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Rafik Guefaifia, Mohamed Abdalla, Tahar Bouali, Fares Kamache, Bahri Belkacem Cherif, Ibrahim Mekawy, "Multiplicity Solutions of Fractional Impulsive -Laplacian Systems: New Result", Journal of Function Spaces, vol. 2021, Article ID 9957952, 15 pages, 2021. https://doi.org/10.1155/2021/9957952
Multiplicity Solutions of Fractional Impulsive -Laplacian Systems: New Result
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.
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.  investigated the existence of three solutions for a class of fractional -Laplacian systems using a variational structure and critical point theory.
In , 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 , , , 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  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,  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 ).
The existence of infinitely many solutions for the system (1) was discussed in  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  under appropriate hypotheses and using variational methods.
Recently, in Reference , 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  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 , 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).
Let be a nonempty set and be two functions. For all , , we define
Theorem 1 (, 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 . 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 ). 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 ). From fractional integration, we have provided that , , and or .
Definition 6 (see ). 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  (Proposition 3.1), we know that for , the space is a reflexive and separable Banach space.
Lemma 8 (see ). 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 .
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