Second-order Dynamic Equations Research Articles

Oscillation criteria for second order quasilinear neutral delay differential equations on time scales

In this paper, we consider the second-order quasilinear neutral dynamic equation ( r ( t ) | Z Δ ( t ) | α − 1 Z Δ ( t ) ) Δ + q ( t ) | x ( δ ( t ) ) | β − 1 x ( δ ( t ) ) = 0 , on a time scale T , where Z ( t ) = x ( t ) + p ( t ) x ( τ ( t ) ) , α , β > 0 are constants, and obtain oscillation criteria for the equation when β > α , β = α and β < α , respectively. Our results extend some known results when p ( t ) = 0 in the literature and contain the continuous and discrete cases as special cases.

Solvability of boundary value problems with Riemann-Stieltjes Δ-integral conditions for second-order dynamic equations on time scales at resonance

In this paper, by making use of the coincidence degree theory of Mawhin, the existence of the nontrivial solution for the boundary value problem with Riemann-Stieltjes Δ-integral conditions on time scales at resonance

Oscillation criteria for second-order half-linear delay dynamic equations with damping on time scales (Ⅱ)

By using the generalized Riccati transformation and the inequality technique,we establish four new oscillation criteria for certain second-order half-linear delay dynamic equations with damping on time scales.Our results extend and improve some known results,but also unify the oscillation of the second-order half-linear delay differential equations with damping and the second-order half-linear delay difference equations with damping.

Alternative solutions of inhomogeneous second-order linear dynamic equations on time scales

We exhibit an alternative method for solving inhomogeneous second-order linear ordinary dynamic equations on time scales, based on reduction of order rather than variation of parameters. Our form extends recent (and long-standing) analysis on to a new form for difference equations, quantum equations and arbitrary dynamic equations on time scales.

Oscillation of second-order quasi-linear neutral functional dynamic equations with distributed deviating arguments

In this paper, some sufficient conditions for the oscillation of second-order nonlinear neutral functional dynamic equation \[( r(t) ( [x(t) + p(t)x[\tau (t)]]^\Delta)^\gamma )^\Delta +\int^b_a q(t; \xi)x^\gamma [g(t; \xi)]\Delta\xi= 0; t \in \mathbb{T}\] are established. An example is given to illustrate an application of our results.

Existence of a spectral measure for second-order delta dynamic equations on semi-infinite time scale intervals

In this study, we prove existence of a spectral measure (or orthogonality measure) for second-order delta dynamic equations on semi-infinite time scale intervals. A Parseval equality and an expansion in eigenfunctions formula are established in terms of the spectral measure. The result obtained unifies the well-known results on existence of a spectral measure for Sturm–Liouville operators on the real semi-axis and for semi-infinite Jacobi matrices, and extends them to variety of numerous time scales which may, in particular, be fractals.

On The Bounded Oscillation of Certain Second-order Nonlinear Neutral Delay Dynamic Equations with Oscillating Coefficients

We investigate the bounded oscillation of the second-order nonlinear neutral delay dynamic equation with oscillating coefficients \[\Big(r(t)\Big|\big[x(t)+p(t)x(\tau(t))\big]^\Delta\Big|^{\alpha-1}\big[x(t)+p(t)x(\tau(t))\big]^\Delta\Big)^\Delta+q(t)|x(t)|^{\beta-1}x(t)=0 \] on an arbitrary time scale $\mathbb{T}$, where $p$ is an oscillating function defined on $\mathbb{T}$ and $\alpha,\beta>0$ are constants, and obtain several sufficient conditions for the oscillation of all bounded solutions of the equation when $\beta>\alpha, \beta=\alpha$ and $\beta<\alpha$, respectively. Our results extend and complement some known results where $p(t)\equiv 0$ and $\alpha,\beta$ are quotients of odd positive integers.

Nonlinear Oscillation and Asymptotic Behavior of Second-order Neutral Dynamic Equations on Time Scales

The paper is to study the oscillation and asymptotic behavior of certain second-order nonlinear neutral dynamic equations on time scales. We obtain some sufficient conditions which ensure that every solution of the equations oscillates or converges to zero. Our results improve and extend some existing results.

Asymptotic iteration technique for second-order dynamic equations on time scales

An asymptotic iteration method for solving second-order homogeneous linear dynamic equation of the form y△△ = a0(t)y△ + b0(t)y is introduced. Where a0(t) and b0(t) are \documentclass[12pt]{minimal}\begin{document}$C_{rd}^{m}( \mathbb {T} )$\end{document}Crdm(T) functions and the positive integer m depends on the differentiability of a0(t), b0(t) and iteration steps.

Oscillation of second-order dynamic equations

In this paper, we consider the second-order nonlinear dynamic equations on an isolated time scale . Our first goal is to establish a relationship between the oscillatory behaviour of these equations. Here we assume that τ : → . We also give two results about the behaviour of the linear form of the latter equation on a general time scale that is unbounded above. We use the Riccati transformation technique to obtain our results.

