| Title | PERIODIC SOLUTIONS OF RETARDED FUNCTIONAL PERTURBATIONS OF AUTONOMOUS DIFFERENTIAL EQUATIONS ON MANIFOLDS |
| Publication Type | Journal Article |
| Year of Publication | 2011 |
| Authors | FURI, MASSIMO, PERA, MARIAPATRIZIA, SPADINI, MARCO |
| Secondary Title | Communications in Applied Analysis |
| Volume | 15 |
| Issue | 3 |
| Start Page | 381 |
| Pagination | 394 |
| Date Published | 08/2011 |
| Type of Work | scientific: mathematics |
| ISSN | 1083–2564 |
| AMS | 34K13, 34K18., 58F32 |
| Abstract | Inspired by [1] and [10] we apply the topological tools of fixed point index and of degree of a tangent vector field to the study of the set of harmonic solutions to periodic perturbations of autonomous ODEs on (smooth) boundaryless differentiable manifolds, allowing the perturbation to contain a distributed, possibly infinite, delay. In order to do so, we construct a Poincar ́e-type T-translation operator on an appropriate function space and, in the unperturbed case, prove a formula for its fixed point index in terms of the degree of the autonomous vector field.
|
| URL | http://www.acadsol.eu/en/articles/15/3/10.pdf |
| Short Title | PERIODIC SOLUTIONS OF RETARDED FUNCTIONAL. . . |
| Refereed Designation | Refereed |
| Full Text | REFERENCES[1] P. Benevieri, A. Calamai, M. Furi, and M. P. Pera, Global branches of periodic solutions for forced delay differential equations on compact manifolds, J. Differential Equations 233 (2007), 404–416.
[2] P. Benevieri, A. Calamai, M. Furi, and M. P. Pera, A continuation result for forced oscillations of constrained motion problems with infinite delay, to appear in Adv. Nonlinear Stud. [3] P. Benevieri, A. Calamai, M. Furi, and M. P. Pera, On General Properties of Retarded Functional Differential Equations on Manifolds. In preparation. [4] K. Borsuk, Theory of retracts, Polish Sci. Publ., Warsaw, 1967. [5] M. Furi and M. P. Pera, A continuation principle for periodic solutions of forced motion equations on manifolds and application to bifurcation theory, Pacific J. Math. 160 (1993), 219–244. [6] M. Furi, M. P. Pera, and M. Spadini, The fixed point index of the Poincar ́e operator on differentiable manifolds, Handbook of topological fixed point theory, (Ed: Brown R. F., Furi M., Gorniewicz L., Jiang B.), Springer, 2005. [7] M. Furi, M. P. Pera, and M. Spadini, A set of axioms for the degree of a tangent vector field on differentiable manifolds, Fixed Point Theory App. 2010, Article Id. 8445631 (2010), 11 pages. [8] M. Furi and M. Spadini, On the fixed point index of the flow and applications to periodic solutions of differential equations on manifolds, Boll. Un. Mat. Ital. A (7) 10 (1996), no. 2, 333–346. [9] M. Furi and M. Spadini, On the set of harmonic solutions of periodically perturbed autonomous differential equations on manifolds, Nonlinear Analysis TMA 29 (1997) n. 8, 963–970. [10] M. Furi and M. Spadini, Periodic perturbations with delay of autonomous differential equations on manifolds, Adv. Nonlinear Stud. 9 No. 2 (2009), 263–276 . [11] A. Granas, The Leray-Schauder index and the fixed point theory for arbitrary ANR’s, Bull Soc. Math. France 100 (1972), 209–228. [12] A. Granas and J. Dugundji, Fixed point theory, Springer-Verlag, New York, 2003. [13] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences 99, Springer-Verlag Berlin, 1993. [14] M. W. Hirsch, Differential topology, Graduate Texts in Math. Vol. 33, Springer Verlag, Berlin, 1976. [15] J. W. Milnor, Topology from the differentiable viewpoint, Univ. press of Virginia, Charlottesville, 1965. [16] R. D. Nussbaum, The fixed point index and fixed points theorems, C.I.M.E. course on topological methods for ordinary differential equations, (Ed: M. Furi and P. Zecca), Lecture notes in Math. 1537, Springer Verlag 1991, 143–205. |