On a System of Riemann–Liouville Fractional Boundary Value Problems with (cid:36) -Laplacian Operators and Positive Parameters

: In this paper, we study the existence and nonexistence of positive solutions of a system of Riemann–Liouville fractional differential equations with (cid:36) -Laplacian operators, supplemented with coupled nonlocal boundary conditions containing Riemann–Stieltjes integrals, fractional derivatives of various orders, and positive parameters. We apply the Schauder ﬁxed point theorem in the proof of the existence result.

We present in this paper sufficient conditions for the functions f and g, and intervals for the parameters c 0 and d 0 such that problem (1) and (2) has at least one positive solution, or it has no positive solutions. We apply the Schauder fixed point theorem in the proof of the main existence result. A positive solution of (1) and (2) is a pair of functions (u, v) ∈ (C([0, 1]; R + )) 2 that satisfy the system (1) and the boundary conditions (2), with u(t) > 0 and v(t) > 0 for all t ∈ (0, 1]. Now, we present some recent results related to our problem. By using the Guo-Krasnosel'skii fixed point theorem, in [1], the authors investigated the system of fractional differential equations D γ 1 0+ (ϕ 1 (D δ 1 0+ u(t))) + λ f (t, u(t), v(t)) = 0, t ∈ (0, 1), D γ 2 0+ (ϕ 2 (D δ 2 0+ v(t))) + µg (t, u(t), v(t)) = 0, t ∈ (0, 1), supplemented with the boundary conditions (2) with c 0 = d 0 = 0, where λ and µ are positive parameters, and f , g ∈ C([0, 1] × R + × R + , R + ). They presented various intervals for λ and µ such that problem (2) and (3) with c 0 = d 0 = 0 has at least one positive solution (u(t) > 0 for all t ∈ (0, 1], or v(t) > 0 for all t ∈ (0, 1]). They also studied the nonexistence of positive solutions. In [2], the author investigated the existence and nonexistence of positive solutions for the system (3) with the uncoupled boundary conditions . . , n, and K j , j = 1, · · · , m are functions of bounded variation. In [3], the authors studied the positive solutions for the system of nonlinear fractional differential equations subject to the coupled integral boundary conditions where n − 1 < α ≤ n, m − 1 < β ≤ m, n, m ∈ N, n, m ≥ 3, a, b, f, g are nonnegative continuous functions, c 0 and d 0 are positive parameters, and H and K are bounded variation functions. In [4], the authors investigated the existence and nonexistence of positive solutions for the system (1) with the nonlocal uncoupled boundary conditions with positive parameters We note that our problem (1) and (2) is different than the problem studied in [4], because of the boundary conditions, which are coupled in (2) and uncoupled in [4]. Based on this difference, here, we will use, for problem (1) and (2), other Green functions, different systems of integral equations, and different operators than those in [4]. We would also like to mention the papers [5][6][7][8][9][10], and the monographs [11][12][13], which contain other recent results for fractional differential equations and systems of fractional differential equations with or without Laplacian operators, and for various applications. The novelties of our problem (1) and (2) with respect to the above papers consist in the consideration of positive parameters c 0 and d 0 in the coupled nonlocal boundary conditions (2) containing fractional derivatives of various orders and Riemann-Stieltjes integrals, combined with the system of fractional differential Equation (1), which has -Laplacian operators.
The paper is structured as follows. In Section 2, we present some auxiliary results, which include the Green functions associated with our problem (1) and (2) and their properties. In Section 3, we give the main theorems for the existence and nonexistence of positive solutions for (1) and (2), and Section 4 contains an example illustrating our results. Finally, in Section 5, we present the conclusions of this work.

Main Results
In this section, we study the existence and nonexistence of positive solutions for problem (1) and (2) under some conditions on a, b, f, and g, when the positive parameters c 0 and d 0 belong to some intervals.
We now give the assumptions that we will use in the next part.
Proof. By assumption (K3) we deduce that there exist s 0 > 0 and t 0 > 0 such that . We define now c 1 and d 1 as follows: • If ∆ 1 = 0 and ∆ 2 = 0, then Let c 0 ∈ (0, c 1 ] and d 0 ∈ (0, d 1 ]. Then, for (h, k) ∈ V and ζ ∈ [0, 1], we have and so By using Lemma 3, we deduce that S i (h, k)(t) ≥ 0, i = 1, 2 for all t ∈ [0, 1] and (h, k) ∈ V. By inequalities (16), for all (h, k) ∈ V, we obtain and I γ 2 Then, by Lemma 2 and the definition of L from (K3), we find Therefore, we find that S(V ) ⊂ V. By using a standard method, we conclude that S is a completely continuous operator. Therefore, by the Schauder fixed point theorem, we deduce that S has a fixed point (h, k) ∈ V, which is a non-negative solution for problem (15), or equivalently, for problem (13) and (14). Hence, (u, v), where u(t) = h(t) + x(t) and v(t) = k(t) + y(t) for all t ∈ [0, 1], is a positive solution of problem (1) and (2). This solution (u, v) satisfies the conditions t δ 1 −1 ∆ (c 0 The second result is the following nonexistence theorem for the boundary value problem (1) and (2).
, with x and y given by (10), is a solution of problem (13) and (14), or equivalently, of system (15). By using Lemma 3, we find that k . By the definition of the functions x and y, we obtain By using Lemma 3 and the above inequalities we find and then We deduce that h ≥ h(η 1 ) > 0. In a similar manner, we obtain and so We deduce that k ≥ k(η 1 ) > 0. In addition, from the above inequalities we have and so Hence, In a similar manner, we deduce and then Hence, by (17) and (18), we conclude that h ≤ 1 2 k ≤ 1 4 h , which is a contradiction (we saw before that h > 0). Therefore, problem (1) and (2) has no positive solution.

Conclusions
In this paper, we studied the system of coupled Riemann-Liouville fractional differential Equation (1) with 1 -Laplacian and 2 -Laplacian operators, subject to the nonlocal coupled boundary conditions (2), which contain fractional derivatives of various orders, Riemann-Stieltjes integrals, and two positive parameters c 0 and d 0 . Under some assumptions for the nonlinearities f and g of system (1), we established intervals for the parameters c 0 and d 0 such that our problem (1) and (2) has at least one positive solution. First, we made a change of unknown functions such that the new boundary conditions have no positive parameters. By using the corresponding Green functions, the new boundary value problem was then written equivalently as a system of integral equations (namely the system (15)). We associated to this integral system an operator (S), and we proved the existence of at least one fixed point for it by applying the Schauder fixed point theorem. Intervals for parameters c 0 and d 0 were also given such that problem (1) and (2) has no positive solution. Finally, we presented an example to illustrate our main results.