Existence of Positive Solutions for a System of Singular Fractional Boundary Value Problems with p -Laplacian Operators

: We investigate the existence and multiplicity of positive solutions for a system of Riemann–Liouville fractional differential equations with singular nonnegative nonlinearities and p -Laplacian operators, subject to nonlocal boundary conditions which contain fractional derivatives and Riemann–Stieltjes integrals.

The paper is organized as follows. In Section 2, we study two nonlocal boundary value problems for fractional differential equations with p-Laplacian operators, and we present some properties of the associated Green functions. Section 3 contains the main existence theorems for the positive solutions for our problem (1) and (2), and in Section 4, we give two examples which illustrate our results.

Auxiliary Results
We consider firstly the nonlinear fractional differential equation with the boundary conditions where . . , p are bounded variation functions, and h ∈ C(0, 1) ∩ L 1 (0, 1). We denote by where the Green function G 1 is given by with Proof. We denote by ϕ r 1 (D β 1 0+ u(t)) = x(t). Then problem (3) and (4) is equivalent to the following two boundary value problems and For the first problem (8), the function is the unique solution x ∈ C[0, 1] of (8). For the second problem (9), if ∆ 1 = 0, then by [7] (Lemma 2.2), we deduce that the function where G 1 is given by (6), is the unique solution u ∈ C[0, 1] of problem (9). Now, by using relations (10) and (11), we find formula (5) for the unique solution u ∈ C[0, 1] of problem (3) and (4).
Next we consider the nonlinear fractional differential equation with the boundary conditions where . . , q are bounded variation functions, and k ∈ C(0, 1) ∩ L 1 (0, 1). We denote by In a similar manner as above we obtain the following result.

Lemma 2.
If ∆ 2 = 0, then the unique solution v ∈ C[0, 1] of problem (12) and (13) is given by where the Green function G 2 is given by with By using the properties of the functions g 1 , g 2i , i = 1, . . . , p, g 3 , g 4i , i = 1, . . . , q given by (7) and (16) (see [7] and [17]), we obtain the following properties of the Green functions G 1 and G 2 that we will use in the next section.

Existence of Positive Solutions
In this section, we investigate the existence of positive solutions for problem (1) and (2) under various assumptions on the functions f and g which may be singular at t = 0 and/or t = 1. We present the basic assumptions that we will use in the main theorems. and where

Remark 1.
We present below two cases in which Λ 1 , Λ 2 ∈ (0, ∞); for other cases see the examples from Section 4. (17) are satisfied with equality. In addition, the conditions Λ 1 , Λ 2 ∈ (0, ∞) are also satisfied, because in this nonsingular case, we obtain In a similar manner we have , then by using the Cauchy inequality we find where ζ 1 2 is the norm of ζ 1 in the space L 2 (0, 1). In a similar manner we obtain Λ 2 ∈ (0, ∞).
By using Lemmas 1 and 2 (the relations (5) and (14)), (u, v) is a solution of problem (1) and (2) if and only if (u, v) is a solution of the nonlinear system of integral equations We consider the Banach space We also define the operators A 1 , (1) and (2) if and only if (u, v) is a fixed point of operator A.
Lemma 5. Assume that (I1) and (I2) hold. Then A : Q → Q is a completely continuous operator (continuous, and it maps bounded sets into relatively compact sets).
Therefore for any t ∈ (0, 1) we deduce We denote by We compute the integral of function θ 1 , by exchanging the order of integration, and we have For the integral of the function θ 2 , we obtain We conclude that θ 2 ∈ L 1 (0, 1). Hence for any t 1 , t 2 ∈ [0, 1] with t 1 ≤ t 2 and (u, v) ∈ S, by (18) and (19), we find By (19), (20) and the absolute continuity of the integral function, we deduce that A 1 (S) is equicontinuous. By a similar approach, we obtain that A 2 (S) is also equicontinuous, and so A(S) is equicontinuous. Using the Ascoli-Arzela theorem, we conclude that A 1 (S) and A 2 (S) are relatively compact sets, and so A(S) is also relatively compact. Besides, we can prove that A 1 , A 2 and A are continuous on Q (see [16] (Lemma 1.4.1)). Then A is a completely continuous operator on Q.
We define now the cone Under the assumptions (I1) and (I2), by using Lemma 4, we obtain A(Q) ⊂ Q 0 , and so A| Q 0 : Q 0 → Q 0 (denoted again by A) is also a completely continuous operator. For r > 0 we denote by B r the open ball centered at zero of radius r, and by B r and ∂B r its closure and its boundary, respectively. Theorem 1. Assume that (I1) and (I2) hold. In addition, the functions χ 1 , χ 2 , f and g satisfy the conditions (I3) There exist µ 1 ≥ 1 and µ 2 ≥ 1 such that g(t, x, y) Then problem (1) and (2) has at least one positive solution (u(t), v(t)), t ∈ [0, 1].

Conclusions
In this paper, we have discussed the existence and multiplicity of positive solutions for a system of Riemann-Liouville fractional differential equations with singular nonnegative nonlinearities and p-Laplacian operators, complemented with nonlocal boundary conditions involving fractional derivatives and Riemann-Stieltjes integrals. Some properties of the associated Green functions are also presented. Two examples are constructed for the illustration of the obtained results.