scholarly journals Periodic solutions of singular systems without the strong force condition

2007 ◽  
Vol 136 (04) ◽  
pp. 1229-1236 ◽  
Author(s):  
Daniel Franco ◽  
Pedro J. Torres
Keyword(s):  
Periodic Solutions ◽  
Singular Systems ◽  
Force Condition ◽  
Strong Force

A relationship between the periodic and the Dirichlet BVPs of singular differential equations

1998 ◽  
Vol 128 (5) ◽  
pp. 1099-1114 ◽  
Author(s):  
Meirong Zhang
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Existence Result ◽  
Linear Growth ◽  
Dirichlet Boundary ◽  
Positive Periodic Solutions ◽  
Singular Differential Equations ◽  
Strong Force ◽  
Linear Growth Condition ◽  
The Difference

In this paper, a relationship between the periodic and the Dirichlet boundary value problems for second-order ordinary differential equations with singularities is established. This relationship may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations. Using this relationship, we give in this paper an existence result of positive periodic solutions to singular differential equations when the singular forces satisfy some strong force condition at the singularity 0 and some linear growth condition at infinity.

scholarly journals Homoclinics for singular strong force Lagrangian systems

2019 ◽  
Vol 9 (1) ◽  
pp. 644-653 ◽  
Author(s):  
Marek Izydorek ◽  
Joanna Janczewska ◽  
Jean Mawhin
Keyword(s):  
Singular Point ◽  
Homoclinic Solutions ◽  
Global Maximum ◽  
Lagrangian Systems ◽  
Infinite Depth ◽  
Force Condition ◽  
Strong Force ◽  
Action Integral ◽  
Smooth Potential ◽  
Single Well

Abstract We study the existence of homoclinic solutions for a class of Lagrangian systems $\begin{array}{} \frac{d}{dt} \end{array} $(∇Φ(u̇(t))) + ∇uV(t, u(t)) = 0, where t ∈ ℝ, Φ : ℝ2 → [0, ∞) is a G-function in the sense of Trudinger, V : ℝ × (ℝ2 ∖ {ξ}) → ℝ is a C1-smooth potential with a single well of infinite depth at a point ξ ∈ ℝ2 ∖ {0} and a unique strict global maximum 0 at the origin. Under a strong force condition around the singular point ξ, via minimization of an action integral, we will prove the existence of at least two geometrically distinct homoclinic solutions u± : ℝ → ℝ2 ∖ {ξ}.

scholarly journals New Periodic Solutions for the Singular Hamiltonian System

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Yi Liao
Keyword(s):  
Hamiltonian System ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Second Order ◽  
Singular Case ◽  
Strong Force ◽  
Second Order Hamiltonian Systems ◽  
Palais Smale Condition ◽  
Singular Hamiltonian System ◽  
Force Potential

By use of the Cerami-Palais-Smale condition, we generalize the classical Weierstrass minimizing theorem to the singular case by allowing functions which attain infinity at some values. As an application, we study certain singular second-order Hamiltonian systems with strong force potential at the origin and show the existence of new periodic solutions with fixed periods.

Positive periodic solutions of first-order singular systems

2012 ◽  
Vol 218 (23) ◽  
pp. 11421-11428 ◽  
Author(s):  
Ruipeng Chen ◽  
Ruyun Ma ◽  
Zhiqian He
Keyword(s):  
Periodic Solutions ◽  
Singular Systems ◽  
Positive Periodic Solutions ◽  
First Order

Periodic Solutions of Symmetric Elliptic Singular Systems: the Higher Codimension Case

2006 ◽  
Vol 6 (1) ◽  
Author(s):  
Flaviano Battelli ◽  
Michal Fečkan
Keyword(s):  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Singular Systems ◽  
Singularly Perturbed ◽  
Singularly Perturbed Systems ◽  
Restoring Force ◽  
Symmetry Properties ◽  
Perturbed Systems ◽  
Existence Of Periodic Solutions

AbstractWe show the existence of periodic solutions of certain singularly perturbed systems having symmetry properties. Our result applies to some singular systems arising in the study of Hamiltonian systems with a strong restoring force and extends to a higher codimension the result obtained in our previous paper [4].

Periodic Solutions of Symmetric Elliptic Singular Systems

2005 ◽  
Vol 5 (2) ◽  
Author(s):  
Flaviano Battelli ◽  
Michal Fečkan
Keyword(s):  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Singular Systems ◽  
Singularly Perturbed ◽  
Singularly Perturbed Systems ◽  
Restoring Force ◽  
Symmetry Properties ◽  
Perturbed Systems ◽  
Existence Of Periodic Solutions

AbstractWe show the existence of periodic solutions of certain singularly perturbed systems having symmetry properties. Our results apply to some singular systems arising in the study of Hamiltonian systems with a strong restoring force.

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