A note on “Stability and periodicity in dynamic delay equations” [Comput. Math. Appl. 58 (2009) 264–273]

AbstractThe purpose of this note is twofold: First we highlight the importance of an implicit assumption in [Murat Adıvar, Youssef N. Raffoul, Stability and periodicity in dynamic delay equations, Computers and Mathematics with Applications 58 (2009) 264–272]. Second we emphasize one consequence of the bijectivity assumption which enables ruling out the commutativity condition δ∘σ=σ∘δ on the delay function

Boundedness and stability in nonlinear systems of differential equations using a modified variation of parameters formula

In this research we introduce a new variation of parameters for systems of linear and nonlinear ordinary differential equations. We use known mathematical methods and techniques including Gronwall’s inequality and fixed point theory to obtain boundedness on all solutions and stability results on the zero solution

Existence of periodic solutions in totally nonlinear delay dynamic equations

By means of a fixed point theorem we offer sufficient conditions for the existence of periodic solutions of totally nonlinear delay dynamic equations, where the solution maps a periodic time scale into another time scale

Boundedness And Stability In Nonlinear Discrete Systems With Nonlinear Perturbation

Abstract We consider the nonlinear Volterra discrete system with nonlinear perturbation Our goal is to use Lyapunov functionals to obtain conditions that guarantee all solutions of the above Volterra equation are bounded and derive conditions that ensure asymptotic stability and exponential stability, in the case x = 0 is a solution

Boundedness in Functional Dynamic Equations on Time Scales

Using nonnegative definite Lyapunov functionals, we prove general theorems for the boundedness of all solutions of a functional dynamic equation on time scales. We apply our obtained results to linear and nonlinear Volterra integro-dynamic equations on time scales by displaying suitable Lyapunov functionals

Positive periodic solutions in neutral nonlinear differential equations

We use Krasnoselskii’s fixed point theorem to show that the nonlinear neutral differential equation with delay d [x(t) − ax(t − τ)] = r(t)x(t) − f(t,x(t − τ)) dt has a positive periodic solution. An example will be provided as an application to our theorems

Analysis of Functional and Neutral Differential Equations via Lyapunov Functionals

We employ Lyapunov types functions and functionals and obtain sufficient conditions that guarantee the boundedness and the exponential decay of solutions, stability and exponential stability of the zero solution in nonlinear delay and neutral differential systems. The theory is illustrated with several examples. 2000 Mathematics Subject Classification: Primary 39A13, 39A23; Secondary 34K4

Necessary and sufficient conditions for stability of Volterra integro-dynamic equation on time scales

summary:In this research we establish necessary and sufficient conditions for the stability of the zero solution of scalar Volterra integro-dynamic equation on general time scales. Our approach is based on the construction of suitable Lyapunov functionals. We will compare our findings with known results and provides application to quantum calculus

General theorems for stability and boundedness for nonlinear functional discrete systems

AbstractWe consider the nonlinear functional discrete system x(n+1)=Gn,x(s);0⩽s⩽ndef=Gn,x(·) and obtain sufficient conditions for stability properties of the zero solution and for the boundedness of solutions