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Homoclinic solutions of 2nth-order difference equations containing both advance and retardation

100%
Long Y., Zhang Y., Shi H.
Open Mathematics
|
2016
|
tom 14
|
nr 1
520-530
EN
By using the critical point method, some new criteria are obtained for the existence and multiplicity of homoclinic solutions to a 2nth-order nonlinear difference equation. The proof is based on the Mountain Pass Lemma in combination with periodic approximations. Our results extend and improve some known ones.
2
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Homoclinic orbits for a class of singular second order Hamiltonian systems in ℝ3

100%
Janczewska J., Maksymiuk J.
Open Mathematics
|
2012
|
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.
3
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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
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.
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