On Existence of Sequences of Weak Solutions of Fractional Systems with Lipschitz Nonlinearity

In this article, the variational method together with two control parameters is used for introducing the proof for the existence of infinitely many solutions for a new class of perturbed nonlinear system having 
 
 p
 
 -Laplacian fractional-order differentiation.


Introduction
One of the main applications of fractional calculus science is the fractional-order differential equations (FDEs). Various natural phenomena are modeled mathematically through the FDEs, and this is evident in numerous areas of physics, engineering, chemistry, and other fields. The fractionalorder partial differential equations have several applications in many fields such as engineering, biophysics, physics, mechanics, chemistry, and biology (see [1][2][3][4][5][6][7]). More and more efforts have been made in the fractional calculus field especially in FDEs (see, for instance, [2,5,[8][9][10][11][12][13][14][27][28][29][30][31][32][33][34][35][36][37][38][39]). Solution existence for a lot of boundary value problems and several nonlinear elementary problems is studied via a huge number of techniques and nonlinear mathematical tools (see [7,[15][16][17][18][19][20][21][22][23]): the theory of critical point, fixed-point theory, technique of monochromatic iterative, theory degree of coincidence, and the change methods. Motivated by multiple works involved in this domain, we concentrate in this paper on the existence of several infinite solutions to the following fractional-order differentiation system: Þ , ⋯, u n t ð ÞÞ + μG u i t, u 1 t ð Þ, u 2 t ð Þ ð Þ , ⋯, u n t ð ÞÞ + h i u i ð Þ, a:e:t ∈ 0, T ½ , λ, μ are positive and nonnegative real parameters, respectively, ðF0ÞF, G : ½0, T × ℝ n ⟶ ℝ are continuous functions according to t ∈ ½0, T for any ðx 1 , x 2 , ⋯, x n Þ∈ℝ n and are C 1 with respect to ðx 1 , x 2 , ⋯, x n Þ∈ℝ n for a.e. t ∈ ½0, T, Fðt, 0, 0, ⋯, 0Þ = 0 and Gðt, 0, 0, ⋯, 0Þ = 0 for it is to say t ∈ ½0, T. Also, F u i and G u i indicates partial derivatives of F and G according to u i , respectively, and h i : ℝ ⟶ ℝ are ðp − 1Þ -order of Lipschitz continuous functions with L i > 0 constants of Lipschizian, 1 ≤ i ≤ n, i.e., In the last few months, we treated the same area of this study, in [18], by using variational methods together with a critical point theory due to Bonano and Marano. We got at least three weak solutions for the following nonlinear dual-Laplace systems with respect to two parameters: In addition, in [24], by using the variational method and Ricceri's critical point theorems, the existence of three weak solutions has been used to investigate the following class of perturbed nonlinear fractional p-Laplacian differential systems: where some necessary conditions on the primitive function of nonlinear terms F u and F v have been considered. Then, in [25], the same last methods have been used for problem (5), the existence of multiplicity of weak solutions for the following perturbed nonlinear fractional differential systems: where Lipschitz nonlinearity order of p − 1 has been used.
Most recently, in [26], the authors proved the existence of infinitely multiple solutions of the following perturbed nonlinear fractional p-Laplacian differential systems: Þ a:e: t ∈ 0, T ½ , Þa:e:t ∈ 0, T ½ , Þa:e:t ∈ 0, T ½ , Journal of Function Spaces where one control parameter with the variational method has been used. Motivated by recently mentioned papers, the main contribution of this article is to use two control parameters and variational method to study a class of a nonlinear perturbed fractional-order p-Laplacian differential system which is defined in (6), where we can prove that the studied system admits sequences of weak different solutions, strongly converge to zero.

