526 research outputs found
Variations on some finite-dimensional fixed-point theorems
We give rather elementary topological proofs of some generalizations of fixed-point theorems in Rⁿ due to Pireddu-Zanolin and Zgliczynski, which are useful in various questions related to ordinary differential equations.Наведено елементарнi топологiчнi доведення деяких узагальнень теорем Пiредду – Занолiна та Зглiчинського про нерухому точку в Rⁿ, якi можуть бути використанi при розглядi рiзних питань, пов’язаних iз звичайними диференцiальними рiвняннями
Resonance and nonlinearity: a survey
Наведено огляд останніх результатів щодо нерезонансних і резонансних періодично збуджуваних нелінійних осциляторів: існування періодичних необмежених або обмежених розв'язків для обмежених нелінійних збурень лінійних та кусково-лінійних осциляторів, а також деяких класів плоских гамільтонових систем.This paper surveys recent results about nonresonant and resonant periodically forced nonlinear oscillators. This includes the existence of periodic, unbounded or bounded solutions for bounded nonlinear perturbations of linear and of piecewise linear oscillators, as well as of some classes of planar Hamiltonian systems
Positive solutions to indefinite Neumann problems when the weight has positive average
We deal with positive solutions for the Neumann boundary value problem
associated with the scalar second order ODE where is positive on and is an indefinite weight. Complementary to previous
investigations in the case , we provide existence results
for a suitable class of weights having (small) positive mean, when
at infinity. Our proof relies on a shooting argument for a suitable equivalent
planar system of the type with
a continuous function defined on the whole real line.Comment: 17 pages, 3 figure
Existence Results for Periodic Boundary Value Problems with a Convenction Term
By using an abstract coincidence point theorem for sequentially weakly continuous maps the existence of at least one positive solution is obtained for a periodic second order boundary value problem with a reaction term involving the derivative of the solution u: the so called convention term. As a consequence of the main result also the existence of at least one positive solution is obtained for a parameter-depending problem
Ground state solutions for non-autonomous dynamical systems
We study the existence of periodic solutions for a second order non-autonomous dynamical system. We allow both sublinear and superlinear problems. We obtain ground state solutions
On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems
This paper deals with periodic solutions of the Hamilton equation with many
parameters. Theorems on global bifurcation of solutions with periods
from a stationary point are proved. The Hessian matrix of the
Hamiltonian at the stationary point can be singular. However, it is assumed
that the local topological degree of the gradient of the Hamiltonian at the
stationary point is nonzero. It is shown that (global) bifurcation points of
solutions with given periods can be identified with zeros of appropriate
continuous functions on the space of parameters. Explicit formulae for such
functions are given in the case when the Hessian matrix of the Hamiltonian at
the stationary point is block-diagonal. Symmetry breaking results concerning
bifurcation of solutions with different minimal periods are obtained. A
geometric description of the set of bifurcation points is given. Examples of
constructive application of the theorems proved to analytical and numerical
investigation and visualization of the set of all bifurcation points in given
domain are provided.
This paper is based on a part of the author's thesis [W. Radzki, ``Branching
points of periodic solutions of autonomous Hamiltonian systems'' (Polish), PhD
thesis, Nicolaus Copernicus University, Faculty of Mathematics and Computer
Science, Toru\'{n}, 2005].Comment: 35 pages, 4 figures, PDFLaTe
Multiplicity of periodic solutions for systems of weakly coupled parametrized second order differential equations
We prove a multiplicity result of periodic solutions for a system of second order differential equations having asymmetric nonlinearities. The proof is based on a recent generalization of the Poincar\ue9\u2013Birkhoff fixed point theorem provided by Fonda and Ure\uf1a
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