On periodic solutions to autonomous retarded functional differential equations. (English) Zbl 0686.34069
Summary: Under the assumption that \(C_ a=C([-r,0],S^{n-1}(a))\) is positively invariant for \(a>0\), two necessary and sufficient conditions are obtained for an autonomous retarded functional differential equation to have a non-trivial periodic solution in \(C_ a\). Moreover, a feasible sufficient condition is given, which is better for \(n=2\) than that given by Dos Reis and Baroni.
MSC:
| 34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |
| 34C25 | Periodic solutions to ordinary differential equations |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
References:
| [1] | Krasnosel’skii, Positive Solutions of Operator Equations (1964) |
| [2] | Dos Reis, Proc. Royal Soc. Edinburgh Section A 102 pp 259– (1986) · Zbl 0616.34066 · doi:10.1017/S0308210500026342 |
| [3] | Hale, Functional Differential Equations (1971) · Zbl 0222.34003 · doi:10.1007/978-1-4615-9968-5 |
| [4] | Hale, Theory of Functional Differential Equations (1977) · Zbl 0352.34001 · doi:10.1007/978-1-4612-9892-2 |
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