Existence of periodic solutions for delay differential equations with state dependent delay. (English) Zbl 0913.34057
The existence of periodic solutions to a delay differential equation, with delay depending indirectly on the state, of the form
\[
\frac {dx}{dt}=-f(x(t-\tau(t))), \quad\text{with}\quad \frac {d\tau}{dt}=h(x(t),\tau(t))
\]
is studied. A fixed point problem related to a Poincaré operator is constructed and solved using an ejective fixed point theorem.
Reviewer: J.Diblík (Brno)
MSC:
| 34K13 | Periodic solutions to functional-differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
References:
| [1] | Aiello, W. G.; Freedman, H. I.; Wu, J., Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52, 855-869 (1992) · Zbl 0760.92018 |
| [2] | O. Arino, M. L. Hbid, R. Bravo de la Parra, A mathematical model of growth of population of fish in the larval stage: Density-dependence effects, Math. Biosc.; O. Arino, M. L. Hbid, R. Bravo de la Parra, A mathematical model of growth of population of fish in the larval stage: Density-dependence effects, Math. Biosc. · Zbl 0938.92028 |
| [3] | Arino, O.; Chérif, A., More on ordinary differential equations which yield periodic solutions of delay differential equations, J. Math. Anal. Appl., 180, 361-385 (1993) · Zbl 0798.34067 |
| [4] | Browder, F., A further generalization of the Schauder fixed point theorem, Duke Math. J., 32, 575-578 (1965) · Zbl 0137.32601 |
| [5] | Chow, S.; Hale, J. K., Periodic solutions of autonomous equations, J. Math. Anal. Appl., 66, 495-506 (1978) · Zbl 0397.34091 |
| [6] | Györi, I.; Ladas, G., Oscillation Theory of Delay Differential Equations with Applications (1991), Clarendon pressOxford Mathematical Monographs: Clarendon pressOxford Mathematical Monographs Oxford · Zbl 0780.34048 |
| [7] | Hadeler, K. P.; Tomiuk, J., Periodic solutions of difference differential equations, Arch. Rat. Mech. Anal., 65, 87-95 (1977) · Zbl 0426.34058 |
| [8] | Hale, J. K., Functional Differential Equations (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0425.34048 |
| [9] | Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional Differential Equations (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0787.34002 |
| [10] | Kuang, Y.; Smith, H. L., Periodic solutions of differential delay equations with threshold-type delays, Oscillations and Dynamics in Delay Equations, Contemp. Math., 129, 153-176 (1992) · Zbl 0762.34044 |
| [11] | P. Magal, A fixed point theorem for maps on cones with application to a population dynamics model; P. Magal, A fixed point theorem for maps on cones with application to a population dynamics model |
| [12] | Mallet-Paret, J.; Nussbaum, R., Boundary layer phenomena for differential delay equations with state-dependent time lags, Arch. Rat. Mech. Anal., 120, 99-146 (1992) · Zbl 0763.34056 |
| [13] | Mallet-Paret, J.; Nussbaum, R.; Paraskevopulos, P., Periodic solutions to functional differential equations with multiple state-dependent time lags, Topological Methods in Nonlinear Analysis, J. Juliusz Schauder Center, 3, 101-162 (1994) · Zbl 0808.34080 |
| [14] | Nussbaum, R., Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Math. Pure Appl., 10, 263-306 (1974) · Zbl 0323.34061 |
| [15] | Nussbaum, R., Periodic solutions of nonlinear autonomous functional differential equations, Lecture Notes in Math., 730, 283-325 (1979) · Zbl 0447.34069 |
| [16] | Wright, E. M., A nonlinear differential difference equation, J. Reine Angew. Math., 194, 66-87 (1955) · Zbl 0064.34203 |
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