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Perturbations of the half-linear Euler differential equation. (English) Zbl 0958.34029

The authors investigate oscillation/nonoscillation properties of the perturbed half-linear Euler differential equation

(x ' n* ) ' +γ 0 t n+1 [n+2(n+1)δ(t)]x n* =0,(*)

where the function δ(t) is piecewise continuous on (t 0 ,), t 0 0, n>0 is a fixed real number and u n* =|u| n sgnu. The number γ 0 =n n (n+1) n+1 is a critical constant, which means that in case δ(t)0 the half-linear Euler differential equation is for γγ 0 nonoscillatory while for γ>γ 0 is oscillatory. Using a Riccati technique and a transformation of the independent variable, there are proved the following main results:

Suppose that there exists the finite limit

lim T t 0 T δ(t)dt t

such that t δ(s)ds s0 for t>t 0 .

(a) If n>1 and the linear equation

z '' +δ(e s )z=0(**)

is nonoscillatory then equation (*) is also nonoscillatory.

(b) If 0<n<1 and equation (*) is nonoscillatory then the linear equation (**) is also nonoscillatory.

In addition, the authors establish an asymptotic form of the solution to (*) provided that the solutions to (**) satisfy two integral inequalities.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
[1]Á. Elbert, Oscillation and nono s dilation theorems for some nonlinear ordinary differential equations, Ordinary and Partial Differential Equations (Dundee, 1982), Lecture Notes in Mathematics, vol. 964, Springer, Berlin – Heidelberg – New York, 1982, pp. 187–212.
[2]Á. Elbert, Asymptotic behaviour of autonomous half-linear differential systems on the plane, Studia Sci. Math. Hungar. 19 (1984), 447–464.
[3]Á. Elbert, Generalized Riccati equation for half-linear second order differential equations, Differential and Integral Equations: Qualitative Theory. Colloq. Math. Soc. J. Bolyai, vol. 47, 1984, pp. 227–249.
[4]Á. Elbert, Qualitative Theory of Half-linear Differential Equations, Dissertation for Doctoral Degree, 1987. (Hungarian)
[5]Á. Elbert and T. Kusano, Principal solutions of non-oscillatory half-linear differential equations, Advances in Mathematical Sciences and Applications 8 (1998), 745–759.
[6]Ph. Hartman, On nonoscillatory linear differential equations of second order, Amer. J. Math. 74 (1952), 389–400. · doi:10.2307/2372004
[7]Ph. Hartman, Ordinary Differential Equations (2nd edit.), Birkhäuser, Boston – Basel – Stuttgart, 1982.
[8]Ph. Hartman and A. Wintner, On the asignment of asymptotic values for the solutions of linear differential equations of second order, Amer. J. Math. 77 (1955), 475–483. · doi:10.2307/2372635
[9]E. Hille, Nonoscillation theorems, Trans. Amer. Math. Soc. 64 (1948), 234–252. · doi:10.1090/S0002-9947-1948-0027925-7
[10]J. D. Mirzov, Principal and nonprincipal solutions of a nonlinear system, Proceedings of I.N. Vekua Institute of Applied Mathematics (Tbilissi State University) 31 (1988), 100–117. (Russian)
[11]A. Wintner, A criterion of oscillatory stability, Quart. Appl. Math. 7 (1949), 115–117.
[12]A. Wintner, On the nonexistence of conjugate points, Amer. J. Maths 73 (1951), 368–380. · doi:10.2307/2372182