Positive Solutions for a System of Coupled Semipositone Fractional Boundary Value Problems with Sequential Fractional Derivatives

: We study the existence and multiplicity of positive solutions for a system of Riemann– Liouville fractional differential equations with sequential derivatives, positive parameters and sign-changing singular nonlinearities, subject to nonlocal coupled boundary conditions which con-tain Riemann–Stieltjes integrals and various fractional derivatives. In the proof of our main existence results we use the nonlinear alternative of Leray–Schauder type and the Guo–Krasnosel’skii ﬁxed point theorem.

where p 1 , p 2 , q 1 , q 2 ∈ R, p 1 ∈ [1, n − 2], p 2 ∈ [1, m − 2], q 1 ∈ [0, p 1 ], q 2 ∈ [0, p 2 ], ξ i , a i ∈ R for all i = 1, . . . , N (N ∈ N), 0 < ξ 1 < · · · < ξ N ≤ 1, η i , b i ∈ R for all i = 1, . . . , M (M ∈ N), 0 < η 1 < · · · < η M ≤ 1, f and g are nonnegative and nonsingular functions was studied in [4]. In [4], the author presented conditions for f and g and intervals for positive parameters λ, µ such that the problem (3) and (4) has at least one positive solution or it has no positive solutions. In [5], the author investigated the existence of solutions for the nonlinear system of fractional differential equations with the coupled nonlocal boundary conditions where α, β ∈ R, α ∈ (n − 1, n], β ∈ (m − 1, m], n, m ∈ N, n ≥ 2, m ≥ 2, θ 1 , is the Riemann-Liouville integral of order ζ (for ζ = θ 1 , σ 1 , θ 2 , σ 2 ), f and g are nonlinear functions, and the integrals from the boundary conditions (BC) are Riemann-Stieltjes integrals with H i for i = 1, . . . , p and K i for i = 1, . . . , q functions of bounded variation. She proved the existence of a unique solution of problem (5) and (6) by using the Banach contraction mapping principle, and five existence results by applying the Leray-Schauder alternative theorem, the Krasnosel'skii theorem for the sum of two operators (for two results), the Schauder fixed point theorem, and the nonlinear alternative of Leray-Schauder type, respectively. In [6], the authors studied the existence of multiple positive solutions for the nonlinear fractional differential equation with the integral-differential boundary conditions . . , m are functions of bounded variation, and the nonlinearity f (t, u) may change sign and may be singular at the points t = 0, 1 and/or u = 0. In the proof of the main theorem, they used various height functions of f defined on special bounded sets, and two theorems from the fixed point index theory. In [7], the authors investigated the existence of positive solutions for the system of fractional differential equations subject to the coupled integral boundary conditions where α, β ∈ R, α ∈ (n − 1, n], β ∈ (m − 1, m], n, m ∈ N, n, m ≥ 3, p, q ∈ R, p ∈ [1, n − 2], q ∈ [1, m − 2], the integrals from (8) are Riemann-Stieltjes integrals with H and K functions of bounded variation, λ and µ are positive parameters, and f and g are sign-changing continuous functions which may be singular at t = 0 and/or t = 1. In [7], the authors present various assumptions on the nonlinearities f and g and intervals for λ and µ such that the problem (7) and (8) has at least one positive solution. In [8], the authors studied the existence and multiplicity of positive solutions for the system (7) with λ = µ = 1, subject to the coupled multi-point boundary conditions , and the functions f and g are nonegative and they can be nonsingular or singular at the points t = 0 and/or t = 1. They used some theorems from the fixed point index theory and the Guo-Krasnosel'skii fixed point theorem. In [9], the author investigated the existence and nonexistence of positive solutions for a system with three Riemann-Liouville fractional differential equations with positive parameters, nonnegative and nonsingular nonlinearities, supplemented with uncoupled multi-point boundary conditions, by using (for the existence) the Guo-Krasnosel'skii fixed point theorem. In [10], the authors studied the existence and nonexistence of positive solutions for the system (7) with nonnegative and nonsingular functions f and g, subject to the coupled boundary conditions where H and K are nondecreasing functions. For other recent studies on fractional differential equations and systems see the papers [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26], and the books [27,28]. We also mention the books [29][30][31][32][33][34][35], and their references, where the authors present applications of the fractional calculus and fractional differential equations in many scientific and engineering domains.
The semipositone boundary value problems are more difficult to solve than other problems with nonnegative and singular/nonsingular nonlinearities. Motivated by [6,7], in the present paper, we consider 1 = 2 = 2 in the system (3), and sign-changing and singular nonlinearities f and g, with the general nonlocal boundary conditions (2). We were able to apply the change of functions (see Section 3 and problem (14) and (15)) only for these values of 1 and 2 . So our paper was also motivated by the application of p-Laplacian operators in various fields such as fluid flow through porous media, nonlinear elasticity, glaciology, etc., (see [36] and its references).
The paper is organized as follows. In Section 2, we study a nonlocal boundary value problem for fractional differential equations with sequential fractional derivatives, and we give some properties of the associated Green functions. Section 3 is devoted to the main existence theorems for the positive solutions of problem (1) and (2). In Section 4, we present two examples which illustrate our results, and Section 5 contains the conclusions for the paper.