Oscillation Theorems for Second‐Order Half‐Linear Advanced Dynamic Equations on Time Scales

This paper is concerned with the oscillatory behavior of the second‐order half‐linear advanced dynamic equation on an arbitrary time scale 𝕋 with sup 𝕋 = ∞, where g(t) ≥ t and . Some sufficient conditions for oscillation of the studied equation are established. Our results not only improve and complement those results in the literature but also unify the oscillation of the second‐order half‐linear advanced differential equation and the second‐order half‐linear advanced difference equation. Three examples are included to illustrate the main results.

Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales

Our aim is to establish some sufficient conditions for the oscillation of the second-order quasilinear neutral functional dynamic equation $$ {\left( {p(t){{\left( {{{\left[ {y(t) + r(t)y\left( {\tau (t)} \right)} \right]}^\Delta }} \right)}^\gamma }} \right)^\Delta } + f\left( {t,y\left( {\delta (t)} \right)} \right. = 0,\quad t \in {\left[ {{t_0},\infty } \right)_\mathbb{T}}, $$ on a time scale $ \mathbb{T} $ , where ∣f(t, u)∣ ≥ q(t)∣u β ∣, r, p, and q are real-valued rd-continuous positive functions defined on $$ \mathbb{T} $$ , and γ and β > 0 are ratios of odd positive integers. Our results do not require that γ = β ≥ 1, p ∆(t) ≥ 0, $$ \int\limits_{{t_0}}^\infty {{{\left( {\frac{1}{{p(t)}}} \right)}^{\frac{1}{\gamma }}}\Delta t = \infty, } \quad \quad {\text{and}}\quad \quad \int\limits_{{t_0}}^\infty {{\delta^\beta }(s)q(s){{\left[ {1 - r\left( {\delta (s)} \right)} \right]}^\beta }\Delta s = \infty .} $$ Some examples are considered to illustrate the main results.

Necessary and Sufficient Conditions for Positive Solutions of Second‐Order Nonlinear Dynamic Equations on Time Scales

This paper is concerned with the existence of nonoscillatory solutions for the nonlinear dynamic equation on time scales. By making use of the generalized Riccati transformation technique, we establish some necessary and sufficient criteria to guarantee the existence. The last examples show that our results can be applied on the differential equations, the difference equations, and the q‐difference equations.

Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with Mixed Nonlinearities

This paper is concerned with some oscillation criteria for the second order neutral delay dynamic equations with mixed nonlinearities of the form where and with . Further the results obtained here generalize and complement to the results obtained by Han et al. (2010). Examples are provided to illustrate the results.

Forced oscillation of second-order superlinear dynamic equations on time scales

In this paper, by constructing a class of Philos type functions on time scales, we investigate the oscillation of the following second-order forced nonlinear dynamic equation x �� (t) − p(t)j x(q(t))j � 1 x(q(t)) = e(t); t 2 T where T is a time scale, p; e : T ! R are right dense continuous functions with p > 0, � > 1 is a constant, and q(t) = t or q(t) = � (t). Our results not only unify the oscillation of second-order forced differential equations and their discrete analogues, but also complement several results in the literature.

Multiple Positive Solutions for Second-Order p-Laplacian Dynamic Equations with Integral Boundary Conditions

We are concerned with the following second-order -Laplacian dynamic equations on time scales , , with integral boundary conditions , . By using Legget-Williams fixed point theorem, some criteria for the existence of at least three positive solutions are established. An example is presented to illustrate the main result.

Oscillation of second-order nonlinear delay dynamic equations on time scales

By using the generalized Riccati transformation and the inequality technique, we establish one new oscillation criterion for the second-order nonlinear delay dynamic equations on time scales. Our results not only extend and improve some known theorems, but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation on time scales.

Oscillation of second-order half-linear delay dynamic equations with damping on time scales

By using the generalized Riccati transformation and the inequality technique, we establish a few new oscillation criterions for certain second-order half-linear delay dynamic equations with damping on a time scale. Our results extend and improve some known results, but also unify the oscillation of the second-order half-linear delay differential equation with damping and the second-order half-linear delay difference equation with damping.

Interval criteria for second-order super-half-linear functional dynamic equations with delay and advance arguments

Interval oscillation criteria are established for second-order forced super half-linear dynamic equations on time scales containing both delay and advance arguments, where the potentials and forcing term are allowed to change sign. Four discrete examples are provided to illustrate the relevance of the results. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.

Oscillation of second-order nonlinear delay dynamic equations with damping on time scales

By using the generalized Riccati transformation and the inequality technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations with damping on a time scale. Our results extend and improve some known theorems, but also unify the oscillation of the second-order nonlinear delay differential equation with damping and the second-order nonlinear delay difference equation with damping.