Non-oscillation of half-linear differential equations with periodic coefficients

We consider half-linear Euler type differential equations with general periodic coefficients. It is well-known that these equations are conditionally oscillatory, i.e., there exists a border value given by their coefficients which separates oscillatory equations from non-oscillatory ones. In this paper, we study oscillatory properties in the border case. More precisely, we prove that the considered equations are non-oscillatory in this case. Our results cover the situation when the periodic coefficients do not have any common period

Modified Riccati technique for half-linear differential equations with delay

We study the half-linear differential equation $$(r(t)\Phi(x'(t)))'+c(t)\Phi(x(\tau(t)))=0,\quad \Phi(x):=|x|^{p-2}x,\ p>1.$$ We formulate new oscillation criteria for this equation by comparing it with a certain ordinary linear or half-linear differential equation. Our proofs are based on a suitable estimate for the solution of the equation studied and on the modified Riccati technique, which, in ordinary case, appeared to be an effective replacement of the well known linear transformation formula

Oscillation and nonoscillation of third order functional differential equations

A qualitative approach is usually concerned with the behavior of solutions of a given differential equation and usually does not seek specific explicit solutions;This dissertation is the analysis of oscillation of third order linear homogeneous functional differential equations, and oscillation and nonoscillation of third order nonlinear nonhomogeneous functional differential equations. This is done mainly in Chapters II and III. Chapter IV deals with the analysis of solutions of neutral differential equations of third order and even order. In Chapter V we study the asymptotic nature of nth order delay differential equations;Oscillatory solution is the solution which has infinitely many zeros; otherwise, it is called nonoscillatory solution;The functional differential equations under consideration are:(UNFORMATTED TABLE OR EQUATION FOLLOWS) (b(ay[superscript]\u27)[superscript]\u27)[superscript]\u27 + (q[subscript]1y)[superscript]\u27 + q[subscript]2y[superscript]\u27 = 0, &(b(ay[superscript]\u27)[superscript]\u27)[superscript]\u27 + q[subscript]1y + q[subscript]2y(t - [tau]) = 0, &(b(ay[superscript]\u27)[superscript]\u27)[superscript]\u27 + qF(y(g(t))) = f(t), &(y(t) + p(t)y(t - [tau]))[superscript]\u27\u27\u27 + f(t, y(t), y(t - [sigma])) = 0, &(y(t) + p(t)y(t - [tau]))[superscript](n) + f(t, y(t), y(t - [sigma])) = 0, and &y[superscript](n) + p(t)f(t, y[tau], y[subscript]sp[sigma][subscript]1\u27,..., y[subscript]sp[sigma][subscript]n[subscript]1(n-1)) = F(t). (TABLE/EQUATION ENDS);The first and the second equations are considered in Chapter II, where we find sufficient conditions for oscillation. We study the third equation in Chapter III and conditions have been found to ensure the required criteria. In Chapter IV, we study the oscillation behavior of the fourth and the fifth equations. Finally, the last equation has been studied in Chapter V from the point of view of asymptotic nature of its nonoscillatory solutions