Abstract
Abstract In this paper, we use the homotopy method to establish the existence and uniqueness of anti-periodic solutions for the nonlinear anti-periodic problem { x ˙ + A ( t , x ) + B x = 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.
Highlights
Anti-periodic problems have important applications in auto-control, partial differential equations and engineering, and they have been studied extensively in the past ten years
The study of anti-periodic solutions for nonlinear evolution equations was initiated by Okochi [ ]
Many authors have been devoted to investigation of the existence of anti-periodic solutions to nonlinear evolution equations in Hilbert spaces
Summary
Anti-periodic problems have important applications in auto-control, partial differential equations and engineering, and they have been studied extensively in the past ten years.
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