Order Half-linear Differential Equations Research Articles

Oscillation criteria for a second order half-linear differential equation with delay, with monotone nondecreasing delay function

Oscillation criteria for a second order half-linear differential equation with delay, with monotone nondecreasing delay function

New Criteria on Oscillatory Behavior of Third Order Half-Linear Functional Differential Equations

This paper deals with some new criteria for the oscillation of third order half-linear differential equations. The purpose of the present paper is the linearization of equation 1.1 in the sense that we would deduce oscillation of studied equation from that of the linear form and to provide new oscillation criteria via comparison with first order equations whose oscillatory behavior are known. The obtained results are new, improve and correlate many of the known oscillation criteria appeared in the literature for equation 1.1. The results are illustrated by some examples.

Extreme and moderate solutions of nonoscillatory second order half-linear differential equations

An existence and asymptotic theory is built for second order half-linear differential equations of the form $${\rm (A)}\qquad (p(t)|x^{\prime }|^\alpha {\rm sgn} \,x^{\prime })^{\prime } =q(t)|x|^\alpha {\rm sgn}\, x, \quad t \geq a \gt 0, $$ where $\alph

Oscillation and Asymptotic Properties of Second Order Half-Linear Differential Equations with Mixed Deviating Arguments

In this paper, we study oscillation and asymptotic properties for half-linear second order differential equations with mixed argument of the form r(t)(y′(t))α′=p(t)yα(τ(t)). Such differential equation may possesses two types of nonoscillatory solutions either from the class N0 (positive decreasing solutions) or N2 (positive increasing solutions). We establish new criteria for N0=∅ and N2=∅ provided that delayed and advanced parts of deviating argument are large enough. As a consequence of these results, we provide new oscillatory criteria. The presented results essentially improve existing ones even for a linear case of considered equations.

Oscillation of second order half linear neutral differential equations

In this research, we study the properties of oscillation for all solutions of the second- order half linear neutral differential equations. Some basic, necessary and sufficient condition are established for every solution of to become oscillatory. Examples are given in support of our results.

Asymptotic Behavior Criteria for Solutions Second Order Half Linear Neutral Differential Equation

In this paper the we study the oscillation and asymptotic behavior of second-order Half linear neutral differential equations. some basic and necessary and sufficient condition are established for every solution of , to insure the convergence of all nonoscillatory solutions to zero or tends to infinity as Some examples are given to illustrate the main results.

Nonlinear Poincaré–Perron theorem

We establish a nonlinear extension of the Poincaré–Perron theorem for a second order half-linear differential equation. Conditions are established which guarantee that all nontrivial solutions y of the equation are such that a proper limit limt→∞y′(t)∕y(t) exists. In addition, we discuss establishing precise asymptotic formulae for these solutions. We employ theory of regular variation and thereby, as a by-product, we complete some results for asymptotics of differential equations considered in this framework. The results can serve for comparison purposes involving more general equations and are of importance also from the stability point of view.

Oscillation of second order half-linear differential equations with deviating arguments of mixed type

In the paper, new criteria for the oscillatory behavior of half-linear second order differential equations with mixed argument of the form (E)r(t)(y′(t))α′=p(t)yα(τ(t)).are established. The presented results essentially improve existing ones even for linear case of considered equations.

Even Order Half-Linear Differential Equations with Regularly Varying Coefficients

We establish nonoscillation criterion for the even order half-linear differential equation (−1)nfn(t)Φx(n)(n)+∑l=1n(−1)n−lβn−lfn−l(t)Φx(n−l)(n−l)=0, where β0,β1,…,βn−1 are real numbers, n∈N, Φ(s)=sp−1sgns for s∈R, p∈(1,∞) and fn−l is a regularly varying (at infinity) function of the index α−lp for l=0,1,…,n and α∈R. This equation can be understood as a generalization of the even order Euler type half-linear differential equation. We obtain this Euler type equation by rewriting the equation above as follows: the terms fn(t) and fn−l(t) are replaced by the tα and tα−lp, respectively. Unlike in other texts dealing with the Euler type equation, in this article an approach based on the theory of regularly varying functions is used. We establish a nonoscillation criterion by utilizing the variational technique.

Oscillation of second order half-linear neutral differential equations with weaker restrictions on shifted arguments

Abstract Neutral differential equations are one of the most important extensions of classical ordinary differential equations and aim to give a better explanation for modeling phenomena where ordinary differential equations are insufficient. Naturally, all the questions studied in the scope of ordinary differential equations attracted the attention also for neutral differential equations. In this paper we study the oscillatory properties of second order half-linear neutral differential equations. We present oscillation criteria derived using a new approach. This approach allows us to reduce common restrictions on the deviations in arguments which are present in the currently known results of this type.

