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

We study the asymptotic behavior of solutions to the second-order evolution equation p(t)u ′′ (t) + r(t)u ′ (t) ∈ Au(t) a.e. t ∈ (0,+∞), u(0) = u0, sup t≥0|u(t)| < +∞, 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).