arXiv subjects
Publications and source records attributed to Milan Tvrdý.
In this contribution we consider the singular periodic problem \[ u''+k\,u=g(u)+e(t),\quad u(0)=u(2π),\,\,u'(0)=u'(2π), \] where $k\in\R,$ $e:\J\to\R$ is Lebesgue integrable and $g:\I\mapsto\R$ is continuous, but can have a singularity at $x=0.$ In particular, I wish to revisit our earlier result regarding the existence of at least two its solutions and present a proof that is somewhat more transparent than the original one.
This paper is devoted to bifurcations of periodic solutions of nonlinear measure differential equations with a parameter. Main tools are nonlinear generalized differential equations (in the sense of Kurzweil) and the Kurzweil gauge type generalized integral. We continue the research started by the first author under the supervision of the second one.
Integral equations of the form $$ x(t)=x(t_0)+\int_{t_0}^t d[A]\,x=f(t)-f(t_0)$$ are natural generalizations of systems of linear differential equations. Their main goal is that they admit solutions which need not be absolutely continuous. Up to now such equations have been considered by several authors starting with J. Kurzweil and T.H. Hildebrandt. These authors worked with several different concepts of the Stieltjes type integral like Young's (Hildebrandt), Kurzweil's (Kurzweil, Schwabik and Tvrdý), Dushnik's (Hönig) or Lebesgue's (Ashordia, Meng and Zhang). Thus an interesting question arises: what are the relationships between all these concepts? Our aim is to give an answer to this question. In addition, we present also convergence results that are new for the Young and Dushnik integrals. Let us emphasize that the proofs of all the assertions presented in this paper are based on rather elementary tools.
We prove new results regarding the existence of positive solutions for a nonlinear periodic boundary value problem related to the Liebau phenomenon. As a consequence we obtain new sufficient conditions for the existence of a pump in a simple model. Our methodology relies on the use of classical fixed point index. Some examples are provided to illustrate our theory. We improve and complement previous results in the literature.
We give some sufficient conditions for existence, non-existence and localization of positive solutions for a periodic boundary value problem related to the Liebau phenomenon. Our approach is of topological nature and relies on the Krasnosel'ski\uı-Guo theorem on cone expansion and compression. Our results improve and complement earlier ones in the literature.