An integral oscillation criterion for even order half-linear differential equations

Integral tests for detection of oscillatory behavior of solutions of even order half-linear differential equations of the form Lnx+q(t)|x|βsgnx=0,t≥a>0,(A)where L1x(t)=x′(t),Li+1x(t)=[pi(t)|Lix(t)|αisgnLix(t)]′,i=1,…n−1,αi,1≤i≤n−1, and β being positive constants and pi,1≤i≤n−1, and q continuous functions from [a,∞) to (0,∞), are established in the case where α1⋯αn−1=β.

Oscillation criteria for fourth order half-linear differential equations

Criteria for oscillatory behavior of solutions of fourth order half-linear differential equations of the form \begin{equation*} \big (|y^{\prime \prime }|^\alpha {\rm sgn\ } y^{\prime \prime }\big )^{\prime \prime } + q(t)|y|^\alpha {\rm sgn}\ y = 0, \quad t \ge a > 0, A \end{equation*} where $\alpha > 0$ is a constant and $q(t)$ is positive continuous function on $[a,\infty )$, are given in terms of an increasing continuously differentiable function $\omega (t)$ from $[a,\infty )$ to $(0,\infty )$ which satisfies $\int _a^\infty 1/(t\omega (t))\,dt < \infty $.

Oscillation criteria for second order half-linear differential equations with delay

Oscillation criteria for second order half-linear differential equations with delay

Oscillation of neutral second order half-linear differential equations without commutativity in deviating arguments

Abstract In this paper we derive oscillation criteria for the second order half-linear neutral differential equation [ r ( t ) Φ ( z ′ ( t ) ) ] ′ + c ( t ) Φ ( x ( σ ( t ) ) ) = 0 , z ( t ) = x ( t ) + b ( t ) x ( τ ( t ) ) , $$ \displaystyle \Bigl[r(t)\Phi(z'(t))\Bigr]'+c(t)\Phi(x(\sigma(t)))=0, \quad z(t)=x(t)+b(t)x(\tau(t)), $$ where Φ(t) = |t| p−2 t, p ≥ 2, is a power type nonlinearity. We improve recent results published in the literature by obtaining better oscillation constants and removing the usual condition σ(τ(t)) = τ(σ(t)). Two methods (comparison method and Riccati equation method) are used.

Precise asymptotic behavior of regularly varying solutions of second order half-linear differential equations

Precise asymptotic behavior of regularly varying solutions of second order half-linear differential equations

Lyapunov-type inequalities for higher order half-linear differential equations

The Lyapunov inequality for second order linear differential equations has been extended to higher order linear differential equations in various forms. It has also been extended to second and third order half-linear equations. However, it has not yet been developed for higher order half-linear differential equations due to the nonlinear feature of the equation and the complexity caused by the multiple p-Laplacian operators in the equation. In this paper, by subtle applications of forward and backward inductions, we establish several Lyapunov-type inequalities for even and odd order half-linear differential equations. These inequalities are applied to determine the nonexistence of nontrivial solutions of higher order boundary value problems and to estimate eigenvalues for higher order eigenvalue problems. Our results cover many results in the literature when the equations become linear.

Oscillations of Third Order Half Linear Neutral Differential Equations

In this paper the oscillation criterion was investigated for all solutions of the third-order half linear neutral differential equations. Some necessary and sufficient conditions are established for every solution of (a(t)[(x(t)±p(t)x(?(t) ) )^'' ]^? )^'+q(t) x^? (?(t) )=0, t?t_0, to be oscillatory. Examples are given to illustrate our main results.

Oscillations of Third Order Half Linear Neutral Differential Equations

In this paper the oscillation criterion was investigated for all solutions of the third-order half linear neutral differential equations. Some necessary and sufficient conditions are established for every solution of (a(t)[(x(t)±p(t)x(?(t) ) )^'' ]^? )^'+q(t) x^? (?(t) )=0, t?t_0, to be oscillatory. Examples are given to illustrate our main results.

Oscillations of even order half-linear impulsive delay differential equations with damping

In this paper, a kind of half-linear impulsive delay differential equations with damping is studied. By employing a generalized Riccati technique and the impulsive differential inequality, we derive several oscillation criteria which are either new or improve several recent results in the literature. In addition, we provide several examples to illustrate the use of our results.

Lyapunov-type inequalities for $(m+1)$th order half-linear differential equations with anti-periodic boundary conditions

In this work, we will establish several new Lyapunov-type inequalities for (m + 1)th order half-linear differential equations with anti-periodic boundary condi- tions, the results of this paper are new and generalize and improve some early results in the literature.