Order Nonlinear Delay Dynamic Equations Research Articles

Retraction notice to ‘‘Oscillation Properties of Third Order Nonlinear Delay Dynamic Equations on Time Scales’’ [J. Amr. Uni. 3 (2023) 75 – 100

In this paper, we shall investigate the oscillatory properties of third order nonlinear delay dynamicequations. Applying suitable comparison theorems and by a Riccati transformation technique, weestablish some new sufficient conditions which insure that every solution of this equation either oscillatesor converges to zero. Our results not only unify the oscillation of third order nonlinear differential anddifference equations but also can be applied to different types of time scales with

Oscillation Properties of Third Order Nonlinear Delay Dynamic Equations on Time Scales

In this paper, we shall investigate the oscillatory properties of third order nonlinear delay dynamic equations. Applying suitable comparison theorems and by a Riccati transformation technique, we establish some new sufficient conditions which insure that every solution of this equation either oscillates or converges to zero. Our results not only unify the oscillation of third order nonlinear differential and difference equations but also can be applied to different types of time scales with

New oscillatory results for non-linear delay dynamic equations with super-linear neutral term

A new approach is introduced for the oscillatory solutions of the second order non-linear delay dynamic equations on time scale with a super-linear neutral term. The result are obtained through various integral and differential relations. The Kamenev and Philos-type criteria for oscillation are also discussed. The results improve and extend the ones reported in the literature in the sense that the inequalities are easy to verify and results cover various kinds of equations by choosing different time scales. Examples for particular time scales are given for illustration.

OSCILLATORY AND ASYMPTOTIC CRITERIA OF THIRD ORDER NONLINEAR DELAY DYNAMIC EQUATIONS WITH DAMPING TERM ON TIME SCALES

In this paper, we are concerned with oscillatory and asymptotic behavior of third order nonlinear delay dynamic equations with damping term on time scales. By using a generalized Riccati function and inequality technique, we establish some new oscillatory and asymptotic criteria. The established results on one hand extend some known results in the literature, on the other hand unify continuous and discrete analysis as two special cases of an arbitrary time scale. We also present some applications for the established results.

Oscillation criteria for higher order nonlinear delay dynamic equations on time scales

Abstract In this paper, we study the oscillation criteria of the following higher order nonlinear delay dynamic equation R n △ ( t , x ( t ) ) + b ( t ) | R n − 1 △ ( t , x ( t ) ) | γ − 1 R n − 1 △ ( t , x ( t ) ) + q ( t ) f ( | x ( τ ( t ) ) | γ − 1 x ( τ ( t ) ) ) = 0 $$R_n^\triangle (t, x(t))+b(t)|R_{n-1}^\triangle(t, x(t))|^{\gamma-1} R_{n-1}^\triangle(t, x(t))+q(t)f(|x(\tau(t))|^{\gamma-1}x(\tau(t)))=0$$ on an arbitrary time scale T with sup T = ∞, where n ≥ 2, γ > 0 is a constant, τ: T → T with τ(t) ≤ t and lim t → ∞ ⁡ τ ( t ) = ∞ $\mathop {\lim }\limits_{t \to \infty } \tau (t) = \infty $ and R k ( t , x ( t ) ) = x ( t ) , if k = 0 , r k ( t ) R k − 1 △ ( t , x ( t ) ) , if 1 ≤ k ≤ n − 1 , r n ( t ) | R n − 1 △ ( t , x ( t ) ) | γ − 1 R n − 1 △ ( t , x ( t ) ) , if k = n , $$R_k(t, x(t))= \begin{cases} x(t),&\text{if}\ \;\;k=0, \\ r_k(t)R^\triangle_{k-1}(t, x(t)),&\text{if}\ \;\;1\leq k\leq n-1, \\ r_n(t)|R^\triangle_{n-1}(t, x(t))|^{\gamma-1} R^\triangle_{n-1}(t, x(t)),&\text{if}\ \;\;k= n, \end{cases}$$ with rk (t) (1 ≤ k ≤ n) are positive rd-continuous functions. We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.

Oscillation criteria for second order nonlinear delay dynamic equations on time scales

By employing the generalized Riccati transformation, we establish some new oscillation criteria for second order delay dynamic equations on time scales. The obtained results essentially improve the well-know oscillation results for half-linear dynamic equations such as Kamenev-type and Philos-type oscillation criteria. Some interesting examples are given to illustrate the versatility of our results.

Nonoscillation of all solutions of a higher order nonlinear delay dynamic equation on time scales

The authors give sufficient conditions for all solutions of a higher order nonlinear delay dynamic equation on time scales to be nonoscillatory. The results cover both the superlinear and sublinear cases and, in the differential equations case, resolve a question that has been open for 30 years. In order to prove their results, the authors first prove a new Bihari type inequality for functions on time scales.

Oscillation of even order nonlinear delay dynamic equations on time scales

One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor’s Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality is sufficient for oscillation of even order dynamic equations on time scales. The arguments are based on Taylor monomials on time scales.

Oscillation theorems for certain third order nonlinear delay dynamic equations on time scales

In this paper, we establish some new oscillation criteria for the third order nonlinear delay dynamic equations

Oscillation criteria for third order nonlinear delay dynamic equations on time scales

By means of Riccati transformation technique, we establish some new oscillation criteria for the third-order nonlinear delay dynamic equations x 3 (t) + p(t)x(τ(t)) = 0 on a time scale T unbounded above, here γ > 0 is a quotient of odd positive integers with p real-valued positive rd-continuous function defined on T. Three examples are given to illustrate the main results.

Oscillation of third order nonlinear delay dynamic equations on time scales

It is the purpose of this paper to give oscillation criteria for the third order nonlinear delay dynamic equation ( a ( t ) { [ r ( t ) x Δ ( t ) ] Δ } γ ) Δ + f ( t , x ( τ ( t ) ) ) = 0 , on a time scale T , where γ ≥ 1 is the quotient of odd positive integers, a and r are positive r d -continuous functions on T , and the so-called delay function τ : T → T satisfies τ ( t ) ≤ t for t ∈ T and lim t → ∞ τ ( t ) = ∞ and f ∈ C ( T × R , R ) . Our results are new for third order delay dynamic equations and extend many known results for oscillation of third order dynamic equation. These results in the special cases when T = R and T = N involve and improve some oscillation results for third order delay differential and difference equations; when T = h N , T = q N 0 and T = N 2 our oscillation results are essentially new. Some examples are given to illustrate the main results.

Oscillation criteria for nonlinear damped dynamic equations on time scales

We present new oscillation criteria for the second order nonlinear damped delay dynamic equation ( r ( t ) ( x Δ ( t ) ) γ ) Δ + p ( t ) ( x Δ σ ( t ) ) γ + q ( t ) f x ( τ ( t ) ) = 0 . Our results generalize and improve some known results for oscillation of second order nonlinear delay dynamic equation. Our results are illustrated with examples.