

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

Acknowledgments

The second author extends their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through a research group program under grant R.G.P. 1/15/42. The authors also acknowledge Prof. Salah Boulaaras for the first and second revisions and kind comments on this work.

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Copyright © 2021 Rafik Guefaifia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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