Table of Contents
ISRN Applied Mathematics
Volume 2014 (2014), Article ID 305718, 13 pages
http://dx.doi.org/10.1155/2014/305718
Research Article

On Second-Order Differential Equations with Nonsmooth Second Member

1DM, UEPB, Campina Grande-PB, Brazil
2IM, UFRJ, Rio de Janeiro, RJ, Brazil

Received 15 January 2014; Accepted 27 February 2014; Published 24 March 2014

Academic Editors: A. Bellouquid and C. Join

Copyright © 2014 M. Milla Miranda et al. 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

In an abstract framework, we consider the following initial value problem: u′′ + μAu + F(u)u = f  in  (0,T), , where is a positive function and f a nonsmooth function. Given u0, u1, and f we determine in order to have a solution u of the previous equation. We analyze two cases of . In our approach, we use the Theory of Linear Operators in Hilbert Spaces, the compactness Aubin-Lions Theorem, and an argument of Fixed Point. One of our two results provides an answer in a certain sense to an open question formulated by Lions in (1981, Page 284).