Existence of positive solutions for p-Laplacian boundary value problems of fractional differential equations

In this paper, we study the existence and multiplicity of ρ-concave positive solutions for a p-Laplacian boundary value problem of two-sided fractional differential equations involving generalized-Caputo fractional derivatives. Using Guo–Krasnoselskii fixed point theorem and under some additional assumptions, we prove some important results and obtain the existence of at least three solutions. To establish the results, Green functions are used to transform the considered two-sided generalized Katugampola and Caputo fractional derivatives. Finally, applications with illustrative examples are presented to show the validity and correctness of the obtained results.

In this work, we study the following p-Laplacian fractional boundary value problem: where ρ 1 ;à + G σ 1 CK and ρ 2 ;ὶ -G σ 2 CK , (ρ 1 , ρ 2 ∈ R \ {1}) are the right-and left-sided Caputo-Katugampola fractional derivatives, 2 < σ 1 , σ 2 ≤ 3, φ p is the p-Laplacian operator, i.e., φ p (ξ ) = |ξ | p-2 ξ , p > 1, , F • is a continuous even function, ℘, are continuous and positive functions. η ∈ (à,ὶ), 0 ≤ μ < 1, and λ ≥ 0. In this paper, we obtain some sufficient conditions ensuring the existence of at least one, two, and three positive solutions for fractional boundary value problem (9). These results can be extended in some works such as [35][36][37]. The rest of the paper is organized as follows. Section 2 presents some basic definitions, lemmas, and preliminary results. In Sect. 3, we derive some conditions on the parameter λ to obtain the existence of at least one positive solution. We derive an interval for λ, which ensures the existence of ρ-concave positive solutions of the fractional boundary value problem in Sect. 4. In Sect. 5, we discuss the existence of multiple positive solutions. Finally, we give some illustrative examples in Sect. 6.

Preliminaries and background material
In addition to the notations introduced with problem (9), let J = [à,ὶ] ⊂ (0, ∞), and ρ > 0, 1: C(J) denotes the Banach space of continuous functions q on J endowed with the norm q C = max τ ∈J |q(τ )|, and 2: AC(J) and C n (J) denote the spaces of absolutely continuous and n times continuously differentiable functions on J respectively. 3: L p (à,ὶ) denotes the space of Lebesgue integrable functions on (à,ὶ). 4: C n ρ (J) is the Banach space of n continuously differentiable functions on J with respect to δ ρ : endowed with the norm

5:
[σ ] is the largest integer less than or equal to σ . Throughout the paper, we use n = [σ ] if σ is an integer and n = [σ ] + 1 otherwise.
Remark 2.1 If r ∈ R + andὶ ≤ (pr) 1/pr , then C(J) → M p r (J) and q M p r ≤ q C for each q ∈ C(J). Now, we recall the Katugampola and Caputo-Katugampola fractional integrals and derivatives [38].

Definition 2.2
The Katugampola left-sided ρ;à + I σ K and right-sided ρ;ὶ -I σ K fractional integrals of noninteger order α > 0 of a function q ∈ M p c (a, T) are defined by The Katugampola fractional derivatives of q are defined by When σ is integer, we consider the ordinary definition.
In the following, we present some properties for left-sided integrals and derivatives. But the same properties are also true for the right-sided ones.

Fixed point theorems
Let E be a real Banach function space, endowed with the infinity norm. A nonempty closed convex set K ⊂ E is called cone (i) if for each q ∈ K and for all λ > 0: λq ∈ K ; (ii) for all q ∈ K , if -q ∈ K , then q = 0. A continuous operator is called completely continuous operator if it maps bounded sets into precompact sets. Let K be a cone, > 0, = q ∈ K : q < , and i is the fixed point index function.
holds, then L has a fixed point in K ∩ ( 2 \ 1 ).
The following technical hypotheses will be used later.

Main results
We present some important lemmas which assist in proving our main results. Consider the linear generalized fractional boundary value problem associated with (9) Lemma 3. 1 For w ∈ C(J), the integral solution of (14) is given by for τ , ξ ∈ J, where and Proof By applying (12), equation (14) becomes for some arbitrary constants l 0 , l 1 , l 2 ∈ R. From the boundary conditions of (14) we get Splitting the second integral in two parts permits us to write The converse follows by direct computation. The proof is completed. Now, consider the generalized p-Laplacian fractional boundary value problem associated with (9) where Proof From Lemma 2.6, equation (18) is equivalent to the equation for some constants l 0 , l 1 , l 2 ∈ R. Using the second boundary condition, we get Consequently, Thus, problem (18) can be written as which, according to Lemma 3.1, has a unique solution of the form (19).

