Positive solutions of fractional p-Laplacian equations with integral boundary value and two parameters

We consider a class of Caputo fractional p-Laplacian differential equations with integral boundary conditions which involve two parameters. By using the Avery–Peterson fixed point theorem, we obtain the existence of positive solutions for the boundary value problem. As an application, we present an example to illustrate our main result.

In [6], Jia et al. consider the fractional-order differential equation integral boundary value problem with disturbance parameters 1], [0, +∞)), disturbance parameter a > 0, and C D δ is the Caputo fractional derivative of order δ. By using an upper and lower solution method, the fixed point index theorem and the Schauder fixed point theorem, sufficient conditions are obtained for the problem to have at least one positive solution, two positive solutions and no solution.
In [25], Wang et al. consider a class of fractional differential equations with integral boundary conditions which involve two disturbance parameters. By using the Guo-Krasnoselskii fixed point theorem, new results on the existence and nonexistence of positive solutions for the boundary value problem are obtained. The problem is given by where D α 0+ is the standard Riemann-Liouville fractional derivative with 3 < α ≤ 4, f : [0, 1] × [0, +∞) → [0, +∞) is a continuous function, g 1 , g 2 ∈ L 1 [0, 1] and a, b ≥ 0.
In [26] Hao et al. consider the existence of positive solutions of higher order fractional integral boundary value problem with a parameter 0+ are the standard Riemann-Liouville fractional derivative, n -1 < η ≤ n, η ≥ 4, 2 ≤ k ≤ n -2, α, β, γ , δ > 0. The purpose of this paper is to establish conditions ensuring the existence of three positive solutions of BVP (1) and give an estimate of these solutions by using the Avery-Peterson fixed point theorem. Our supposed problem is different from the problems studied before and mentioned above. Our result is new and our work extends the application of the theorem.
In this paper, a positive solution x = x(t) of BVP (1) means a solution of (1) satisfying Throughout this paper, we always assume that the following condition is satisfied:
Here we present some necessary basic results that will be used.

Lemma 2.1 (see [2])
The Caputo fractional derivative of order n -1 < α < n for t β is given by has an unique solution Denote From (L 0 ), we know, for t ∈ (0, 1), Thus Thus, the following lemma holds.

Lemma 2.7
Assume (L 0 ) hold, then the function ω(t) satisfies the following properties: where ρ is given by (15). and Hence, To finish this section, we present the well-known Avery-Peterson fixed point theorem as follows.
Let γ and θ be nonnegative continuous convex functionals on P, ϕ be a nonnegative continuous concave functional on P, and ψ be a nonnegative continuous functional on P. For A, B, C, D > 0, we define the following convex set: and a closed set Then T has at least three fixed point

Main results
In this section, we prove the existence of positive solution of BVP (1) by applying the following Avery-Peterson fixed point theorem.
We consider the Banach space 1] x(t) , max 1] x(t) , then P is a cone in E. Define the operator T : P → E by Lemma 3.1 Assume (L 0 ) hold, then T : P → P is a completely continuous operator.
Proof For x ∈ P, it is easy to see that T is continuous operator and Tx(t) ≥ 0. By (14), we have From Lemma 2.5 and Lemma 2.6 and Lemma 2.7, similar to Lemma 3.1 in [19], we can easily prove that T is a completely continuous operator.
To sum up, the conditions of Lemma 2.8 are all verified and we notice that x i (t) ≥ ω(0) > 0. Hence, BVP (1) has at least three positive solutions x 1 , x 2 , x 3 satisfying (18) and (19).