Prolongation of solutions of measure differential equations and dynamic equations on time scales

In this paper, we prove the results on existence and uniqueness of the maximal solutions for measure differential equations, considering more general conditions on functions f and g by using the correspondence between the solutions of these equations and the solutions of generalized ODEs. Moreover, we prove these results for the dynamic equations on time scales, using the correspondence between the solutions of these last equations and the solutions of the measure differential equations.     

A note on continuous dependence of solutions of dynamic equations on time scales

We give a precise formulation and a proof as constructive as possible of the widely accepted claim that solutions of a dynamic equation depend continuously on the base time scale. Our approach to this problem is via Euler polygons which opens possibilities for development of numerical analysis of dynamic equations on time scales.     

A unified analysis of linear quaternion dynamic equations on time scales

Over the last years, considerable attention has been paid to the role of the quaternion differential equations (QDEs) which extend the ordinary differential equations. The theory of QDEs was recently well established and has wide applications in physics and life science. This paper establishes a systematic frame work for the theory of linear quaternion dynamic equations on time scales (QDETS), which can be applied to wave phenomena modeling, fluid dynamics and filter design. The algebraic structure of the solutions to the QDETS is actually a left- or right- module, not a linear vector space. On the non-commutativity of the quaternion algebra, many concepts and properties of the classical dynamic equations on time scales (DETS) can not be applied. They should be redefined accordingly. Using $q$-determinant, a novel definition of Wronskian is introduced under the framework of quaternions which is different from the standard one in DETS. Liouville formula for QDETS is also analyzed. Upon these, the solutions to the linear QDETS are established. The Putzer''s algorithms to evaluate the fundamental solution matrix for homogeneous QDETS are presented. Furthermore, the variation of constants formula to solve the nonhomogeneous QDETs is given. Some concrete examples are provided to illustrate the feasibility of the proposed algorithms.     

Almost automorphic solutions of dynamic equations on time scales

In the present work, we introduce the concept of almost automorphic functions on time scales and present the first results about their basic properties. Then, we study the nonautonomous dynamic equations on time scales given by x^Δ(t)=A(t)x(t)+f(t)$x^{\mathrm{\Delta }}(t)=A(t)x(t)+f(t)$ and x^Δ(t)=A(t)x(t)+g(t,x(t))$x^{\mathrm{\Delta }}(t)=A(t)x(t)+g(t,x(t))$, t∈T$t\in \mathbb{T}$ where T$\mathbb{T}$ is a special case of time scales that we define in this article. We prove a result ensuring the existence of an almost automorphic solution for both equations, assuming that the associated homogeneous equation of this system admits an exponential dichotomy. Also, assuming that the function g satisfies the global Lipschitz type condition, we prove the existence and uniqueness of an almost automorphic solution of the nonlinear dynamic equation on time scales. Further, we present some applications of our results for some new almost automorphic time scales. Finally, we present some interesting models in which our main results can be applied.     

Nonlinear oscillation of second-order dynamic equations on time scales

Interval oscillation criteria are established for a second-order nonlinear dynamic equation on time scales by utilizing a generalized Riccati technique and the Young inequality. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.     

Eigenvalue problems for second-order nonlinear dynamic equations on time scales

This paper is concerned with the existence and nonexistence of positive solutions of the second-order nonlinear dynamic equation uΔΔ(t)+λa(t)f(u(σ(t)))=0, t∈[0,1], satisfying either the conjugate boundary conditions u(0)=u(σ(1))=0 or the right focal boundary conditions u(0)=uΔ(σ(1))=0, where a and f are positive. We show that there exists a λ^∗>0 such that the above boundary value problem has at least two, one and no positive solutions for 0<λ<λ^∗, λ=λ^∗ and λ>λ^∗, respectively. Furthermore, by using the semiorder method on cones of the Banach space, we establish an existence and uniqueness criterion for positive solution of the problem. In particular, such a positive solution uλ(t) of the problem depends continuously on the parameter λ, i.e., uλ(t) is nondecreasing in λ, limλ→⁰⁺‖uλ‖=0 and limλ→+∞‖uλ‖=+∞.     

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

This paper is concerned with the oscillation of second-order nonlinear neutral dynamic equations of the form

(r(t)((y(t)+p(t)y(τ(t)))^Δ)^γ)^Δ+f(t,y(δ(t)))=0,     

Existence of nonoscillatory solutions to neutral dynamic equations on time scales

In this paper, we give an analogue of the Arzela-Ascoli theorem on time scales. Then, we establish the existence of nonoscillatory solutions to the neutral dynamic equation Δ[x(t)+p(t)x(g(t))]+f(t,x(h(t)))=0 on a time scale. To dwell upon the importance of our results, three interesting examples are also included.     

Kamenev-type oscillation criteria for second order nonlinear dynamic equations on time scales

The purpose of this paper to establish oscillation criteria for second order nonlinear dynamic equation

(r(t)(x^Δ(t))^γ)^Δ+f(t,x(g(t)))=0,     

A note on oscillation criteria for second-order neutral dynamic equations on isolated time scales

The principal goal of this paper is to amend oscillation results obtained in the recent paper by Saker and O’Regan (2011) [9].     

Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation

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 half-linear dynamic equations on time scales

This paper is concerned with oscillation of the second-order half-linear dynamic equation

(r(t)(xΔγ⁾Δ⁾+p(t)xγ(t)=0,     

Oscillation criteria for higher‐order nonlinear dynamic equations with Laplacians and a deviating argument on time scales

In this paper, we study the n th‐order nonlinear dynamic equation with Laplacians and a deviating argument on an above‐unbounded time scale, where n ?2, New oscillation criteria are established for the cases when n is even and odd and when α  > γ ,α  = γ , and α  < γ , respectively, with α  = α ₁?α _{n  ? 1}. Copyright © 2017 John Wiley & Sons, Ltd.     

Time scales in linear delayed differential equations

The aim of this paper is to apply and justify the so-called aggregation of variables method for reduction of a complex system of linear delayed differential equations with two time scales: slow and fast. The difference between these time scales makes a parameter ε>0 to appear in the formulation, being a mathematical problem of singular perturbations. The main result of this work consists of demonstrating that, under some hypotheses, the solution to the perturbed problem converges when ε→0 to the solution of an aggregated system whose construction is proposed.     

时间尺度上具阻尼项的二阶半线性时滞动力方程振动性的新结果

本文讨论一类具阻尼项的二阶半线性时滞动力方程解的振动性质, 利用广义Riccati 变换和不等式技巧, 在一定条件下, 建立了4 个新的振动准则, 其结果改进和推广了已知的一些结果.     

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.     

Exponential stability for set dynamic equations on time scales

A new theory known as set dynamic equations on time scales has been built. The criteria for the equistability, equiasymptotic stability, uniform and uniformly asymptotic stability were developed in Hong (2010) [1]. In this paper, we consider the exponential stability, exponentially asymptotic stability, uniform and uniformly exponentially asymptotic stability for the trivial solution of set dynamic equations on time scales by using Lyapunov-like functions.