On solutions of some fractional m-point boundary value problems at resonance

Zhanbing Bai

doi:10.14232/ejqtde.2010.1.37

Abstract

In this paper, the following fractional order ordinary differential equation boundary value problem: D � ( �)

Highlights

The subject of fractional calculus has gained considerable popularity and importance during the past decades or so, due mainly to its demonstrated applications in numerous seemingly and widespread fields of science and engineering
M-point integer-order differential equation boundary value problems have been studied by many authors, see [4, 12, 13, 14]
In [7], we investigated the boundary value problem at resonance m−2

Summary

Introduction

The subject of fractional calculus has gained considerable popularity and importance during the past decades or so, due mainly to its demonstrated applications in numerous seemingly and widespread fields of science and engineering. The purpose of this paper is to study the existence of solution for boundary value problem (1.1), (1.2) at resonance case, and establish an existence theorem under nonlinear growth restrictions of f. Let Y, Z be real Banach spaces, L : dom(L) ⊂ Y → Z be a Fredholm map of index zero and P : Y → Y, Q : Z → Z be continuous projectors such that Im(P ) = Ker(L), Ker(Q) = Im(L) and Y = Ker(L) ⊕ Ker(P ), Z = Im(L) ⊕ Im(Q). It follows that L|dom(L)∩Ker(P ) : dom(L)∩Ker(P ) → Im(L) is invertible.

Background materials and preliminaries

Existence result

An example