Multiple Positive Solutions for a System of Caputo Fractional p -Laplacian Boundary Value Problems

In this work, we are pleased to investigate multiple positive solutions for a system of Caputo fractional p -Laplacian boundary value problems, and we also provide an example for illustrating our main results.


Introduction
In this work, we are pleased to discuss the positive solutions for the following system of Caputo fractional p-Laplacian boundary value problems: where c D μ i t , c D ] i t (i � 1, 2) are the fractional derivatives of Caputo sense with μ i , ] i ∈ (1, 2); φ p is the p-Laplacian, i.e., φ p (x) � |x| p− 2 x with p > 1, x ∈ R; and the constants α i , β i , c i , δ i , b i , ξ i (i � 1, 2) and the functions g i (i � 1, 2) satisfy the following conditions: As a generalization of integer-order equations, fractional-order equations can effectively describe various materials and physical processes with memory and genetic properties. It has a large number of applications in our society, such as biology, chemical kinetics, electromagnetics, transmission and diffusion, and automatic control. Recently, there many researchers pay their attentions to studying the existence of solutions for various types of fractional-order equations by use of fixed point theorems, upper and lower solution methods, and monotone iterative techniques. For instance, we refer the readers to  and references therein.
In [1], by using the method of upper and lower solutions and the Schauder fixed point theorem, Vong investigated the positive solutions for the following nonlocal fractional boundary value problem: where n ≥ 2, α ∈ (n − 1, n), and μ(s) is a function of bounded variation. f may be singular at t � 1.
In [2], Wang and Yang studied the integral boundary value problem of Caputo sense: By the Leggett-Williams fixed point theorem, they obtained multiple positive solutions when the nonlinearity f is bounded from below. Moreover, when f is asymptotically linear at infinity, they also obtained an existence theorem.
In [3], Wang et al. adopted the theory of mixed monotone operators to obtain a unique positive solution for the mixed fractional boundary value problems involving the p-Laplacian: where c D α t is the Caputo fractional derivative and Recently, coupled systems of fractional differential equations have also been investigated by many authors.
In [11], Wang utilized the Guo-Krasnosel'skii fixed point theorem to investigate the multiple positive solutions for the mixed fractional p-Laplacian differential system: where D In [12], Rao studied the system of fractional p-Laplacian differential equations: When the nonlinearities f i satisfy some appropriate conditions, the author made use of the Avery-Henderson fixed point theorem and the six functionals' fixed point theorem to obtain some existence theorems of multiple positive solutions.
Inspired by the aforementioned results, in this work, we study the solvability for (1) and establish the existence results of multiple positive solutions via the six functional fixed point theorem under some bounded conditions for g i (i � 1, 2). Finally, we also provide an example to illustrate our main results.

Preliminaries
In this section, we only recall the definition of Caputo fractional derivative, for more details, see the book [26].
e fractional derivative of f in the Caputo sense is defined as where n � [μ] + 1, [μ] denotes the integer part of the number μ. Now, we calculate Green's functions associated with (1). Let and then by the boundary conditions in (1), we have Consequently, substituting (8) and (9) into (1), we obtain We next translate (10) into an equivalent system of integral equations. By the similar arguments as in [2], we have the following result. (10) is equivalent to the following system of Hammerstein-type integral equations:

Lemma 1. Problem
where Proof. We only need to consider the case i � 1 (by the similar method, the case i � 2 can be easily proved). Using Lemma 2.5 of [2], we have where c i ∈ R, i � 0, 1. x(0) � 0 implies that c 0 � 0, and by , v(s))ds, Complexity erefore, is completes the proof.
Note from (8) and Lemma 1, we have and with the boundary conditions

Complexity
where Lemma 3 (see [2], Lemma 2.8). e functions G i , H i (i � 1, 2) have the following properties: is a real Banach space and P is a cone on E. Moreover, E × E is a Banach space with the norm ‖(u, v)‖ � ‖u‖ + ‖v‖, and P × P a cone on E × E. From Lemmas 1 and 2, we define operators T i (i � 1, 2) and T as follows: en, we obtain that T i (i � 1, 2): P × P ⟶ P, T: P × P ⟶ P × P are completely continuous operators, and if there exists then (u, v) is a positive solution for (1).
Lemma 4 (see [27]). Let P be a cone in a real Banach space E. Suppose that α, ψ, and ζ are nonnegative continuous concave functionals on P, β, σ, and θ are nonnegative continuous convex functionals on P, and there are nonnegative constants l, l ′ , r, r ′ , R, and R ′ such that A: Q(β, R) ⟶ P is a completely continuous operator and If the following claims hold: en, A has at least three fixed points u 1 , u 2 , and u 3 in Q(β, R) such that σ(u 1 ) ≤ l, r ≤ (u 2 ) with β(u 2 ) ≤ R, and l < σ(u 3 ) with α(u 3 ) < r.
Complexity Theorem 1. Suppose that there exist positive real numbers l, l ′ , r, r ′ , R, and R ′ such that g i (i � 1, 2) satisfy the following conditions: Proof. Note, from a standard calculus argument, we obtain the set Q(β, R) is bounded. Since if (u, v) ∈ Q(β, R), and then max t∈I and Q(σ, l) ∩ Q(α, β, r, R) � ∅. Also, it can easily be shown that Hence, these sets are nonempty. As a result, (B1)-(B5) of Lemma 4 are satisfied. Now, we verify the functional conditions.