Table of Contents
ISRN Mathematical Analysis
Volume 2011, Article ID 276701, 13 pages
http://dx.doi.org/10.5402/2011/276701
Research Article

Application of Spectral Methods to Boundary Value Problems for Differential Equations

Faculty of Tourism and Commercial Management Constanta, “Dimitrie Cantemir” Christian University Bucharest, Romania

Received 13 January 2011; Accepted 12 March 2011

Academic Editor: G. Mantica

Copyright © 2011 Ene Petronela. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝐽 acting between two Banach spaces 𝑋 and π‘Œ , which takes into account the analytical and topological properties of a given operator 𝐹 , although the original definitions have been given only in the case 𝑋 = π‘Œ 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.