Preliminaries
In this section, we introduce some notations, lemmas that are required for the subsequential. Then, a variational framework is constructed; then, the critical point theory is applied to explore the existence of infinite solutions for the system given in (6).
We denote Y X the class of all functionals ϕ : X ⟶ ℝ, where X is real Banach space which has the following properties.
If fw n g is a sequence in X converge weakly to w ∈ X and lim n⟶∞ inf ϕðw n Þ ≤ ϕðwÞ, thus fw n g has a subsequence that strongly converge to w.
As an example, suppose a uniformly convex X with S : ½ 0,+∞Þ ⟶ ℝ is an increasing, continuous strictly function, then the functional w ⟶ SðkwkÞ∈Y X .
Definition 1 (see [4]). Let u be a defined function on ½a, b. The left and right Riemann-Liouville fractional derivatives of order α > 0, respectively, are given as where the right-hand sides are pointwise defined over ½a, b, ∀ t ∈ ½a, b, n − 1 ≤ α < n and n ∈ ℕ: The gamma function, ΓðαÞ, is given by As familiar, we denote C n−1 ð½a, b, ℝÞ the mappings set indicates ðn − 1Þ-times continuously differentiable on ½a, b: Definition 2 (see [22]). Let 0 < α i ≤ 1 (1 ≤ i ≤ n, 1 < p < ∞), we introduce the space of the fractional-order derivative as follows: then, the norm of E p α i can be defined ∀u ∈ E p α i , as the following Lemma 3 (see [3]).
Also, if α i > p and 1/p + 1/q = 1, then Hence, the operator E p α i according to the norm can be considered for 1 ≤ i ≤ n, that is equivalent to (13).
Definition 4. Suppose a Cartesian product X of n spaces E p α ; that is to say, provided with the norm where ku i k α i is defined in (16).

Principle Results
This section deals with stating and proving our main results. For assistance, suggest For a given constant θ ∈ ð1/p, 0Þ, set For any ρ > 0, we set Then, for each λ ∈ Λ ≔ λ 1 , λ 2 ½ where 5 Journal of Function Spaces for each nonnegative function G : ½0, T × ℝ n ⟶ ℝ achieving the constrain and for every μ ∈ ½0, μ G,λ ½ where system (1) has an unbounded sequence of weak solutions in space X.
Proof. The main aim here is applying Lemma 6 (see [16]) over system (1). For this purpose, fix λ ∈ Λ, and let G be a function that satisfies our hypotheses. Since λ < λ 2 , we claim for every t ∈ ½0, T and ξ = ðξ 1 , ⋯, ξ n Þ ∈ ℝ n . We construct the mappings ϕ, Ψ : X ⟶ ℝ as follows: ∀u = ðu 1 , ⋯, u n Þ ∈ X and determine Let us prove that ϕ & Ψ satisfy the required constrains. Since X is compactly embedded in ðCð½0, T, ℝÞÞ n , it is well known that Ψ is well-defined Gateaux differentiable functional whose Gateaux derivative at u ∈ X is the functional Ψ ′ , given by ∀v = ðv 1 , ⋯, v n Þ ∈ X. Moreover, Ψ is sequentially weakly continuous.
The functional Φ is a Gateaux differentiable functional with the differential at u ∈ X, for every v ∈ X. Moreover, ϕ is sequentially weakly lower semicontinuous, strongly continuous, and coercive functional on X.
Obviously, the weak solutions of system (1) are precisely the critical points of the functional I λ ≤ μ . Furthermore, since (3) holds for every x 1 , ⋯, x n ∈ ℝ and h 1 ð0Þ = ⋯ = h n ð0Þ = 0, Journal of Function Spaces one has |h i ðxÞ | ≤L i jxj p−1 , 1 ≤ i ≤ n∀x ∈ ℝ. It obtained from (14) and (15) the following: ∀u ∈ X, and as a result for this, ϕ is coercive. Now, allow us to verify that λ < 1/γ. Assume fξ k g is a positive number sequence such that ξ k ⟶ ∞ as k ⟶ ∞ and Put r k ≔ k′ξ p k /pk p n p ∀k ∈ ℕ. Since max t∈½0,T | u i ðtÞ | ≤k ku i k α i for all u i ∈ E α i 0 ð½0, TÞ and 1 ≤ i ≤ n, we have for each u = ðu 1 , ⋯, u n Þ ∈ X. So, from (45) and (47), we have Consequently, taking into the description that ϕð0, ⋯, 0 Þ = Ψð0, ⋯, 0Þ = 0, for all k big enough, one has φ r k ð Þ = inf Moreover, from assumption (h2) and (37), one has which implies Therefore, The assumption μ ∈ 0, μ G, λ ½ immediately yields λ < 1/γ: The succeeding step is to confirm that for a fixed λ, the functional I λ, μ has no global minimum. Let us verify that I λ, μ is unbounded from below. Since consider fη k g is a real sequence and τ is a positive constant such that η k ⟶ ∞ as k ⟶ ∞, and Then, for each λ ∈ λ 3 , λ 4 ½, where a sequence of weak solutions for system (system (1) exists, and it strongly converges to zero in the space X.