A Granas type approach to some continuation theorems and periodic boundary value problems with impulse


In this paper we study periodic solutions of a second order differential equation
x^{\prime\prime} = f(t, x, x^{\prime}) \quad for \ a.e. \ t\in [0, 1],
subject to some impulses at certain points.

Our work was inspired by a paper by Capietto–Mawhin–Zanolin [1], where the case of no impulses was treated.

The major difference between paper [1] and ours is that instead of topological degree, we use the elementary method based on essential maps. In this context, we also give some new contributions to Granas’ theory of continuation principles.


Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania



Paper coordinates

R. Precup, A Granas type approach to some continuation theorems and periodic boundary value problems with impulses, Topological Methods in Nonlinear Analysis, 5 (1995) no. 2, 385-396.


About this paper

Print ISSN
Online ISSN


MR: 97a:34028

google scholar link

[1] A. Capietto, J. Mawhin and F. Zanolin, A continuation approach to superlinear periodic boundary value problems, J. Differential Equations 88 (1990), 347–395.
[2] , Boundary value problems for forced superlinear second order ordinary differential equations, SISSA Ref. 106/92/M (1992), 1–10.
[3] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
[4] L. Erbe and W. Krawcewicz, Existence of solutions to boundary value problems for impulsive second order differential inclusions, Rocky Mountain J. Math. 22 (1992), 519–540.
[5] M. Frigon, Application de la theorie de la transversalite topologique a des problemes non lineaires pour des equations differentielles ordinaires, Dissertationes Math. 296 (1990).
[6] A. Granas, Homotopy extension theorem in Banach spaces and some of its applications to the theory of non-linear equations, Bull. Acad. Polon. Sci. 7 (1959), 387–394.
[7] A. Granas, R. Guenther and J. Lee, Nonlinear boundary value problems for ordinary differential equations, Dissertationes Math. 244 (1985).
[8] W. Krawcewicz, Contribution a la theorie des equations non lineaires dans les espaces de Banach, Dissertationes Math. 273 (1988).
[9] J. Leray et J. Schauder, Topologie et equations fonctionnelles, Ann. Sci. Ecole Norm. Sup. (3) 51 (1934), 45–78.
[10] D. O’Regan, Second and higher order systems of boundary value problems, J. Math. Anal. Appl. 156 (1991), 120–149.
[11] R. Precup, Generalized topological transversality and existence theorems, Libertas Math. 11 (1991), 65–79.
[12] , On the topological transversality principle, Nonlinear Anal. 20 (1993), 1–9.
[13] H. Schaefer, Uber die Methode der a priori Schranken , Math. Ann. 129 (1955), 415– 416.
[14] R. Schoneberg, Leray-Schauder principles for condensing multivalued mappings in topological linear spaces, Proc. Amer. Math. Soc. 72 (1978), 268–270.

Related Posts