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

Abstract

We study the asymptotic behavior of solutions to the second-order evolution equation a.e. , , where is a maximal monotone operator in a real Hilbert space with nonempty, and and are real-valued functions with appropriate conditions that guarantee the existence of a solution. We prove a weak ergodic theorem when 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 or the function .