Oscillation Criteria for Second Order Nonlinear Differential Equations Involving General Means

AbstractConsider the second order nonlinear differential equation (E) y″+a(t)f(y)=0 where a(t)∈C[0,∞), f(y)∈C1(−∞,∞), f′(y)≥0, and yf(y)>0 for y≠0. Furthermore, f(y) also satisfies either a superlinear or a sublinear condition, which covers the nonlinear function f(y)=y|y|γ−1 with γ>1 and 0<γ<1, respectively, commonly known as the Emden–Fowler case. Here the coefficient function a(t) is allowed to be negative for arbitrarily large values of t. Kamenev type oscillation criteria involving integral averages for the linear equations (L) y″+a(t)y=0 are extended to the nonlinear equation (E) by using more general means. The results extend similar results on general means by Philos for the linear equation (L) and also results based upon Kamenev's integral averaging method concerning the nonlinear equation (E)

Integral averaging technique for the interval oscillation criteria of certain second-order nonlinear differential equations

AbstractWe present new interval oscillation criteria related to integral averaging technique for certain classes of second-order nonlinear differential equations which are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0,∞), rather than on the whole half-line. They generalize and improve some known results. Examples are also given to illustrate the importance of our results

Oscillation of Second Order Nonlinear Dynamic Equations on Time Scales

By means of Riccati transformation techniques, we establish some oscillation criteria for a second order nonlinear dynamic equation on time scales in terms of the coefficients. We give examples of dynamic equations to which previously known oscillation criteria are not applicable

A remark on Philos-type oscillation criteria for differential equations

The purpose of this short note is to call attention to expressions of Philos-type criteria for the oscillation of solutions of a simple differential equation. A perfect square expression is used to obtain an evaluation that plays an essential role in the proof of Philos-type oscillation theorems. The required condition is pointed out when using the perfect square expression. To simplify the discussion, here we deal with two second-order linear differential equations, but its content is also applied to a variety of equations