Existence of a periodic solution for implicit nonlinear equations

Abstract

The solvability of the periodic problem $$ v'(t)+\mathcal{B}(t)u(t)+\mathcal {C}(t)u(t)\ni \tilde f(t), \ \ v(t)\in A(t)u(t),\ \, v(0)=v(T), $$ will be investigated. It will be shown that the solution exists if $\mathcal {B}t)$ are maximal monotone operators, $\mathcal {A}(t)$ are subdifferentials, compact and possibly degenerated, $\mathcal {C}(t)$ are Lipschitzian and compact, and $\mathcal{B}(t)+\mathcal {C}(t)$ are coercive. The proof consists of several limit processes for an approximating periodic problem which has a solution.