Existence and uniqueness of positive solutions for boundary value problems of a fractional differential equation with a parameter

Year 2015, Volume: 44 Issue: 3, 659 - 667, 01.06.2015

Chen Yang

Abstract

In this paper, we are concerned with the existence and uniqueness
of positive solutions for the following nonlinear fractional two-point
boundary value problem

D
α
0+u(t) + λf(t, u(t), u(t)) = 0, 0 < t < 1, 2 < α ≤ 3,
u(0) = u
0
(0) = u
0
(1) = 0,
where D
α
0+ is the standard Riemann-Liouville fractional derivative, and
λ is a positive parameter. Our analysis relies on a fixed point theorem
and some properties of eigenvalue problems for a class of general mixed
monotone operators. Our results can not only guarantee the existence
of a unique positive solution, but also be applied to construct an iterative scheme for approximating it. An example is given to illustrate the
main results.

Keywords

Riemann-Liouville fractional derivative; Fractional differential equation; Positive solution; Existence and uniqueness; Mixed monotone operator

References

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