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-epsilon mu)(ap), (P-mu)(ap), and (E-00), involve operators that are odd, maximal monotone, and possibly set-valued. The second- and first-order differential inclusions are parameterized by two nonnegative constants, epsilon and mu, which affect the behavior of the differential terms. We establish the existence and uniqueness of strong solutions for the problems (P-epsilon mu)(ap) and (P-mu)(ap), as well as for the algebraic inclusion (E-00). Additionally, we prove the continuous dependence of the solution to problem (P-epsilon mu)(ap) on parameters epsilon and mu. We also provide approximation results for the solutions to (P-mu)(ap) and (E-00) as the parameters epsilon and mu approach zero. Finally, we discuss some applications of our theoretical results.