Abstract
We study the existence and nonexistence of the positive solutions for the integral boundary value problem of the fractional differential equations with the disturbance parameterain the boundary conditions and the impact of the disturbance parameteraon the existence of positive solutions. By using the upper and lower solutions method, fixed point index theory and the Schauder fixed point theorem, we obtain sufficient conditions for that the problem has at least one positive solution, two positive solutions and no solutions. Under certain conditions, we also obtain the demarcation point which divides the disturbance parameters into two subintervals such that the boundary value problem has positive solutions for the disturbance parameter in one subinterval while no positive solutions in the other.
Highlights
IntroductionWe are concerned with the existence and nonexistence of positive solutions for the boundary value problem of the fractional differential equations
One says u is a solution of the boundary value problem (1) if u ∈ AC2(J), CDδu(t) ∈ C(J) and satisfies (1)
If w is a positive solution of the boundary value problem (57), min w (t) ≥ γ ‖w‖
Summary
We are concerned with the existence and nonexistence of positive solutions for the boundary value problem of the fractional differential equations. In paper [35, 36], the authors studied nonlinear nonlocal boundary value problem with nonhomogeneous boundary conditions u (t) + f (t, u (t) , u (t)) = 0, t ∈ (0, 1) , m u (0) − ∑ aiu (ti) = λ1, i=1. The purpose of this paper is to study the impact of the disturbance parameter a on the existence of positive solutions and obtain sufficient conditions for the boundary value problem (1) to have at least one positive solution, at least two solutions, and no solutions.
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