Asymptotic behavior of bounded solutions to a nonhomogeneous second-order evolution equation of monotone type

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 asymptotic behavior of solutions to the following system of second-order nonhomogeneous evolution equations: { u ″ ( t ) ∈ A u ( t ) + f ( t ) a.e. t ∈ ( 0 , + ∞ ) u ( 0 ) = u 0 , sup t ≥ 0 | u ( t ) | < + ∞ where A is a maximal monotone operator in a real Hilbert space H , and f : R + → H is a suitable function. We investigate weak and strong convergence theorems for solutions to this system. Our results extend previous results of 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 f ≡ 0 by assuming that A − 1 ( 0 ) ≠ ϕ , as well as previous results of the authors [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 homogeneous case without this additional assumption.