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the application of Walsh functions in communication and signal processing (see, among others, [ 1,12,14,16,17 ]). In [ 8 ] Corrington developed a method to solve n th order linear differential equations using previously prepared huge tables of the

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Došlý , O. and Hasil , P., Critical oscillation constant for half-linear differential equations with periodic coefficients, Ann. Mat. Pura Appl. , 190 (2011), no. 3, pp. 395–408. Hasil P

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. and Došlý, O. , Principal Solution of Half-linear Differential Equation: Limit and Integral Characterization , Proceedings of the 8th Colloquium on the Qualitative Theory of Differential Equations (Szeged, 2008), no. 6, pp. 1

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Summary  

We establish new comparison theorems on the oscillation of solutions of a class of perturbed half-linear differential equations. These improve the work of Elbert and Schneider [6] in which connections are found between half-linear differential equations and linear differential equations. Our comparison theorems are not of Sturm type or Hille--Wintner type which are very famous. We can apply the main results in combination with Sturm's or Hille--Wintner's comparison theorem to a half-linear differential equation of the general form (|x'|α-1x')' + a(t) |x|α-1x = 0.

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Abstract  

Several comparison theorems with respect to powers in nonlinearities for half-linear differential equations are presented. The Riccati transformation and the reciprocity principle are utilized. Some examples and an integral extension of the classical comparison result are presented as well.

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Abstract  

This paper deals with the second-order half-linear differential equation

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(\phi _p (\dot x))^ \cdot + a(t)\phi _p (\dot x) + b(t)\phi _p (x) = 0.$$ \end{document}
Here ϕ p(z):= |z|p−2 z is the so-called one-dimensional p-Laplacian operator. Our main purpose is to establish new criteria for all nontrivial solutions to be oscillatory and for those to be nonoscillatory. In our theorems, the parametric curve given by (a(t), b(t)) plays a critical role in judging whether all solutions are oscillatory or nonoscillatory. This paper takes a di-erent approach from most of the previous research. Our results are new even in the linear case (p = 2). The method used here is mainly phase plane analysis for a system equivalent to the half-linear differential equation. Some suitable examples are included to illustrate the main results. Global phase portraits are also attached.

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] D ošlý , O. and F unková , H. , Euler type half-linear differential equation with periodic coefficients , Abstr. Appl. Anal. , 2013 , article ID 714263, 1 – 6 . [6

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Abstract  

We consider a nonoscillatory half-linear second order differential equation

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(r(t)\Phi (x'))' + c(t)\Phi (x) = 0,\Phi (x) = \left| x \right|^{p - 2} x,p > 1,$$ \end{document}
((*)) and suppose that we know its solution h. Using this solution we construct a function d such that the equation
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(r(t)\Phi (x'))' + [c(t) + \lambda d(t)]\Phi (x) = 0$$ \end{document}
((**))
is conditionally oscillatory. Then we study oscillations of the perturbed equation (**). The obtained (non)oscillation criteria extend existing results for perturbed half-linear Euler and Euler-Weber equations.

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