Multiplicity Solutions of Fractional Impulsive p-Laplacian Systems: New Result

Department of Mathematics and Computer Science, Larbi Tebessi University, Tebessa, Algeria Mathematics Department, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt Department of Mathematics, College of Sciences, Jazan University, P.O. Box 277, Jazan, Saudi Arabia Preparatory Institute for Engineering Studies in Sfax, Tunisia Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Saudi Arabia

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  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][19][20][34][35][36][37][38][39][40][41][42][43][44][45][46] and the references therein. For example, Kamache et al. [40] investigated the existence of three solutions for a class of fractional p-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 D α uðtÞ = ð 0 D α t uÞðtÞ = ð1/ðΓð2 − αÞÞÞðd/dtÞ Ð t 0 ðt − τÞ −α uðτÞdτ is the standard Riemann-Liouville fractional derivative, D 2α u = D α ðD α uÞ is the sequential Riemann-Liouville fractional derivative presented by Miller and Ross on p. 209 of [14], 0 < t 0 < t 1 < ⋯<t m = 1, I j , I j ∈ Cðℝ, ℝÞðj = 1, ⋯, mÞ, and f is continuous at every point ðt, u, vÞ ∈ ½0, 1 × ℝ × ℝ. 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 μ = 0 and I ij = 0 for i = 1, ⋯, n and j = 1, ⋯, m.
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 μ = 0, 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 λ and μ. 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 (n = 1), 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.

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Journal of Function Spaces 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).

Preliminaries
Let X be a nonempty set and Φ, Ψ : X ⟶ ℝ be two functions. For all r 1 , r 2 , ð12Þ . Let X be a reflexive real Banach space; let Φ : X ⟶ ℝ be a coercive and continuously Gateaux differentiable and sequentially weakly lower semicontinuous functional whose Gateaux derivative admits a continuous inverse on X * , where X * is the dual space of X, and let Ψ : X ⟶ ℝ be a continuously Gateaux differentiable functional whose Gateaux derivative is compact, such that ða 1 Þ: Φ is convex, and inf X Φ = Φð0Þ = Ψð0Þ = 0. ða 2 Þ: for every u 1 , u 2 ∈ X such that Ψðu 1 Þ ≥ 0 and Ψðu 2 Þ ≥ 0, one has Assume that there are three positive constants r 1 , r 2 , and r 3 with r 1 < r 2 , such that Then, for each λ ∈ 1/βðr 1 , r 2 Þ, 1/αðr 1 , r 2 , r 3 Þ½, the functional Φ − λΨ admits three distinct critical points u 1 , u 2 , and u 3 such that u 1 Now, we introduce some important fractional calculus concepts and properties that will be used in this paper.
Let C ∞ 0 ð½0, T, ℝ n Þ be the set of all functions x ∈ C ∞ 0 ð½0, T, ℝ n Þ with xð0Þ = xðTÞ = 0 and the norm Denote the norm of the space The following lemma yields the boundedness of the Riemann-Liouville fractional integral operators from the space Definition 2 [35]. The left and right Riemann-Liouville fractional derivatives of order α i for the function u are defined in the following forms, respectively, where u is a function defined on ½0, T and α i > 0 for 1 ≤ i ≤ n, and Γðα i Þ is the standard gamma function given by Definition 3 (see [40]). Let α i ≥ 0 for 1 ≤ i ≤ n and n ∈ ℕ. respectively.
Definition 6 (see [40]) with respect to the weighted norm for every u i ∈ H a i 0 and for 1 ≤ i ≤ n.
Moreover, if α i > 1/p, then where ð1/pÞ + ð1/qÞ = 1. Upon using (23), we observe that for 1 ≤ i ≤ n, which is equivalent to (15). Then, we have and if α i > 1/p, then with Now, we let X be the Cartesian product of n Sobolev spaces H , which is a reflexive Banach space endowed with the norm Obviously, X is compactly embedded in ðC 0 ð½0, TÞÞ n .

Main Results
In this section, we present our key findings regarding the existence of at least three weak system solutions (1). For any ς > 0, we denote by QðςÞ the set fðx i , ⋯, x n Þ ∈ ℝ n : ð1/pÞ ∑ n i=1 jx i j p ≤ ςg. For positive constants θ and η, set For the rest of this article, positive constants will be used (θ and η), and let Θ and η be the vectors in ℝ n defined by respectively. Set for 0 < γ < 1/p, and Fixing four positive constants θ 1 , θ 2 , θ 3 , and η, put for 0 < γ < 1/p.
Proof. Our aim is to apply Theorem 1 to the system (1). We take X = H α 1 0 × ⋯ × H α n 0 and introduce the functionals Φ and Ψ for u = ðu 1 , u 2 , ⋯, u 3 Þ ∈ X, as follows: 5 Journal of Function Spaces and we put Clearly, Φ and Ψ are continuously Gateaux differentiable functionals whose Gateaux derivatives at the point u ∈ X are given by for every v = ðv 1 , v 2 , ⋯, v n Þ ∈ X. Clearly, Φ ′ , Ψ ′ ∈ X * , and we easily observe that inf X Φ = Φð0Þ = Ψð0Þ = 0: We can show by (42) that Φ is sequentially weakly lower semicontinuous. Indeed, taking the sequentially weakly lower semicontinuity property of the norm into account and since H i is continuous for i = 1, ⋯, n, it is enough to prove that is weakly continuous in X. In fact, for fu k = ðu 1k , ⋯, u nk Þg ⊂ X, if fu k g converges to u in X, then there exists S 1 > 0 such that ku k k ∞ ≤ S 1 . Therefore, we have where S 2 = max i∈f1,⋯,ng,j∈f,⋯,mg,|s|≤S 1 I ij ðsÞ. So, we have |Φ ðu k Þ − ΦðuÞ | ⟶0; thus, Φ is weakly continuous. Hence, Φ is sequentially weakly lower semicontinuous in X. We show what is required. Since h i ð0Þ = 0, one has jh i ðx i Þj ≤ L i jx i j p−1 for i = 1, ⋯, n; from (43) and the condition ðH2Þ, we see that and bearing the condition ðH3Þ in mind, it follows lim kuk⟶∞ ΦðuÞ = +∞; namely, Φ is coercive and convex.