Abstract
The existence of the nontrivial periodic solutions for the nonautonomous first order delay differential equation $x'(t)=-[f(t, x(t-1))+f(t, x(t-2))+\cdots+f(t, x(t-(2N-1)))]$ is investigated, where $f\in C({\mathbf{R}}\times{\mathbf{R}}, {\mathbf{R}})$ is 2N-periodic in t and odd in x, N is a positive integer. We prove several new existence results by some recent critical point theorems.
Highlights
It is well known that critical point theory is a powerful tool that deals with the multiplicity of periodic solutions to ordinary differential systems as well as partial differential equations
We prove several new existence results by some recent critical point theorems
1 Introduction It is well known that critical point theory is a powerful tool that deals with the multiplicity of periodic solutions to ordinary differential systems as well as partial differential equations
Summary
It is well known that critical point theory is a powerful tool that deals with the multiplicity of periodic solutions to ordinary differential systems as well as partial differential equations (see [ – ]). In the past several years, some results on the existence of periodic solutions for the functional differential equation were obtained by critical point theory (see [ – ]). Periodic solutions of the Hamiltonian systems are still obtained as critical points of a function φ over a Hilbert space E. The function φ is invariant and φ is equivariant about the compact group G This allows to find critical points of φ on a subspace of E which is invariant under the group G. We can apply Theorem A to obtain periodic solutions in this subspace, which surely have the required symmetric structure and give solutions to We denote by M+(·), M–(·) and M (·) the positive definite, negative definite, and null subspaces of the self-adjoint linear operator defining it, respectively
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