Order Linear Dynamic Equations Research Articles

Oscillation condition for first order linear dynamic equations on time scales

In this paper, we deal with the first-order dynamic equations withnonmonotone arguments\begin{equation*}y^{\Delta }(t)+\underset{i=1}{\overset{m}{\sum }}p_{i}(t)y\left( \tau_{i}(t)\right) =0,\text{ }t\in \lbrack t_{0},\infty )_{\mathbb{T}}\end{equation*}where $p_{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},%\mathbb{R}^{+}\right) ,$ $\tau _{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{%T}},\mathbb{T}\right) $ and $\tau _{i}(t)\leq t,\ \lim_{t\rightarrow \infty}\tau _{i}(t)=\infty $ for $1\leq i\leq m$. Also, we present a new sufficient condition for theoscillation of delay dynamic equations on time scales. Finally, we give anexample illustrating the result.

First order linear fuzzy dynamic equations on time scales

In this paper, we study the concept of generalized differentiability for fuzzy-valued functions on time scales. Usingthe derivative of the product of two functions, we provide solutions to first order linear fuzzy dynamic equations. Wepresent some examples to illustrate our results.

Many Types of Stability of Abstract First and Second Order Linear Dynamic Equations on Time Scales

In this paper we investigate sufficient conditions for many types of stability of both of the abstract first order linear dynamic equations on time scales of the form and the second order linear dynamic equations of the form Where , the space of all bounded linear operators from a Banachspace 𝕏 into itself, and is rd–continuous from a time scale 𝕋to 𝕏. Some givenillustrative examples show the applicability of the main results. Mathematics Subject Classifications:26E70, 34N05, 34K20, 39A30.

The Ulam Stability of First Order Linear Dynamic Equations on Time Scales

In the present paper, the Ulam stability of the first order linear dynamic equation and its adjoint equation on a time scale is established by using the integrating factor method. Meantime, two examples are provided to illustrate our main results.

Hyers-Ulam stability of abstract second order linear dynamic equations on time scales

In this paper we investigate the Hyers-Ulam Stability of the abstract dynamic equation of the form

Oscillation of Dynamic Equation on Time Scales

In this paper,we consider the oscillation of second order linear dynamic equation on time scales, and obtain sufficient condition for oscillation of solutions.

Oscillation of Higher Order Linear Impulsive Dynamic Equations on Time Scales

In this paper, we will establish some oscillation criteria for the higher order linear dynamic equation on time scale in term of the coefficients and the graininess function. We illustrate our results with an example.

A Wong-type oscillation theorem for second order linear dynamic equations on time scales

We obtain Wong-type oscillation theorems for second order linear dynamic equations on a time scale. The results obtained extend and are motivated by oscillation results due to Wong [15]. As a particular application of our results, we show that the difference equation is oscillatory iff |b|>1.

Forced Oscillation of Second‐Order Half‐Linear Dynamic Equations on Time Scales

We will establish a new interval oscillation criterion for second‐order half‐linear dynamic equation (r(t)[xΔ(t)] α) Δ + p(t)xα(σ(t)) = f(t) on a time scale 𝕋 which is unbounded, which is a extension of the oscillation result for second order linear dynamic equation established by Erbe et al. (2008). As an application, we obtain a sufficient condition of oscillation of the second‐order half‐linear differential equation ([x′(t)] α) ′ + csintxα(t) = cos t, where α = p/q, p, q are odd positive integers.

Oscillation and nonoscillation of solutions of second order linear dynamic equations with integrable coefficients on time scales

We obtain Willett–Wong-type oscillation and nonoscillation theorems for second order linear dynamic equations with integrable coefficients on a time scale. The results obtained extend and are motivated by oscillation and nonoscillation results due to Willett [20] and Wong [21] for the second order linear differential equation. As applications of the new results obtained, we give the complete classification of oscillation and nonoscillation for the difference equations Δ 2 x ( n ) + b ( - 1 ) n t c x ( n + 1 ) = 0 and Δ 2 x ( n ) + a t c + 1 + b ( - 1 ) n t c x ( n + 1 ) = 0 for t ∈ N , a , b , c ∈ R . We also improve a nonoscillation result of Mingarelli [17] and extend an oscillation result of Del Medico and Kong [7].

A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales

This paper is mainly concerned with a class of optimal controlproblems of systems governed by the first order linear dynamicequations on time scales with quadratic cost functionals.Introducing the weak solutions of the first order linear dynamicequations and presenting the Arzela-Ascoli theorem on time scales,we prove the existence of solution to a class of linear quadraticoptimal control problems on time scales.

Wong's comparison theorem for second order linear dynamic equations on time scales

We obtain Wong-type comparison theorems for second order linear dynamic equations on a time scale. The results obtained extend and are motivated by Wong's comparison theorems. As a particular application of our results, we show that the difference equation Δ 2 x ( n ) + b ( − 1 ) n n c x ( n + 1 ) = 0 where c < 1 , b ≠ 0 , is oscillatory.

Delay dynamic equations with stability

We first give conditions which guarantee that every solution of a first order linear delay dynamic equation for isolated time scales vanishes at infinity. Several interesting examples are given. In the last half of the paper, we give conditions under which the trivial solution of a nonlinear delay dynamic equation is asymptotically stable, for arbitrary time scales.

Comparison theorems for linear dynamic equations on time scales

We obtain several comparison theorems for second order linear dynamic equations on a time scale. These results extend comparison theorems for the continuous case and provide some new results in the discrete case, as well as other more general situations.

First and second order linear dynamic equations on time scales

We consider first and second order linear dynamic equations on a time scale. Such equations contain as special cases differential equations, difference equationsq— difference equations, and others. Important properties of the exponential function for a time scale are presented, and we use them to derive solutions of first and second order linear dyamic equations with constant coefficients. Wronskians are used to study equations with non—constant coefficients. We consider the reduction of order method as well as the method of variation of constants for the nonhomogeneous case. Finally, we use the exponential function to present solutions of the Euler—Cauchy dynamic equation on a time scale.

Higher Order Dynamic Equations on Measure Chains: Wronskians, Disconjugacy, and Interpolating Families of Functions

This paper introduces generalized zeros and hence disconjugacy of nth order linear dynamic equations, which cover simultaneously as special cases (among others) both differential equations and difference equations. We also define Markov, Fekete, and Descartes interpolating systems of functions. The main result of this paper states that disconjugacy is equivalent to the existence of any of the above interpolating systems of solutions and that it is also equivalent to a certain factorization representation of the operator. The results in this paper unify the corresponding theories of disconjugacy for nth order linear ordinary differential equations and for nth order linear difference equations.