On some differential inclusions with anti-periodic solutions

In this paper, we investigate a class of second- and first-order differential inclusions, along with an algebraic inclusion, all subject to anti-periodic boundary conditions in a real Hilbert space. These problems, denoted as (P_ɛμ)_ap, (P_µ)_ap, and (E₀₀), involve operators that are odd, maximal monotone, and possibly set-valued. The second- and first-order differential inclusions are parameterized by two nonnegative constants, ɛ and µ, which affect the behavior of the differential terms.

We establish the existence and uniqueness of strong solutions for the problems (P_ɛµ)_ap and (P_µ)_ap, as well as for the algebraic inclusion (E₀₀). Additionally, we prove the continuous dependence of the solution to problem (P_ɛµ)_ap on parameters ɛ and µ. We also provide approximation results for the solutions to (P_µ)_ap and (E₀₀) as the parameters ɛ and µ approach zero. Finally, we discuss some applications of our theoretical results.