The second order nonlinear periodic problem at resonance with impulses
In this contribution, based on the joint paper with P. DrВґabek , we consider the second order nonlinear equations with periodic boundary conditions and with impulses in the derivative at fixed times. The impulses depend on the actual value of the solution in a nonlinear way. We concentrate on the resonance problems and provide sufficient condition for the existence of a solution in terms of the forcing term, restoring force and the impulse functions. Our condition not only generalizes the classical Landesman-Lazer condition but also implies the existence results for problems with vanishing or oscillating nonlinearities. To prove our result Saddle Point Theorem due to P. Rabinowitz  is applied. The practical importance of models the solutions of which include instantaneous impulses depending on the position that result in jump discontinuities in velocity, but with no change in position, was stressed in different fields of applied research.
 P. Drabek, M. LangerovВґa, On the second order periodic problem at resonance with impulses, J. Math. Anal. Appl. 428 (2015), 1339вЂ“1353.
 P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Amer. Math. Soc., Providence, RI, 1986.
The journal Biomath Communications is an open access journal. All published articles are immeditely available online and the respective DOI link activated. All articles can be access for free and no reader registration of any sort is required. No fees are charged to authors for article submission or processing. Online publications are funded through volunteer work, donations and grants.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).