Journal of Inequalities and Applications
Volume 2007 (2007), Article ID 72931, 8 pages
doi:10.1155/2007/72931
Research Article

Asymptotic Behavior of Solutions to Some Homogeneous Second-Order Evolution Equations of Monotone Type

Behzad Djafari Rouhani1 and Hadi Khatibzadeh2

1Department of Mathematical Sciences, University of Texas at El Paso, El Paso 79968, TX, USA
2Department of Mathematics, Tarbiat Modares University, P.O. Box 14115-175, Tehran, Iran

Received 7 November 2006; Accepted 12 April 2007

Academic Editor: Andrei Ronto

Copyright © 2007 Behzad Djafari Rouhani and Hadi Khatibzadeh. 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 study the asymptotic behavior of solutions to the second-order evolution equation p(t)u″(t)+r(t)u′(t)∈Au(t) a.e. t∈(0,+∞), u(0)=u0, supt≥0|u(t)|<+∞, where A is a maximal monotone operator in a real Hilbert space H with A−1(0) nonempty, and p(t) and r(t) are real-valued functions with appropriate conditions that guarantee the existence of a solution. We prove a weak ergodic theorem when A is the subdifferential of a convex, proper, and lower semicontinuous function. We also establish some weak and strong convergence theorems for solutions to the above equation, under additional assumptions on the operator A or the function r(t).