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1
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Almost homoclinic solutions for a certain class of mixed type functional differential equations

100%
Janczewska J.
Annales Polonici Mathematici
|
2011
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tom 100
|
nr 1
13-24
EN
We shall be concerned with the existence of almost homoclinic solutions for a class of second order functional differential equations of mixed type: $q̈(t)+V_{q}(t,q(t))+u(t,q(t),q(t-T),q(t+T)) = f(t)$, where t ∈ ℝ, q ∈ ℝⁿ and T>0 is a fixed positive number. By an almost homoclinic solution (to 0) we mean one that joins 0 to itself and q ≡ 0 may not be a stationary point. We assume that V and u are T-periodic with respect to the time variable, V is C¹-smooth and u is continuous. Moreover, f is non-zero, bounded, continuous and square-integrable. The main result provides a certain approximative scheme of finding an almost homoclinic solution.
2
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Bifurcation in the solution set of the von Kármán equations of an elastic disk lying on an elastic foundation

100%
Janczewska J.
Annales Polonici Mathematici
|
2001
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tom 77
|
nr 1
53-68
EN
We investigate bifurcation in the solution set of the von Kármán equations on a disk Ω ⊂ ℝ² with two positive parameters α and β. The equations describe the behaviour of an elastic thin round plate lying on an elastic base under the action of a compressing force. The method of analysis is based on reducing the problem to an operator equation in real Banach spaces with a nonlinear Fredholm map F of index zero (to be defined later) that depends on the parameters α and β. Applying the implicit function theorem we obtain the following necessary condition for bifurcation: if (0,p) is a bifurcation point then $dim KerF'_{x}(0,p) > 0$. Next, we give a full description of the kernel of the Fréchet derivative of F. We study in detail the situation when the dimension of the kernel is one. We prove that (0,p) is a bifurcation point by the use of the Lyapunov-Schmidt finite-dimensional reduction and the Crandall-Rabinowitz theorem. For a one-dimensional bifurcation point, analysing the Lyapunov-Schmidt branching equation we determine the number of families of solutions, their directions and asymptotic behaviour (shapes).
3
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Local properties of the solution set of the operator equation in Banach spaces in a neighbourhood of a bifurcation point

100%
Janczewska J.
Open Mathematics
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2004
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tom 2
|
nr 4
561-572
EN
In this work we study the problem of the existence of bifurcation in the solution set of the equation F(x, λ)=0, where F: X×R k →Y is a C 2-smooth operator, X and Y are Banach spaces such that X⊂Y. Moreover, there is given a scalar product 〈·,·〉: Y×Y→R 1 that is continuous with respect to the norms in X and Y. We show that under some conditions there is bifurcation at a point (0, λ0)∈X×R k and we describe the solution set of the studied equation in a small neighbourhood of this point.
4
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Homoclinic orbits for a class of singular second order Hamiltonian systems in ℝ3

64%
Janczewska J., Maksymiuk J.
Open Mathematics
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2012
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tom 10
|
nr 6
1920-1927
EN
We consider a conservative second order Hamiltonian system $$\ddot q + \nabla V(q) = 0$$ in ℝ3 with a potential V having a global maximum at the origin and a line l ∩ {0} = ϑ as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system.
5
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The shadowing chain lemma for singular Hamiltonian systems involving strong forces

64%
Izydorek M., Janczewska J.
Open Mathematics
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2012
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tom 10
|
nr 6
1928-1939
EN
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.
6
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Degree of T-equivariant maps in ℝⁿ

64%
Janczewska J., Styborski M.
Banach Center Publications
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2007
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tom 77
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nr 1
147-159
EN
A special case of G-equivariant degree is defined, where G = ℤ₂, and the action is determined by an involution $T:ℝ^p ⊕ ℝ^q → ℝ^p ⊕ ℝ^q$ given by T(u,v) = (u,-v). The presented construction is self-contained. It is also shown that two T-equivariant gradient maps $f,g:(ℝⁿ,S^{n-1}) → (ℝⁿ,ℝⁿ∖{0})$ are T-homotopic iff they are gradient T-homotopic. This is an equivariant generalization of the result due to Parusiński.
7
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Approximative sequences and almost homoclinic solutions for a class of second order perturbed Hamiltonian systems

64%
Izydorek M., Janczewska J.
Banach Center Publications
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2014
|
tom 101
|
nr 1
87-92
EN
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.
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