Oscillation criteria for third order delay nonlinear differential equations

The purpose of this paper is to give oscillation criteria for the third order delay nonlinear differential equation \begin{equation*} \lbrack a_{2}(t)\{(a_{1}(t)(x^{\prime }(t))^{\alpha _{1}})^{\prime}\}^{\alpha _{2}}]^{\prime }+q(t)f(x(g(t)))=0, \end{equation*} via comparison with some first differential equations whose oscillatory characters are known. Our results generalize and improve some known results for oscillation of third order nonlinear differential equations. Some examples are given to illustrate the main results

Oscillation criteria for third order delay nonlinear differential equations

The purpose of this paper is to give oscillation criteria for the third order delay nonlinear differential equation \begin{equation*} \lbrack a_{2}(t)\{(a_{1}(t)(x^{\prime }(t))^{\alpha _{1}})^{\prime}\}^{\alpha _{2}}]^{\prime }+q(t)f(x(g(t)))=0, \end{equation*} via comparison with some first differential equations whose oscillatory characters are known. Our results generalize and improve some known results for oscillation of third order nonlinear differential equations. Some examples are given to illustrate the main results

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

Asymptotic properties of solutions of certain third-order dynamic equations

AbstractIn this paper, the well known oscillation criteria due to Hille and Nehari for second-order linear differential equations will be generalized and extended to the third-order nonlinear dynamic equation (r2(t)((r1(t)xΔ(t))Δ)γ)Δ+q(t)f(x(t))=0 on time scale T, where γ≥1 is a ratio of odd positive integers. Our results are essentially new even for third-order differential and difference equations, i.e., when T=R and T=N. Two examples of dynamic equations on different time scales are given to show the applications of our main results

Properties of Third-Order Nonlinear Functional Differential Equations with Mixed Arguments

The aim of this paper is to offer sufficient conditions for property (B) and/or the oscillation of the third-order nonlinear functional differential equation with mixed arguments [()[″()]]′=()([()])+()ℎ([()]). Both cases ∫∞−1/()d=∞ and ∫∞−1/()d<∞ are considered. We deduce properties of the studied equations via new comparison theorems. The results obtained essentially improve and complement earlier ones

Oscillation Criteria of Third-Order Nonlinear Impulsive Differential Equations with Delay

This paper deals with the oscillation of third-order nonlinear impulsive equations with delay. The results in this paper improve and extend some results for the equations without impulses. Some examples are given to illustrate the main results

Oscillation Results for Emden–Fowler Type Differential Equations

AbstractThe third order nonlinear differential equationx‴+a(t)x′+b(t)f(x)=0, (∗)is considered. We present oscillation and nonoscillation criteria which extend and improve previous results existing in the literature, in particular some results recently stated by M. Greguš and M. Greguš, Jr., (J. Math. Anal. Appl.181, 1994, 575–585). In addition, contributions to the classification of solutions are given. The techniques used are based on a transformation which reduces (∗) to a suitable disconjugate form. To this aim auxiliary results on the asymptotic behavior of solutions of a second order linear differential equation associated to (∗) are stated. They are presented in an independent form because they may be applied also to simplify and improve other qualitative problems concerning differential equations with quasiderivatives

ON THE OSCILLATION OF A THIRD ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH NEUTRAL TYPE

In this article, we investigate that oscillation behavior of the solutions of the third-order nonlinear differential equation with neural type of the form$$\Big(a_{1}(t)\big(a_{2}(t)Z^{\prime}(t)\big)^{\prime}\Big)^{\prime}+ q(t) f\big(x(\sigma(t))\big) = 0, \quad t\geq t_0 > 0,$$where $Z(t) := x(t)+p(t)x^{\alpha}(\tau(t))$. Some new oscillation results are presented that extend those results given in the literature

Judėjimo lygčių atskyrimas dvimatės gravitacinės švytuoklės modelyje

We solve the classical problem-model of the oscillation of the gravitational pendulum on the elastic thread. This task was formulated, but it is not completely solved in the classical K. Magnusmonograph “Wavering”. With additional assumptions about the analyzed model in question (massconservation law and considerations about certain physic) system of the differential equations of thetwo second-order nonlinear oscillation transformed into a third order nonlinear differential equationwhich is solved numerical approximate using the Maple program. The solution is compared withasymptotic in a long time range.Nagrinėjame gravitacinės švytuoklės ant tampraus siūlo svyravimų modelį.Šis modelis susiveda į dvejų netiesinių gana komplikuotų lygčių sistemą [2], kurioje išsamiai nagrinėjami tik atskiri artėjimai stacionarios būsenos atžvilgių. Naudojant papildomasfizikines prielaidas apie nagrinėjamą modelį ir energijos tvermės dėsnį, dviejų antros eilėsnetiesinių svyravimo diferencialinių lygčių sistemą pavyko pertvarkyti į paprastesnę sistemą,kurioje viena iš lygčių yra trečios eilės netiesinė diferencialinė lygtis, kuri sprendžiama skaitiškai, naudojant Maple programą. Sprendinys lyginamas su asimptotiniu artiniu ilgajamelaiko intervale