Abstract
Positive solutions for higher-order nonlinear fractional differential equation with integral boundary condition
Highlights
We are interested in the following nonlinear fractional differential equation n−1
There are some papers which deal with the existence and multiplicity of solutions for nonlinear FDE boundary value problems by using techniques of topological degree theory
Since only positive solutions are useful for many applications, we investigate the existence and multiplicity of positive solutions for BVP (1.1)-(1.2) in this paper
Summary
We are interested in the following nonlinear fractional differential equation n−1. subject to the boundary conditions n−1 u(1) − u(0) = ai[I0i+u(t)]t=1, u(k)(0) = bk, k = 1, 2, · · · , n − 1, i=1. There are some papers which deal with the existence and multiplicity of solutions for nonlinear FDE boundary value problems (in short:BVPs) by using techniques of topological degree theory (see [12,13,14,15,20,21] and the references therein). Zhang discussed the existence of solutions of the nonlinear FDE cD0α+u(t) = f (t, u(t)), 0 < t < 1, 1 < α ≤ 2 with the boundary conditions (1.5). We can see two facts: the first, the BVPs of nonlinear FDE have been studied by some authors, to the best of our knowledge, higher-order fractional equations with integral boundary conditions are seldom considered; the second, the author in [15] studied the BVP with integral conditions, those results can’t ensure the solutions to be positive. Two examples are given to demonstrate our results
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