Existence and Multiplicity of Positive Solutions
In this section, we investigate the existence and multiplicity of positive solutions for problem (1) and (2) under various assumptions on the sign-changing nonlinearities f and g which may be singular at t = 0 and/or t = 1, and for some intervals for the parameters λ and µ. We present the assumptions that we will use in our results.
, f , g may be singular at t = 0 and/or t = 1, and there exist functions ζ 1 , (I6) There exists . We consider the system of nonlinear fractional differential equations with the boundary conditions where is the solution of the system of fractional differential equations with the boundary conditions Under the assumptions (I1) and (I2), or (I1) and (I4) we have ξ 1 (t) ≥ 0, ξ 2 (t) ≥ 0 for all t ∈ [0, 1]. We shall prove that there exists a solution (x, y) for the boundary value problem (14) and (15) represent a positive solution of the boundary value problem (1) and (2). Indeed, by (14)-(17) we have for t ∈ (0, 1) So, in what follows we shall investigate the boundary value problem (14) and (15). By using Lemma 1 (relations (11)), the problem ( (14) and (15)) is equivalent to the system We consider the Banach space X = C[0, 1] with the supremum norm · , and the Banach space We define the cone For λ, µ > 0, we introduce the operators A 1 , A 2 : It is easy to see that (x, y) ∈ P is a solution of problem (14) and (15) if and only if (x, y) is a fixed point of operator A. Proof. The operators A 1 and A 2 are well-defined. To prove this, let (x, y) ∈ P be fixed with (x, y) Y = L, that is x + y = L. Then, we have If (I1) and (I2) hold, we deduce If (I1) and (I4) hold, we obtain for all t ∈ [0, 1] In a similar manner we find where J 30 = max s∈[0,1] J 3 (s), J 40 = max s∈[0,1] J 4 (s). Therefore A 1 (x, y) and A 2 (x, y) are well-defined. Besides, by Lemma 3, we deduce that . We obtain (A 1 (x, y), A 2 (x, y)) ∈ P, and hence A(P ) ⊂ P. By using standard arguments, we conclude that operator A : P → P is a completely continuous operator. Theorem 1. We suppose that (I1) − (I3) hold. Then there exist the constants λ 0 > 0 and µ 0 > 0 such that for any λ ∈ (0, λ 0 ] and µ ∈ (0, µ 0 ], the boundary value problem (1) and (2) has at least one positive solution.

Now we define
We consider firstly that f ∞ = ∞, so we have Let (x, y) ∈ P ∩ ∂Ω 2 . So (x, y) Y = R 2 or equivalently x + y = R 2 . This last relation gives us x ≥ R 2 /2 or y ≥ R 2 /2. We suppose firstly that x ≥ R 2 /2. Then we obtain Therefore we deduce Then by (20) and (22), we find Hence for any t ∈ [θ 1 , θ 2 ] we obtain Then A 1 (x, y) ≥ (x, y) Y and therefore we conclude If for (x, y) ∈ P ∩ ∂Ω 2 one has the case y ≥ R 2 /2, then Therefore for any t ∈ [θ 1 , Then by (20) and (25), we find It follows that for any t ∈ [θ 1 , θ 2 ] we have Then A 1 (x, y) ≥ (x, y) Y and we obtain again the relation (23).