Solvability for the ψ -Caputo-Type Fractional Differential System with the Generalized p -Laplacian Operator

: In this article, by combining a recent critical point theorem and several theories of the ψ -Caputo fractional operator, the multiplicity results of at least three distinct weak solutions are obtained for a new ψ -Caputo-type fractional differential system including the generalized p -Laplacian operator. It is noted that the nonlinear functions do not need to adapt certain asymptotic conditions in the paper, but, instead, are replaced by some simple algebraic conditions. Moreover, an evaluation criterion of the equation without solutions is also provided. Finally, two examples are given to demonstrate that the ψ -Caputo fractional operator is more accurate and can adapt to deal with complex system modeling problems by changing different weight functions.


Introduction
As a popular research object in recent years, fractional differential equations (FDEs) play an important role in modeling many practical problems of science and engineering, such as fluid flow, anomalous diffusion, viscoelastic mechanics, epidemiology, etc. (see [1][2][3][4][5][6]). There are various definitions of fractional integration and differentiation, including the most widely used classical definitions of Riemann-Liouville, Caputo, Hadamard and others (see [7][8][9][10]). Currently, these classical definitions are employed in many fields, such as fractional boundary and initial value problems (see [11][12][13][14][15]). In order to overcome the inconvenience arising from a large number of definitions, Kilbas et al. advanced a new and more general form, called the ψ-Caputo-type fractional derivative (cf. [7]). By drawing into the weight function ψ(t), different definitional forms of fractional calculus were generalized and unified into a whole expression. In 2017, Almeida [16] investigated the relevant properties of the new operator and provided a theoretical basis for studying ψ-Caputo-type FDEs in depth.
When the weight function ψ(t) is specified as certain functions, the ψ-Caputo fractional derivative can be degenerated into certain classical functions. Therefore, based on ψ-Caputo fractional integration and differentiation, the modeling accuracy of practical problems is greatly improved. Most recently, some existence results for ψ-Caputo FDEs were achieved by applying fixed-point theorems in topological methods (see [17][18][19][20]). For instance, ref. [18] considered the solvability of the ψ-Caputo-type FDE by taking advantage of a novel fixedpoint theorem. In [19], the authors derived the existence and uniqueness of solutions for a ψ-Caputo fractional initial value problem by applying some standard fixed-point theorems.
However, so far as is known to the authors, there are few studies which have focused on solvability for ψ-Caputo FDEs based on variational methods and critical point theory.
In light of this point, in this paper, we consider a new ψ-Caputo-type fractional differential system, including the generalized p-Laplacian operator.
a + z(t))) + |z(t)| p−2 z(t) = ξg(t, z(t)) + λ f (t, z(t)), t ∈ [a, b], z(a) = z(b) = 0, (1) where 0 < α ≤ 1, 0 ≤ a < b < +∞, λ > 0, ξ ≥ 0, 1 < p < ∞, and the right and left α-order ψ-Caputo fractional derivatives are C D α,ψ b − and C D α,ψ a + . The weight function What is particularly noteworthy is that the nonlinear functions f and g in this article do not need to adapt certain asymptotic conditions; we can acquire the multiplicity of at least three distinct solutions only by imposing algebraic conditions on the nonlinearities. This work is a generalization of several results reported in the literature which are concerned with classical fractional operators.

Fractional Calculus and Critical Point Theorem
In this section, we present the definitions of some kinds of fractional integrals and differentials, as well as related properties, and one effective critical point theorem.
. The left and right ψ-Riemann-Liouville fractional integrals of a function z are defined, respectively, by Let n = [α] + 1 for α / ∈ N, n = α for α ∈ N. The left and right ψ-Riemann-Liouville fractional derivatives of a function z are, respectively, defined by Especially, for 0 < α < 1, Obviously, the classical Riemann-Liouville fractional derivative can be acquired by choosing the weight function ψ(t) = t. Definition 2 ([7,16]). Let −∞ < a < b < +∞, z(t), ψ(t) ∈ C n [a, b], such that ψ is increasing and ψ (t) = 0. Define the left and right ψ-Caputo fractional derivatives of a function z by Obviously, the classical Caputo fractional derivative can be acquired by choosing the weight function ψ(t) = t.
This paper deals mainly with the Caputo-type fractional derivative with the weight function ψ. In what follows, an important and proper fractional derivative space is defined, which is crucial for the system (1) to establish a variational structure.
Proof. Due to Property 1 and z(a) = z(b) = 0, we can obtain the desired conclusion directly.
Based on the inequality (7), we can observe that the norm (6) and norm z (α,ψ, are equal in form.
Proof. Taking advantage of (3), (4), and the Dirichlet boundary value in system (1), At this point, we multiply both sides of system (1) by ψ (t)y(t), and then integrate both ends from a to b simultaneously. Following (11), we can obtain the relationship (10).
Next, we recall an interesting and useful three critical points theorem provided by Bonanno and Candito. This theorem provides the critical theory technology to obtain the multiplicity results for system (1) in our work.
Let H be a nonempty set, and Φ, Ψ : H → R be two functions. For any ρ, ρ 1 , Assume that there are three positive constants ρ 1 ,
It is not difficult to see that the critical point of the functional F is consistent with the weak solution of system (1).

Lemma 7.
The functional F 1 is a continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on H * (α,ψ,p) .
For simplicity of discussion, we introduce some notations before describing the main theorems.
Thus, uniting Lemma 7 and Theorem 1, for every and ξ ∈ [0, σ (λ,G) [, the functional F has three critical points z 1 , z 2 , z 3 on H (α,ψ,p) and Then, consider the fact that the critical points of the functional F are consistent with weak solutions of system (1), we obtain the main conclusion.
for system (1). We claim that z 1 , z 2 , z 3 are non-negative. In fact, let z be a nontrivial weak solution of system (1). We assume the set Θ = {t ∈ (a, b] : z(t) < 0} is non-empty with the positive measure. For any t ∈ [a, b], define y * (t) = min{0, z(t)}. Obviously, y * (t) ∈ H (α,ψ,p) and satisfies b a Since f , g are non-negative, due to (29), one has which means that z ≡ 0 in Θ, which is a contradiction. Therefore, we get the desired result.

Theorem 4.
Assume that there exists a constant C 0 , such that . Then, the system (1) does not include any nontrivial weak solution.

Conclusions
This paper considered a new ψ-Caputo-type fractional differential system including the generalized p-Laplacian operator. By means of a three critical points theorem given by Bonanno and Candito, and several properties of the ψ-Caputo fractional operator, the existence of at least three distinct non-negative weak solutions was studied. Due to a mild condition, an evaluation criterion for the equation without solutions was given. What is noteworthy is that the nonlinear functions f , g do not need to adapt certain asymptotic conditions-the multiplicity results were established only by imposing algebraic conditions on nonlinear functions. This work represents a generalization of several results reported in the literature which concern classical fractional operators.