Positive solutions for a system of higher-order singular nonlinear fractional differential equations with nonlocal boundary conditions

The paper deals with the existence and multiplicity of positive solutions for a system of higher-order singular nonlinear fractional differential equations with nonlocal boundary conditions. The main tool used in the proof is fixed point index theory in cone. Some limit type conditions for ensuring the existence of positive solutions are given.

Motivated by the above mentioned works and continuing the paper [27], in this paper, we present some limit type conditions and discuss the existence and multiplicity of positive solutions of the singular system (1.1)-(1.2) by using of fixed point index theory in cone.Our conditions are applicable for more functions, and the results obtained here are different from those in [7,9,10,17,24,29,30,33].Some examples are also provided to illustrate our main results.
. Then for any g ∈ C[0, 1], the unique solution of the following boundary value problem: is given by where is the Green's function of the integral equation (2.3).
Proof.The equation (2.1) is equivalent to an integral equation:

S. L. Xie (H 6 )
There exists s ∈ (0, +∞) such that (H 7 ) There exists r > 0 such that (H 8 ) f 1 (x, u, v) and f 2 (x, u, v) are increasing with respect to u and v, there exists R > r > 0 such that Then E × E is a real Banach space and P × P is a positive cone of E × E. By (H 1 ), (H 2 ), we can define operators Similar to the proof of Lemma 3.1 in [2], it follows from (H 1 ), (H 2 ) that A j : P × P → P is a completely continuous operator and A(P × P) ⊂ P × P.

Lemma 2.5 ([8]
).Let E be a Banach space, P be a cone in E and Ω ⊂ E be a bounded open set.Assume that A : Ω ∩ P → P is a completely continuous operator.If there exists u 0 ∈ P \ {0} such that u = Au + λu 0 , ∀ λ ≥ 0, u ∈ ∂Ω ∩ P, then the fixed point index i(A, Ω ∩ P, P) = 0.
Lemma 2.6 ([8,14]).Let E be a Banach space, P be a cone in E and Ω ⊂ E be a bounded open set with 0 ∈ Ω. Assume that A : Ω ∩ P → P is a completely continuous operator.
In the following, we adopt the convention that C 1 , C 2 , C 3 , . . .stand for different positive constants.Let Positive solutions of higher-order singular fractional differential equations 7 3 Existence of a positive solution Theorem 3.1.Assume that the conditions (H 1 ), (H 2 ) are satisfied and that (H 3 ), (H 4 ) or (H 7 ), (H 8 ) hold.Then the system (1.1)-(1.2) has at least one positive solution.
So A has a fixed point on (Ω R \ Ω r ) ∩ (P × P).This means that the system (1.1)-(1.2) has at least one positive solution.
Proof.By (H 5 ), there are where We affirm that the set for some λ ≥ 0. We have by (3.16), By the monotonicity and concavity of p(•) as well as Jensen's inequality, (3.18) implies that Hence p( v ) ≤ C 13 .By (1) and (3) of the condition (H 5 ), we know that lim v→+∞ p(v) = +∞, thus there exists C 14 > 0 such that v ≤ C 14 .This shows W is bounded.Then there exists a sufficiently large Q > 0 such that On the other hand, by (H 6 ), there is a σ > 0 and sufficiently small ρ > 0 such that where
Positive solutions of higher-order singular fractional differential equations 13 Hence A has a fixed point on (Ω Q \ Ω R ) ∩ (P × P) and (Ω R \ Ω ρ ) ∩ (P × P), respectively.This means the system (1.1)-(1.2) has at least two positive solutions.

The nonexistence of positive solutions
Theorem 5.1.Assume that the conditions (H 1 ) and (H 2 ) hold, and Then the system (1.1)-(1.2) has no positive solution.
Similarly, we can obtain the following result.