Svatoslav Stanĕk

Nonlocal Boundary Value Problems for Fractional Differential Equations

abstract:
We present the existence principle which can be used for a large class of nonlocal fractional boundary value problems of the form $(^c\kern -2pt D^{\alpha}x)(t)=f(t,x(t),x'(t), (^c\kern -2pt D^{\mu}x)(t))$, $\Lambda(x)=0$, $\Phi(x)=0$, where $^c\kern -2pt D$ is the Caputo fractional derivative. Here, $\alpha \in (1,2)$, $\mu \in (0,1)$, $f$ is a $L^q$-Carathéodory function, $q>\frac{1}{\alpha-1}$, and $\Lambda, \Phi: C^1[0,T] \to \R$ are continuous and bounded ones. The proofs are based on the Leray-Schauder degree theory. Applications of our existence principle are given.

Mathematics Subject Classification: 26A33, 34B16

Key words and phrases: Fractional differential equation, Caputo fractional derivative, nonlocal boundary condition, existence principle, Leray-Schauder degree