HOMOCLINIC SOLUTION FOR A CLASS OF SECOND-ORDER \bigtriangledown-DIFFERENTIAL EQUATIONS ON TIME SCALES

2017 ◽  
Vol 102 (10) ◽  
pp. 2199-2217
Author(s):  
Xiuli Nong ◽  
Li Yang
Keyword(s):  
Differential Equations ◽  
Time Scales ◽  
Homoclinic Solution ◽  
Second Order

scholarly journals Generalized multitime expansions for equations with slowly varying coefficients

1984 ◽  
Vol 7 (1) ◽  
pp. 151-158
Author(s):  
L. E. Levine ◽  
W. C. Obi
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Small Parameter ◽  
Time Scales ◽  
Constant Coefficient ◽  
Variable Coefficients ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Varying Coefficients

The successive terms in a uniformly valid multitime expansion of the solutions of constant coefficient differential equations containing a small parameterϵmay be obtained without resorting to secularity conditions if the time scalesti=ϵit(i=0,1,…)are used. Similar results have been achieved in some cases for equations with variable coefficients by using nonlinear time scales generated from the equations themselves. This paper extends the latter approach to the general second order ordinary differential equation with slowly varying coefficients and examines the restrictions imposed by the method.

scholarly journals Existence and Uniqueness of Homoclinic Solution for a Class of Nonlinear Second-Order Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Lijuan Chen ◽  
Shiping Lu
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Homoclinic Orbit ◽  
Limit Point ◽  
Existence And Uniqueness ◽  
Homoclinic Solution ◽  
Second Order ◽  
Continuation Theorem ◽  
Mawhin’S Continuation Theorem ◽  
Second Order Differential Equations

The authors study the existence and uniqueness of a set with2kT-periodic solutions for a class of second-order differential equations by using Mawhin's continuation theorem and some analysis methods, and then a unique homoclinic orbit is obtained as a limit point of the above set of2kT-periodic solutions.

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

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