PROPERTY (X^+) FOR SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATIONS OF GENERALIZED EULER TYPE

摘要

In this paper the generalized nonlinear Euler diferential 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 sufcient 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.