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Conley index in Hilbert spaces and a problem of Angenent and van der Vorst

Marek Izydorek, Krzysztof P. Rybakowski (2002)

Fundamenta Mathematicae

In a recent paper [9] we presented a Galerkin-type Conley index theory for certain classes of infinite-dimensional ODEs without the uniqueness property of the Cauchy problem. In this paper we show how to apply this theory to strongly indefinite elliptic systems. More specifically, we study the elliptic system - Δ u = ∂ v H ( u , v , x ) in Ω, - Δ v = ∂ u H ( u , v , x ) in Ω, u = 0, v = 0 in ∂Ω, (A1) on a smooth bounded domain Ω in ℝ N for "-"-type Hamiltonians H of class C² satisfying subcritical growth assumptions on their first order derivatives....

Conley type index and Hamiltonian inclusions

Zdzisław Dzedzej (2010)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

This paper is based mainly on the joint paper with W. Kryszewski [Dzedzej, Z., Kryszewski, W.: Conley type index applied to Hamiltonian inclusions. J. Math. Anal. Appl. 347 (2008), 96–112.], where cohomological Conley type index for multivalued flows has been applied to prove the existence of nontrivial periodic solutions for asymptotically linear Hamiltonian inclusions. Some proofs and additional remarks concerning definition of the index and special cases are given.

Connected branches of asymptotically equivalent solutions to non-linear eigenvalue problems

Allan L. Edelson, Maria Patrizia Pera (1987)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We prove an existence theorem for connected branches of solutions to nonlinear operator equations in Banach spaces. This abstract result is applied to the asymptotically equivalent solutions to nonlinear ordinary differential equations.

Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula

Hans Weber (2007)

Open Mathematics

Starting from the Rodrigues representation of polynomial solutions of the general hypergeometric-type differential equation complementary polynomials are constructed using a natural method. Among the key results is a generating function in closed form leading to short and transparent derivations of recursion relations and addition theorem. The complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others and obey Rodrigues formulas....

Consensus of a two-agent system with nonlinear dynamics and time-varying delay

Ye Cheng, Bao Shi, Liangliang Ding (2021)

Applications of Mathematics

To explore the impacts of time delay on nonlinear dynamics of consensus models, we incorporate time-varying delay into a two-agent system to study its long-time behaviors. By the classical 3/2 stability theory, we establish a sufficient condition for the system to experience unconditional consensus. Numerical examples show the effectiveness of the proposed protocols and present possible Hopf bifurcations when the time delay changes.

Consequences of the meromorphic equivalence of standard matrix differential equations.

Hans-Joachim Zwiesler (1997)

Publicacions Matemàtiques

In this article we investigate the question [of] how meromorphic differential equations can be simplified by meromorphic equivalence. In the case of equations of block size 1, which generalizes the case of distinct eigenvalues, we identify a class of equations which are simplest possible in the sense that they carry the smallest number of parameters whithin their equivalence classes. We also discuss conditions under which individual equations can be simplified. Particular attention is paid to the...

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