Multiplicity results on periodic solutions to higher-dimensional differential equations with multiple delays. (English) Zbl 1311.34153
Summary: This paper continues our study on the existence and multiplicity of periodic solutions to delay differential equations of the form
\[
\dot{z}(t)=-f(z(t-1))-f(z(t-2))-\cdots -f(z(t- n+1)),
\]
where \(z\in\mathbb{R}^N\), \(f\in C(\mathbb{R}^N, \mathbb{R}^N)\) and \(n>1\) is an odd number. By using the Galerkin approximation method and the \(S^1\)-index theory in the critical point theory, some known results for Kaplan-Yorke type differential delay equations are generalized to the higher-dimensional case. As a result, the Kaplan-Yorke conjecture is proved to be true in the case of higher-dimensional systems.
MSC:
| 34K13 | Periodic solutions to functional-differential equations |
| 34K07 | Theoretical approximation of solutions to functional-differential equations |
| 58E50 | Applications of variational problems in infinite-dimensional spaces to the sciences |
Keywords:
periodic solutions; higher-dimensional differential equations; multiple delays; the \(S^1\)-index theory; Galerkin approximation methodReferences:
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