Anti‐periodic solutions for a gradient system with resonance via a variational approach

Abstract

In this paper, we investigate a second‐order resonance anti‐periodic boundary value problem {q̈(t)+λmq(t)+∇F(t,q(t))=0,t∈[0,T],q(0)=−q(T),q̇(0)=−q̇(T),where λm is the m‐th eigenvalue of the corresponding eigenvalue problem. By using the dual least action principle, we obtain an existence result. In addition, we obtain the existence of 2T‐periodic solutions for q̈(t)+λmq(t)+∇F(t,q(t))=0,t∈R.

Existence of solutions for nonlinear nonmonotone evolution equations in Banach spaces with anti-periodic boundary conditions

“…The existence of anti-periodic solutions to first order evolution problems has been considered by many mathematicians under several assumptions on nonlinear terms; we invite the readers to refer to [2], [12], [7], [8], [10], [9], [11], [23], [24] for further reading. Second order problems involving anti-periodic boundary conditions are considered in [1], [2], [3], [17], [18], [25], [26], [27].…”