Positive solutions of higher order boundary value problems of nonlinear fractional differential equations.

摘要

Fractional differential equations and boundary value problems are used in many fields of science and engineering. This thesis will first introduce the definition and properties of the Riemann-Liouville and Grunwald-Letnikov fractional differential operator, and basic knowledge on ordinary and fractional boundary value problems. This will be the basis for introducing new results in the existence, uniqueness, and number of positive solutions for higher order boundary value problems of nonlinear fractional differential equations.;The main work in this thesis is about the higher order fractional differential equation [special characters omitted] with the boundary condition [special characters omitted].;where 1 < alpha < 2, m ∈ N, D(alpha/0+) is the Riemann-Liouville fractional differential operator. (D(alpha/0+)0 = I, the identity operator, and (D(alpha/0+)j +1 = D(alpha/0+) (D(alpha/0+ )j for j = 0, ..., m -- 1. The Green's function and fixed point theory on cones are used to establish criteria for fractional boundary value problems to have one, two, an arbitrary number, and even an infinite number of positive solutions.