Asymptotic behavior of bounded solutions to a class of second order nonhomogeneous evolution equations

By using previous results of Djafari Rouhani [B. Djafari Rouhani, Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph. D. Thesis, Yale University, 1981, part 1, pp. 1-76; B. Djafari Rouhani, Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl. 147 (1990) 465-476; B. Djafari Rouhani, Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl. 151 (1990) 226-235] for dissipative systems, we study the weak and strong convergence of solutions to the following class of second order nonhomogeneous evolution equations {u ''(t) - cu'(t) is an element of Au(t) + f(t) a,e,t is an element of (0, +infinity) u(0) = u(0), sup(t >= 0)|u(t)| < +infinity where A is a monotone operator in a real Hilbert space H, c >= 0, and f : R+ -> H is a given function. Our results extend previous results by Morosanu [G. Morosanu, Nonlinear Evolution Equations and Applications, Editura Academiei Romane (and D. Reidel Publishing Company), Bucharest, 1988; G. Morosanu, Asymptotic behaviour of solutions of differential equations associated to monotone operators, Nonlinear Anal. 3 (1979) 873-883] and Mitidieri [E. Mitidieri, Some remarks on the asymptotic behaviour of the solutions of second order evolution equations, J. Math. Anal. Appl. 107 (1985) 211-221] who studied the case c = 0 and f equivalent to 0 by assuming that A is maximal monotone and A(-1) (0) not equal phi, as well as previous results by Veron [L. Veron, Problemes d'evolution du second ordre associes A des operateurs monotones, C. R. Acad. Sci. Paris Ser. A 278 (1974) 1099-1101; L Veron, Equations d'evolution du second ordre associees des operateurs maximaux monotones, Proc. Roy. Soc. Edinburgh Sect. A 75 (1975-76) 131-147] and the authors [B. Djafari Rouhani, H. Khatibzadeh, Asymptotic behavior of solutions to some homogeneous second order evolution equations of monotone type J. C Inequal. Appl. (2007) Art. ID72931, 8 pp; B. Djafari Rouhani, H. Khatibzadeh, Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Mat h. (in press)] who studied the case f equivalent to 0. Some applications are also presented. (C) 2008 Elsevier Ltd. All rights reserved.