Viable periodic solutions in state-dependent impulsive problems

• Autores: Grzegorz Gabor
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 66, Fasc. 3, 2015, págs. 351-365
• Idioma: inglés
• DOI: 10.1007/s13348-015-0139-x
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• Resumen

  • Periodic solutions of state-dependent impulsive ODEs in a prescribed set of constraints are examined. The so-called impulsive index is introduced, as a topological tool, and its properties are studied. Some sufficient conditions for its homotopy property are discussed in detail. In a construction of the impulsive index the fixed point index on ANRs is applied to an induced discrete semidynamical system on a barrier where jumps occur. Several illustrative examples are added.

• Referencias bibliográficas
  • Azevedo, K.A.G., Bonotto, E.M.: On asymptotic stability in impulsive semidynamical systems. J. Dyn. Control Syst. 19(3), 359–380 (2013)
  • Bader, R., Kryszewski, W.: Fixed-point index for compositions of set-valued maps with proximally \infty-connected values on arbitrary ANR’s....
  • Bainov, D., Simeonov, P.: Impulsive differential equations: periodic solutions and applications. Pitman Monographs and Surveys in Pure and...
  • Becker, J.C., Gottlieb, D.H.: Vector fields and transfers. Manuscripta Math. 72(2), 111–130 (1991)
  • Bonotto, E.M., Ferreira, J.C.: Uniform attractors of discontinuous semidynamical systems. Collect. Math. 65(1), 47–59 (2014)
  • Brézis, H.: On a characterization of flow-invariant sets. Commun. Pure Appl. Math. 23, 261–263 (1970)
  • Cardaliaguet, P., Gabor, G., Quincampoix, M.: Equilibria and strict equilibria of multivalued maps on noninvariant sets. Ann. Polon. Math....
  • Cardinali, T., Servadei, R.: Periodic solutions of nonlinear impulsive differential inclusions with constraints. Proc. Am. Math. Soc. 132(8),...
  • Cardinali, T., Servadei, R.: On the existence of solutions for nonlinear impulsive periodic viable problems. Cent. Eur. J. Math. 2(4), 573–583...
  • Ciesielski, K.: On semicontinuity in impulsive dynamical systems. Bull. Pol. Acad. Sci. Math. 52(1), 71–80 (2004)
  • Conley, C.: A new statement of Ważewski’s theorem and an example. Ordin. Part. Diff. Equ. In: Proc. 4th Conf., Dundee 1976, Lect. Notes Math,...
  • Gabor, G., Quincampoix, M.: On existence of solutions to differential equations or inclusions remaining in a prescribed closed subset of a...
  • Gabor, G.: Homotopy index for multivalued flows on sleek sets. Set Valued Anal. 13(2), 125–149 (2005)
  • Gabor, G.: The existence of viable trajectories in state-dependent impulsive systems. Nonlinear Anal. 72, 3828–3836 (2010)
  • Górniewicz, L.: Topological Fixed Point Theory of Multivalued Mappings. Springer, Dordrecht (2006)
  • Granas, A., Dugundji, J.: Fixed Point Theory. Springer Monographs in Mathematics. Springer, New York (2003)
  • Kryszewski, W., Plaskacz, S.: Periodic solutions to impulsive differential inclusions with constraints. Nonlinear Anal. 65(9), 1794–1804 (2006)
  • Mil’man, V.D., Myškis, A.D.: On the stability of motion in the presence of impulses. (Russian) Sibirsk. Mat. Ž. 1, 233–237 (1960)
  • Nagumo, M.: Über die Lage der Integralkurven gewühnlicher Differentialgleichungen. Proc. Phys.-Math. Soc. Japan, III. Ser., vol. 24, pp. 551–559....
  • Ważewski, T.: Sur un principe topologique de l’examen de l’allure asymptotique des intégrales des équations différentielles ordinaires. Ann....