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The shadowing chain lemma for singular Hamiltonian systems involving strong forces

100%

Izydorek M. , Janczewska J.

Open Mathematics

2012

tom 10

nr 6

1928-1939

We consider a planar autonomous Hamiltonian system :q+∇V(q) = 0, where the potential V: ℝ2 \{ζ}→ ℝ has a single well of infinite depth at some point ζ and a strict global maximum 0at two distinct points a and b. Under a strong force condition around the singularity ζ we will prove a lemma on the existence and multiplicity of heteroclinic and homoclinic orbits - the shadowing chain lemma - via minimization of action integrals and using simple geometrical arguments.

Multiple solutions of indefinite elliptic systems via a Galerkin-type Conley index theory

100%

Izydorek M. , Rybakowski K.

nr 3

233-249

Let Ω be a bounded domain in $ℝ^{N}$ with smooth boundary. Consider the following elliptic system: $-Δu = ∂_{v}H(u,v,x)$ in Ω, $-Δv = ∂_{u}H(u,v,x)$ in Ω, u = 0, v = 0 in ∂Ω. (ES) We assume that H is an even "-"-type Hamiltonian function whose first order partial derivatives satisfy appropriate growth conditions. We show that if (0,0) is a hyperbolic solution of (ES), then (ES) has at least 2|μ| nontrivial solutions, where μ = μ(0,0) is the renormalized Morse index of (0,0). This proves a conjecture by Angenent and van der Vorst.

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

100%

Izydorek M. , Rybakowski K.

Fundamenta Mathematicae

2002

tom 173

nr 1

77-100

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 $-Δu = ∂_{v}H(u,v,x)$ in Ω, $-Δv = ∂_{u}H(u,v,x)$ 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. As shown by Angenent and van der Vorst in [1], the solutions of (A1) are equilibria of an abstract ordinary differential equation ż = f(z) (A2) defined on a certain Hilbert space E of functions z = (u,v). The map f: E → E is continuous, but, in general, not differentiable nor even locally Lipschitzian. The main result of this paper is a Linearization Principle which states that whenever z₀ is a hyperbolic equilibrium of (A2) then the Conley index of {z₀} can be computed by formally linearizing (A2) at z₀. As a particular application of the Linearization Principle we obtain an elementary, Conley index based proof of the existence of nontrivial solutions of (A1), a result previously established in [1] via Morse-Floer homology. Further applications of our method to existence and multiplicity results for strongly indefinite systems appear in [3] and [10].

Approximative sequences and almost homoclinic solutions for a class of second order perturbed Hamiltonian systems

100%

Izydorek M. , Janczewska J.

Banach Center Publications

2014

tom 101

nr 1

87-92

In this work we will consider a class of second order perturbed Hamiltonian systems of the form $q̈ + V_q(t,q) = f(t)$, where t ∈ ℝ, q ∈ ℝⁿ, with a superquadratic growth condition on a time periodic potential V: ℝ × ℝⁿ → ℝ and a small aperiodic forcing term f: ℝ → ℝⁿ. To get an almost homoclinic solution we approximate the original system by time periodic ones with larger and larger time periods. These approximative systems admit periodic solutions, and an almost homoclinic solution for the original system is obtained from them by passing to the limit in $C²_{loc}(ℝ,ℝⁿ)$ when the periods go to infinity. Our aim is to show the existence of two different approximative sequences of periodic solutions: one of mountain pass type and the second of local minima.

On the Conley index in Hilbert spaces in the absence of uniqueness

100%

Izydorek M. , Rybakowski K.

Fundamenta Mathematicae

2002

tom 171

nr 1

31-52

Consider the ordinary differential equation (1) ẋ = Lx + K(x) on an infinite-dimensional Hilbert space E, where L is a bounded linear operator on E which is assumed to be strongly indefinite and K: E → E is a completely continuous but not necessarily locally Lipschitzian map. Given any isolating neighborhood N relative to equation (1) we define a Conley-type index of N. This index is based on Galerkin approximation of equation (1) by finite-dimensional ODEs and extends to the non-Lipschitzian case the ℒ𝓢-Conley index theory introduced in [9]. This extended ℒ𝓢-Conley index allows applications to strongly indefinite variational problems ∇Φ(x) = 0 where Φ: E → ℝ is merely a C¹-function.