## Multiplicity results for periodic solutions to delay differential equations via critical point theory.(English)Zbl 1095.34043

Consider the system of delay differential equations $x'(t)=-f(x(t-r)), \tag{*}$ where $$x\in \mathbb R^n$$, $$f\in C(\mathbb R^n,\mathbb R^n)$$, and $$r>0$$ is a given constant. The authors give a lower bound for the number of nontrivial periodic solutions with period $$4r$$. First, a variational structure for (*) with periodic boundary value condition is established. It is shown that under some suitable assumptions, the existence of $$4r$$-periodic solutions $$x(t)$$ to (*) satisfying $$x(t+2r)=-x(t)$$ is equivalent to the existence of critical points of the variational functional. Then, the main result is obtained by using the $$\mathbb Z_2$$-geometrical index theory and related pseudo-index theory.

### MSC:

 34K13 Periodic solutions to functional-differential equations 34K18 Bifurcation theory of functional-differential equations 47J30 Variational methods involving nonlinear operators 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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