Stability criteria for set dynamic equations on time scales

Very recently, a new theory known as set dynamic equations on time scales has been built. In this paper, notions of stability for the solutions of set dynamic equations on time scales, using Lyapunov-like functions are considered. Criteria for the equistability, equiasymptotic stability, uniform and uniform asymptotic stability are developed.     

Stability for first order delay dynamic equations on time scales

In this paper, we establish some sufficient conditions for the uniform stability and the uniformly asymptotical stability of the first order delay dynamic equation

where is a time scale, p(.) is rd-continuous and positive, the delay function . Our results unify the corresponding ones for differential and difference equations. To the best of our knowledge, this is the first time to discuss the asymptotical behavior of delay dynamic equations on time scales.     

Existence of multiple positive solutions for functional differential equations

In this paper, the existence of at least three positive solutions for the boundary value problem (BVP) of second-order functional differential equation with the form y″(t) + f(t, y^t) = 0, for t ε [0,1], y(t) -βy′(t) =η(t), for t ε [−τ,0], −γy(t) + Δy′(t) = ζ(t), for t ε [1, 1 + a], is studied. Moreover, we investigate the existence of at least three partially symmetric positive solutions for the above BVP with Δ = βγ.     

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 theorems for 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 criteria 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.     

Stability radius of implicit dynamic equations with constant coefficients on time scales

This paper deals with the stability radii of implicit dynamic equations on time scales when the structured perturbations act on both the coefficient of derivative and the right-hand side. Formulas of the stability radii are derived as a unification and generalization of some previous results. A special case where the real stability radius and the complex stability radius are equal is studied. Examples are derived to illustrate results.     

Existence and roughness of exponential dichotomies of linear dynamic equations on time scales

In this paper, we define the exponential dichotomy of linear dynamic equations on time scales, then we present perturbation theorems on the roughness of exponential dichotomy, and develop several explicit sufficient criteria for linear dynamic equations to have an exponential dichotomy. As applications of the criteria of exponential dichotomy, we derive some new sufficient conditions for the existence of periodic solutions of semi-linear dynamic equations and nonlinear dynamic equations on time scales.     

Multiple positive solutions for singular BVPs on the positive half-line

In this work, we are concerned with the existence of multiple positive solutions to a second-order nonlinear singular boundary value problem set on the positive half-line. We mainly use the Krasnozels’ki and Leggett–Williams fixed point theorems in cones to prove existence of one positive solution, two positive solutions and three positive solutions. The results complement, extend and correct some recent ones.     

The bounds on the solutions of certain two-dimensional delay dynamic systems on time scales

By means of a new linear Gronwall–Bellman inequality on time scales and an elementary inequality, the bounds on the solutions of a class of new nonlinear two-dimensional dynamic systems on time scales are obtained.     

Multiple solutions for higher-order difference equations

This paper establishes the existence of twin nonnegative solutions to (i) conjugate, and (ii) (n,p), higher-order discrete boundary value problems.     

Stability radii for positive linear time-invariant systems on time scales

We deal with dynamic equations on time scales, where we characterize the positivity of a system. Uniform exponential stability of a system is determined by the spectrum of its matrix. We investigate the corresponding stability radii with respect to structured perturbations and show that, for positive systems, the complex and the real stability radius coincide.     

Existence of triple positive solutions of two-point right focal boundary value problems on time scales

We consider the following boundary value problem, (−1)ⁿ⁻¹y^Δn(t)=(−1)^p+1F(t,y(σⁿ⁻¹(t))),t[a,b]∩T, y^Δn(a)=0,0≤i≤p−1, y^Δn(σ(b))=0,p≤i≤n−1,where n ≥ 2, 1 ≤ p ≤ n - 1 is fixed and T is a time scale. By applying fixed-point theorems for operators on a cone, existence criteria are developed for triple positive solutions of the boundary value problem. We also include examples to illustrate the usefulness of the results obtained.     

Some integral inequalities useful in the theory of certain partial dynamic equations on time scales

Our aim in this paper is to investigate some integral inequalities in two independent variables on time scales, which unify and extend some integral inequalities and their corresponding discrete analogues. The inequalities given here can be used as handy tools to study the properties of certain partial dynamic equations on time scales.