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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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