Periodic solutions of differential inclusions on compact subsets of \({\mathbb{R}}^ n\). (English) Zbl 0705.34040
Let M be a compact subset of \({\mathbb{R}}^ n\) such that there exists an open set U (M\(\subset U)\) and a continuous function r: \(U\to M\) such that \(dist(x,M)=| x-r(x)|\) for every \(x\in U\). It is shown that the Bouligand contingent cone valued map \(T_ M: M\to {\mathbb{R}}^ n\) is lower semi-continuous.
Suppose that the Euler characteristic of M is not equal to zero and a Carathéodory type convex valued map F: \([0,\infty)\times M\to {\mathbb{R}}^ n\) satisfies \(F(t,x)\cap T_ M(x)\neq \emptyset\) for all (t,x)\(\in [0,\infty)\times M\). If the map F is periodic with respect to the first variable then the differential inclusion \(x'(t)\in F(t,x(t))\) possesses a periodic solution. The problem of existence of periodic solutions is reduced to the problem of existence of fixed points of the set valued translation operator along trajectories of the differential inclusion.
Suppose that the Euler characteristic of M is not equal to zero and a Carathéodory type convex valued map F: \([0,\infty)\times M\to {\mathbb{R}}^ n\) satisfies \(F(t,x)\cap T_ M(x)\neq \emptyset\) for all (t,x)\(\in [0,\infty)\times M\). If the map F is periodic with respect to the first variable then the differential inclusion \(x'(t)\in F(t,x(t))\) possesses a periodic solution. The problem of existence of periodic solutions is reduced to the problem of existence of fixed points of the set valued translation operator along trajectories of the differential inclusion.
Reviewer: S.Plaskacz
MSC:
| 34C25 | Periodic solutions to ordinary differential equations |
| 34A60 | Ordinary differential inclusions |
| 37G99 | Local and nonlocal bifurcation theory for dynamical systems |
References:
| [1] | Aubin, J. P.; Cellina, A., Differential Inclusions (1982), Springer-Verlag: Springer-Verlag New York/Berlin |
| [2] | R. Bielawski, L. Górniewicz, and S. Plaskacz\(R^n Reported Dynamics \); R. Bielawski, L. Górniewicz, and S. Plaskacz\(R^n Reported Dynamics \) |
| [3] | Clarke, F., Generalized gradients and applications, Trans. Amer. Math. Soc., 205, 247-262 (1975) · Zbl 0307.26012 |
| [4] | De Blasi, F. S.; Myjak, J., On the solution sets for differential inclusions, Bull. Acad. Polon. Sci., 33, 17-23 (1985) · Zbl 0571.34008 |
| [5] | Dylawerski, G.; Górniewicz, L., A remark on the Krasnosielskii’s translation operator along trajectories of ordinary differential equations, Serdica, 9, 102-107 (1983) · Zbl 0533.34033 |
| [6] | Fryszkowski, A., Carathéodory type selectors of set-valued maps, Bull. Acad. Polon. Sci., 25, 41-46 (1977) · Zbl 0358.54002 |
| [7] | Furi, M.; Pera, M. P., Global branches of periodic solutions for forced differential equation on nonzero Euler characteristic manifolds, Boll. Un. Mat. Ital. C, 3, No. 1, 157-170 (1984), (6) · Zbl 0541.58042 |
| [8] | Górniewicz, L., Homological methods in fixed point theory of multivalued maps, Dissertationes Math., 129, 1-71 (1976) · Zbl 0324.55002 |
| [9] | Haddad, G.; Lasry, J. M., Periodic solutions of functional differential inclusions and fixed points of σ-slectionable correspondences, J. Math. Anal. Appl., 96, 295-312 (1983) · Zbl 0539.34031 |
| [10] | Himmelberg, C. J., Measurable relations, Fund. Math., 87, 53-72 (1975) · Zbl 0296.28003 |
| [11] | Jarnik, J.; Kurzweil, J., On conditions on right hand sides of differential relations, C̆asopis Pěst. Mat., 102, 334-349 (1972) · Zbl 0369.34002 |
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