Property (X+) for Second-Order Nonlinear Differential Equations of Generalized Euler Type

In this paper the generalized nonlinear Euler differential equation t2k(tu′)u″ + t(f(u)+ k(tu′))u′ + g(u) = 0 is considered. Here the functions f(u), g(u) and k(u) satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub(super...

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Bibliographic Details
Published in:Acta mathematica scientia 2013-09, Vol.33 (5), p.1398-1406
Main Authors: AGHAJANI, Asadollah, ROOMI, Vahid
Format: Article
Language:English
Subjects:
34C10
35Q31
Differential equations
Equivalence
Functions (mathematics)
Initial value problems
Liénard system
Mathematical analysis
nonlinear differential equations
Nonlinearity
Oscillations
property (X+)
Uniqueness
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Summary:In this paper the generalized nonlinear Euler differential equation t2k(tu′)u″ + t(f(u)+ k(tu′))u′ + g(u) = 0 is considered. Here the functions f(u), g(u) and k(u) satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub(super) linearity are assumed. We present some necessary and sufficient conditions and some tests for the equivalent planar system to have or fail to have property (X+), which is very important for the existence of periodic solutions and oscillation theory.
ISSN:0252-9602
1572-9087
DOI:10.1016/S0252-9602(13)60091-0