Hybrid systems with time scales and impulses

In this paper, we describe hybrid systems introducing special time scales and impulsive effects so as to provide a broader perspective for hybrid systems.     

Uniform eventual Lipschitz stability of impulsive systems on time scales

In this paper, we discuss the uniform eventual Lipschitz stability of impulsive system on time scales. By using comparison method, Lyapunov function and analysis technology, some criteria of such stability for system with impulses on time scales are obtained. An example is presented to illustrate the efficiency of proposed results.     

First order dynamic inclusions on time scales

In this paper, we study existence of solutions of first order dynamic inclusions on time scales with general boundary conditions. Both the ∇-derivative and Δ-derivative cases are considered. Examples are presented to illustrate that the Δ-derivative case needs more restrictive assumptions.     

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.     

Existence of solutions to boundary value problems for dynamic systems on time scales

Here, we investigate systems of boundary value problems for dynamic equations on time scales. Using a generalized relationship between the boundary conditions and a certain subset of the solution space, the existence of solutions is established through topological arguments. The main tools used are Leray-Schauder and Brouwer degree theory.     

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‖=+∞.     

Triple positive solutions for a class of -point dynamic equations on time scales with -Laplacian

This paper deals with a class of second-order nonlinear m-point dynamic equation on time scales with one-dimensional p-Laplacian. Using a fixed point theorem for operators on a cone, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The interesting point is that the nonlinear term f is involved with the first-order delta derivative explicitly. Meanwhile, an example is worked out to demonstrate the main results.     

Some new oscillation results for a nonlinear dynamic system on time scales

We study the oscillation of a system of two first order nonlinear equations on time scales. This form includes the classical Emden-Fowler differential equation and many of its extensions. We generalize some well-known results of Atkinson, Bohner, Erbe, Peterson and others. We illustrate the results by several examples, including a superlinear Emden-Fowler dynamic system.     

Frequency measures on time scales with applications

The concept of frequency measures of subsets of a time scale is introduced and the relevant properties are discussed. Then, frequent oscillation is defined to strengthen the classical concept of oscillation. Applications are shown by deriving oscillation criteria for first-order dynamic equations on time scales.     

Maximal regularity of evolution equations on discrete time scales

We consider the question of Lp-maximal regularity for inhomogeneous Cauchy problems in Banach spaces using operator-valued Fourier multipliers. This follows results by L. Weis in the continuous time setting and by S. Blunck for discrete time evolution equations. We generalize the later result to the case of some discrete time scales (discrete problems with nonconstant step size). First we introduce an adequate evolution family of operators to consider the general problem. Then we consider the case where the step size is a periodic sequence by rewriting the problem on a product space and using operator matrix valued Fourier multipliers. Finally we give a perturbation result allowing to consider a wider class of step sizes.     

The logarithm on time scales

We briefly present the well-studied exponential function on a time scale and pose the problem of finding an appropriate logarithm function on a time scale.     

Some new dynamic inequalities on time scales

In this paper, based on some known dynamic inequalities, we investigate certain new dynamic inequalities on time scales, which provide explicit bounds on unknown functions. Our results unify and extend some continuous inequalities and their corresponding discrete analogues.     

Noether's theorem on time scales

We show that for any variational symmetry of the problem of the calculus of variations on time scales there exists a conserved quantity along the respective Euler-Lagrange extremals.     

On existence of optimal control governed by a class of the first-order linear dynamic systems on time scales

In this paper, by using the approximation of classical solution, we introduce the definition of solution and prove the existence and uniqueness of solutions of the first-order linear dynamic systems on time scales. Existence of Lagrange optimal control problem governed by the first-order linear dynamic systems on time scales is also presented. For illustration, some examples of optimal control problems on time scales are also discussed.     

On inequalities of Lyapunov for linear Hamiltonian systems on time scales

In this paper, we establish several new Lyapunov type inequalities for linear Hamiltonian systems on an arbitrary time scale T when the end-points are not necessarily usual zeroes, but rather, generalized zeroes, which generalize and improve all related existing ones including the continuous and discrete cases.     

Some nonlinear dynamic inequalities on time scales

The aim of this paper is to investigate some nonlinear dynamic inequalities on time scales, which provide explicit bounds on unknown functions. The inequalities given here unify and extend some inequalities in (B G Pachpatte, On some new inequalities related to a certain inequality arising in the theory of differential equation, J. Math. Anal. Appl. 251 (2000) 736–751).     

Line integrals and Green's formula on time scales

In this paper we study curves parametrized by a time scale parameter, introduce line delta and nabla integrals along time scale curves, and obtain an analog of Green's formula in the time scale setting.     

Instability results for slowly time varying linear dynamic systems on time scales

We develop eigenvalue criteria under which the solutions of a “slowly” time varying linear dynamic system of the form xΔ(t)=A(t)x(t) are unstable.