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

Abstract

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 I, 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 ) − c u ′ ( t ) ∈ A u ( t ) + f ( t ) a.e. t ∈ ( 0 , + ∞ ) u ( 0 ) = u 0 , sup t ≥ 0 | u ( t ) | < + ∞ 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 ≡ 0 by assuming that A is maximal monotone and A − 1 ( 0 ) ≠ ϕ , as well as previous results by Véron [L. Véron, Problèmes d’évolution du second ordre associés à des opérateurs monotones, C. R. Acad. Sci. Paris Sér. A 278 (1974) 1099–1101; L. Véron, Equations d’évolution du second ordre associées à des opérateurs 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. ID 72931, 8 pp; B. Djafari Rouhani, H. Khatibzadeh, Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Math. (in press)] who studied the case f ≡ 0 . Some applications are also presented.