Lemma 3.3
The functions G 1 , G 2 , and H, equations (16), (17), and (20) satisfy the following: and (v) For all (τ , ξ ) ∈ (à,ὶ) 2 , we have Proof Using the definitions of G 1 , G 2 , and H, (i) and (ii) are obtained straightforwardly. For property (iii), we only consider the case ξ ≤ τ as the other case is straightforward. When ξ ≤ τ , we have Thus, G 1 (τ , ξ ) is increasing with respect to τ ∈ J, and therefore and for ξ ≤ τ , we have On the other hand, when τ ≥ ξ , for σ > 0, we obtain For τ ≤ ξ , we have which is a nonincreasing function as σ 1 ≥ 0. Consequently, Using similar techniques, one can prove that forà ≤ ξ , τ <ὶ. Therefore (iv) of Lemma 3.3 holds. Finally, for property (v), we can consider two cases. Nevertheless, we prove the results for the case ξ ≤ τ only. The simpler casè a ≤ τ ≤ ξ <ὶ can be treated with similar arguments. When ξ ≤ τ , we have . Consequently, On the other hand, .
Thus, the proof is completed.
Now, consider the Banach space E = C 3 ρ 1 (J). Suppose that ρ 1 ;à + G σ 1 CK q(τ ) is continuous on J for all q ∈ E, then from Definition 2.6 and Lemma 2.4 we can define the norm on E as follows: in which and the cone K = {q ∈ E : q is nonnegative, increasing, and ρ 1 -concave}. (H2) and let q be the unique solution of fractional boundary value problem (18) associated with given w(τ ) ∈ C + (J). Then q ∈ K and the following inequalities

Lemma 3.4 Assume
where Proof From Lemma 3.2, we have (1) The functions G 1 , G 2 , and H are nonnegative (Lemma 3.3(iii)). In addition, F • (v) is nonnegative for v ≥ 0 (thanks to (H2)). Thus, q is also nonnegative. Furthermore, as G 1 is increasing w.r.t. τ (Lemma 3.3(iv)), so it is the function q. To prove that q is ρ 1 -concave, we need to show that δ 1 ρ 1 q(τ ) is decreasing on J (Remark 2.2), which can be obtained from the negativity of the derivative (2) As q is nonnegative and increasing, we have For τ ∈ [à • ,ὶ • ], using (iv) of Lemma 3.3 and the fact that Consequently, and thus (23) holds.

Thus, we obtain (24). (4) A straightforward calculus gives
Then we get By multiplying both sides of the previous inequality by Multiplying both sides by G 1 (τ , ξ ) and integrating over J w.r.t. ξ , we get (21), one can see that

Lemma 3.6 Assume (H2) is true. Then N λ : ϒ → ϒ is continuous and compact.
Proof The continuity of N λ is a consequence of the continuity and positiveness of G 1 , G 2 , H, , and ℘. To prove that N λ is compact, let us consider a bounded subset ⊂ ϒ. Then there exists L > 0 such that for any q ∈ we have |℘(q(τ ))| ≤ L. For any q ∈ , as N q is positive and G 1 is increasing w.r.t. τ , we have Consequently, using the previous inequality and hypothesis (H2), we get Then, as in Lemma 3.4, we obtain N λ q ≤M 3L , wherȇ Hence, N λ ( ) is uniformly bounded. Furthermore, by using Lemmas (3.2), (3.5), (3.3), and the Lebesgue dominated convergence theorem, we deduce the equicontinuity of N λ ( ). Therefore, N λ is completely continuous by the Arzelà-Ascoli theorem.
Thus N λ (K(q 0 )) ⊆ K(q 0 ). Now, Schauder's fixed point theorem implies that there exists a fixed point q ∈ K(q 0 ) such that it is a positive solution of (9). The proof is completed. (H1) and (H2) hold. Assume that ℘ also satisfies:

Theorem 4.4 Suppose that conditions
. Then fractional boundary value problem (9) has at least one ρ 1 -concave positive solution for λ small enough.

Several solutions in a cone
In order to show the existence of multiple solutions, we will use the Leggett-Williams fixed point theorem [43]. For this, we define the following subsets of a cone K : where the constants m 2 and m 1 are defined in (33). Then fractional boundary value problem (9) has at least three positive ρ 1 -concave solutions q 1 , q 2 , and q 3 satisfying q 1 <å, γ b < ϕ(q 2 ), and q 3 >å with ϕ(q 3 ) < bγ for λ small enough.
Proof By the induction method, we get the proof.

Conclusion
The paper presents a new p-Laplacian boundary value problem of two-sided fractional differential equations involving generalized Caputo fractional derivatives, and we investigate the existence and multiplicity of ρ-concave positive solutions of it. We made some