Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Asymptotic Behavior of Solutions to Some Homogeneous Second-Order Evolution Equations of Monotone Type | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 72931 8 pages doi 2007 72931 Research Article Asymptotic Behavior of Solutions to Some Homogeneous Second-Order Evolution Equations of Monotone Type Behzad Djafari Rouhani and Hadi Khatibzadeh Received 7 November 2006 Accepted 12 April 2007 Recommended by Andrei Ronto We study the asymptotic behavior of solutions to the second-order evolution equation p t ú t r t u t e Au t . t e 0 to u 0 u0 supt 01u t I TO 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 . Copyright 2007 B. D. Rouhani and H. 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. 1. Introduction Let H be a real Hilbert space with inner product and norm I I. We denote weak convergence in H by and strong convergence by - . We will refer to a nonempty subset A of H X H as a nonlinear possibly multivalued operator in H. A is called monotone resp. strongly monotone if y2 - y1 x2 - x1 0 resp. y2 - y1 x2 - x1 p x1 - x2I2 for some p 0 for all xi yi e A i 1 2. A is called maximal monotone if A is monotone and R I A H where I is the identity operator on H. Existence as well as asymptotic behavior of solutions to second-order evolution equations of the form p t u t r t u t e Au t . on R u 0 u0 sup u t TO t 0 2 Journal of Inequalities and Applications in the special case p t 1 and r t 0 were studied by many .
