scholarly journals Two remarks on global solutions of ordinary differential equations in the real line

1976 ◽  
Vol 55 (1) ◽  
pp. 111-111 ◽  
Author(s):  
Giovanni Vidossich
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Real Line ◽  
Global Solutions ◽  
The Real

scholarly journals An integral equation associated with linear homogeneous differential equations

1986 ◽  
Vol 9 (2) ◽  
pp. 405-408 ◽  
Author(s):  
A. K. Bose
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Equations ◽  
Real Line ◽  
Homogeneous Differential Equation ◽  
The Real ◽  
Linear Homogeneous Differential Equation

Associated with each linear homogeneous differential equationy(n)=∑i=0n−1ai(x)y(i)of ordernon the real line, there is an equivalent integral equationf(x)=f(x0)+∫x0xh(u)du+∫x0x[∫x0uGn−1(u,v)a0(v)f(v)dv]duwhich is satisfied by each solutionf(x)of the differential equation.

scholarly journals Generalized Exponential Trichotomies for Abstract Evolution Operators on the Real Line

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Nicolae Lupa ◽  
Mihail Megan
Keyword(s):  
Differential Equations ◽  
Real Line ◽  
Exponential Dichotomy ◽  
Natural Extension ◽  
Evolution Operators ◽  
The Real ◽  
Exponential Trichotomy ◽  
Whole Space

This paper considers two trichotomy concepts in the context of abstract evolution operators. The first one extends the notion of exponential trichotomy in the sense of Elaydi-Hajek for differential equations to abstract evolution operators, and it is a natural extension of the generalized exponential dichotomy considered in the paper by Jiang (2006). The second concept is dual in a certain sense to the first one. We prove that these types of exponential trichotomy imply the existence of generalized exponential dichotomy on both half-lines. We emphasize that we do not assume the invertibility of the evolution operators on the whole spaceX(unlike the case of evolution operators generated by differential equations).

scholarly journals Comparison of Noether Symmetries and First Integrals of Two-Dimensional Systems of Second Order Ordinary Differential Equations by Real and Complex Methods

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1180
Author(s):  
Muhammad Safdar ◽  
Asghar Qadir ◽  
Muhammad Umar Farooq
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
First Integrals ◽  
Second Order ◽  
Complex Method ◽  
Two Dimensional ◽  
Noether Symmetries ◽  
The Real ◽  
Complex Methods

Noether symmetries and first integrals of a class of two-dimensional systems of second order ordinary differential equations (ODEs) are investigated using real and complex methods. We show that first integrals of systems of two second order ODEs derived by the complex Noether approach cannot be obtained by the real methods. Furthermore, it is proved that a complex method can be extended to larger systems and higher order.

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