Abstract
The existence of the nontrivial periodic solutions for nonautonomous second-order delay differential equation is investigated, where , , . Multiple periodic solutions are obtained by some recent critical point theorems. MSC:34K13, 34K18, 58E50.
Highlights
It is well known that the critical point theory is a powerful tool to deal with the multiplicity of periodic solutions to ordinary differential systems as well as partial differential equations
Motivated by the work of [, ], we consider a class of nonautonomous second-order delay differential equation x (t) + λx(t) = –f t, x(t), x(t – τ ), ( . )
We denote by TS the topology on E induced by the semi-norm family {ps}, and let ω and ω* denote the weak-topology and weak*-topology, respectively
Summary
It is well known that the critical point theory is a powerful tool to deal with the multiplicity of periodic solutions to ordinary differential systems as well as partial differential equations (see [ – ]). In , Li and He [ ] first applied the critical point theory to study the multiplicity of periodic solutions for delay differential equations. In the past several years, some results on the existence of periodic solutions for the functional differential equation have been obtained by the critical point theory (see [ – ]). Most of these functional differential equations are autonomous, the results on the non-autonomous functional differential equations are relatively few (see [ , ]). We denote by TS the topology on E induced by the semi-norm family {ps}, and let ω and ω* denote the weak-topology and weak*-topology, respectively. ∂y and H(t, x, y) ∈ C (R × R , R)
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