Application of Spectral Methods to Boundary Value Problems for Differential Equations

Ene Petronela

doi:10.5402/2011/276701

Abstract

We try to generalize the concept of a spectrum in the nonlinear case starting from its splitting into several subspectra, not necessarily disjoint, following the classical decomposition of the spectrum. To obtain an extension of spectrum with rich properties, we replace the identity map by a nonlinear operator acting between two Banach spaces and , which takes into account the analytical and topological properties of a given operator , although the original definitions have been given only in the case and . The FMV spectrum reflects only asymptotic properties of , while the Feng's spectrum takes into account the global behaviour of and gives applications to boundary value problems for ordinary differential equations or for the second-order differential equations, which are referred to as three-point boundary value problems with the classical or the periodic boundary conditions.

Highlights

Let us first recall the concept of a spectrum for linear operators acting in a complex Banach space X
To obtain an extension of spectrum with rich properties, we replace the identity map by a nonlinear operator J acting between two Banach spaces X and Y, which takes into account the analytical and topological properties of a given operator F, the original definitions have been given only in the case X Y and J I
We denote by L X the algebra of all bounded linear operators on X and the resolvent set of L is defined by ρ L λ ∈ /⊂/ λI − L −1 ∈ L X

Summary

Ene Petronela

Faculty of Tourism and Commercial Management Constanta, “Dimitrie Cantemir” Christian University Bucharest, Romania. We try to generalize the concept of a spectrum in the nonlinear case starting from its splitting into several subspectra, not necessarily disjoint, following the classical decomposition of the spectrum. To obtain an extension of spectrum with rich properties, we replace the identity map by a nonlinear operator J acting between two Banach spaces X and Y , which takes into account the analytical and topological properties of a given operator F, the original definitions have been given only in the case X Y and J I. The FMV spectrum reflects only asymptotic properties of F, while the Feng’s spectrum takes into account the global behaviour of F and gives applications to boundary value problems for ordinary differential equations or for the second-order differential equations, which are referred to as three-point boundary value problems with the classical or the periodic boundary conditions

Introduction

Ez t

We put

Then L is invertible on Y with inverse

We have