Geometric characterization of separable second-order differential equations

1993 ◽  
Vol 113 (1) ◽  
pp. 205-224 ◽  
Author(s):  
Eduardo Martínez ◽  
José F. Cariñena ◽  
Willy Sarlet
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Geometric Characterization ◽  
One Dimensional ◽  
Practical Procedure ◽  
Second Order Differential Equations ◽  
Necessary And Sufficient ◽  
Second Order Equations

AbstractWe establish necessary and sufficient conditions for the separability of a system of second-order differential equations into independent one-dimensional second-order equations. The characterization of this property is given in terms of geometrical objects which are directly related to the system and relatively easy to compute. The proof of the main theorem is constructive and thus yields a practical procedure for constructing coordinates in which the system decouples.

Geometric characterization of linearisable second-order differential equations

1996 ◽  
Vol 119 (2) ◽  
pp. 373-381 ◽  
Author(s):  
Eduardo Martínez ◽  
José F. Cariñena
Keyword(s):  
Differential Equations ◽  
Tangent Bundle ◽  
Sufficient Conditions ◽  
Linear Connection ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Ehresmann Connection ◽  
Local Coordinates ◽  
Necessary And Sufficient

AbstractGiven an Ehresmann connection on the tangent bundle τ: TM → M we define a linear connection on the pull-back bundle τ*(TM). With the aid of this tool, necessary and sufficient conditions are derived for the existence of local coordinates in which a system of second-order differential equations is linear.

scholarly journals Multiple Bounded Positive Solutions to Integral Type BVPs for Singular Second Order ODEs on the Whole Line

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Yuji Liu
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Second Order ◽  
Integral Type ◽  
One Dimensional ◽  
Second Order Differential Equations ◽  
Type Boundary ◽  
The One

This paper is concerned with the integral type boundary value problems of the second order differential equations with one-dimensionalp-Laplacian on the whole line. By constructing a suitable Banach space and a operator equation, sufficient conditions to guarantee the existence of at least three positive solutions of the BVPs are established. An example is presented to illustrate the main results. The emphasis is put on the one-dimensionalp-Laplacian term[ρ(t)Φ(x’(t))]’involved with the functionρ, which makes the solutions un-concave.

ON THE INVERSE PROBLEM OF LAGRANGIAN SUPERMECHANICS

1993 ◽  
Vol 08 (20) ◽  
pp. 3565-3576 ◽  
Author(s):  
L. A. IBORT ◽  
G. LANDI ◽  
J. MARÍN-SOLANO ◽  
G. MARMO
Keyword(s):  
Inverse Problem ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Lagrangian Function ◽  
Necessary And Sufficient Conditions ◽  
Recursion Operators ◽  
Second Order Differential Equations ◽  
Necessary And Sufficient

The inverse problem of Lagrangian supermechanics is investigated. A set of necessary and sufficient conditions for a system of second order differential equations in superspace to derive from a (possibly nonregular) super-Lagrangian function are given. The harmonic superoscillator is revisited and a family of even and odd alternative super-Lagrangians are constructed for it. Finally, we comment on the existence of recursion operators.

scholarly journals New Theorems for Oscillations to Differential Equations with Mixed Delays

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 367
Author(s):  
Shyam Sundar Santra ◽  
Debasish Majumder ◽  
Rupak Bhattacharjee ◽  
Omar Bazighifan ◽  
Khaled Mohamed Khedher ◽  
...  
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Mixed Delays ◽  
Second Order Differential Equations ◽  
Neutral Term ◽  
The Right ◽  
Necessary And Sufficient

The oscillation of differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of this equations. The purpose of this article is to establish new oscillatory properties which describe both the necessary and sufficient conditions for a class of nonlinear second-order differential equations with neutral term and mixed delays of the form p(ι)w′(ι)α′+r(ι)uβ(ν(ι))=0,ι≥ι0 where w(ι)=u(ι)+q(ι)u(ζ(ι)). Furthermore, examining the validity of the proposed criteria has been demonstrated via particular examples.

scholarly journals Second-order impulsive differential systems with mixed and several delays

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shyam Sundar Santra ◽  
Apurba Ghosh ◽  
Omar Bazighifan ◽  
Khaled Mohamed Khedher ◽  
Taher A. Nofal
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Oscillation Theory ◽  
Second Order ◽  
Neutral Delay ◽  
Differential Systems ◽  
Impulsive Differential Systems ◽  
Several Delays ◽  
Necessary And Sufficient

AbstractIn this work, we present new necessary and sufficient conditions for the oscillation of a class of second-order neutral delay impulsive differential equations. Our oscillation results complement, simplify and improve recent results on oscillation theory of this type of nonlinear neutral impulsive differential equations that appear in the literature. An example is provided to illustrate the value of the main results.

scholarly journals New Conditions for the Oscillation of Second-Order Differential Equations with Sublinear Neutral Terms

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1159
Author(s):  
Shyam Sundar Santra ◽  
Omar Bazighifan ◽  
Mihai Postolache
Keyword(s):  
Fluid Dynamics ◽  
Neural Networks ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Qualitative Behavior ◽  
Neutral Differential Equations ◽  
Second Order Differential Equations ◽  
Delay Differential

In continuous applications in electrodynamics, neural networks, quantum mechanics, electromagnetism, and the field of time symmetric, fluid dynamics, neutral differential equations appear when modeling many problems and phenomena. Therefore, it is interesting to study the qualitative behavior of solutions of such equations. In this study, we obtained some new sufficient conditions for oscillations to the solutions of a second-order delay differential equations with sub-linear neutral terms. The results obtained improve and complement the relevant results in the literature. Finally, we show an example to validate the main results, and an open problem is included.

scholarly journals Periodic solutions for periodic second-order differential equations with variable potentials

2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

scholarly journals Boundedness of Solutions for Second Order Differential Equations

2020 ◽  
Vol 12 (4) ◽  
pp. 58
Author(s):  
Daniel C. Biles
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Boundedness Of Solutions ◽  
Second Order Differential Equations ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential

We present new theorems which specify sufficient conditions for the boundedness of all solutions for second order non-linear differential equations. Unboundedness of solutions is also considered.

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