Yu-Qing Chen

Anti-periodic solutions for semilinear evolution equations

In this paper, we use the homotopy method to establish the existence and uniqueness of anti-periodic solutions for the nonlinear anti-periodic problem � ˙ + A(t, x )+ Bx = f (t) a.e. t ∈ R, x(t + T )=- x(t), where A(t, x) is a nonlinear map and B is a bounded linear operator from R N to R N. Sufficient conditions for the existence of the solution set are presented. Also, we consider the nonlinear evolution problems with a perturbation term which is multivalued. We show that, for this problem, the solution set is nonempty and weakly compact in W 1,2 (I, R N ) for the case of convex valued perturbation and prove the existence theorems of anti-periodic solutions for the nonconvex case. All illustrative examples are provided.

NOTE ON THE RESULTS WITH LOWER SEMI-CONTINUITY

In this paper, we introduce the concept of lower semi-continuous from above functions and show that many well-known results, such as Eklands and Caristis theorems, remain also true under lower semi-continuous from above functions.

Existence of periodic solutions for first-order evolution equations without coercivity

Abstract In this paper, we study the existence problem of periodic solutions for the following first-order nonlinear evolution equation u′(t)+A(t)u(t)+F t,u(t) ∋0, t∈R, u(t+T)=u(t), t∈R, in a Hilbert space H, where A is a monotone type operator and F is a nonlinear operator. Existence results are obtained without assuming the coercivity condition.