Abstract:
In this article we revisit the perturbation of exponential trichotomy of linear difference equation in Banach space by using a Perron-Lyapunov fixed point formulation for the perturbed evolution operator. This approach allows us to directly re-construct the perturbed semiflow without using shift spectrum arguments. These arguments are presented to the case of linear autonomous discrete time dynamical system. This result is then coupled to Howland semigroup procedure to obtain the persistence of exponential trichotomy for non-autonomous difference equations as well as for linear random difference equations in Banach spaces.

Abstract:
Consider the linear dynamic equation on time scales (1)
where , , is a rd-continuous function, T is a time scales. In this paper, we shall investigate some results for the exponential stability of the dynamic Equation (1) by combinating the first approximate method and the second method of Lyapunov.

Abstract:
We present a new approach to exponential functions on time scales and to timescale analogues of ordinary differential equations. We describe in detail the Cayley-exponential function and associated trigonometric and hyperbolic functions. We show that the Cayley-exponential is related to implicit midpoint and trapezoidal rules, similarly as delta and nabla exponential functions are related to Euler numerical schemes. Extending these results on any Pad\'e approximants, we obtain Pad\'e-exponential functions. Moreover, the exact exponential function on time scales is defined. Finally, we present applications of the Cayley-exponential function in the q-calculus and suggest a general approach to dynamic systems on Lie groups.

Abstract:
Making use of the generalized time scales exponential function, we give a new definition for the exponential stability of solutions for dynamic equations on time scales. Employing Lyapunov-type functions on time scales, we investigate the boundedness and the exponential stability of solutions to first-order dynamic equations on time scales, and some sufficient conditions are obtained. Some examples are given at the end of this paper.

Abstract:
The aim of this paper is to give several characterizations for nonuniform exponential trichotomy properties of linear difference equations in Banach spaces. Well-known results for exponential stability and exponential dichotomy are extended to the case of nonuniform exponential trichotomy.

Abstract:
The switched system consisting of a family of subsystems with delay is considered. In the time-delay situation, the multiple Lyapunov function approach is generalized to investigate the effect of switching and state-delay on stability. Using the Halanay inequality, the delay-dependent constraint condition on switching sequence is derived to guarantee the exponential stability. In the situation without time delay, the correspondent analysis result of switching effect on stability is a special case of our conclusions. Numerical examples are given to demonstrate the proposed approach.

Abstract:
The aim of this paper is to give necessary and sufficient conditions for the uniform exponential trichotomy property of nonlinear evolution operators in Banach spaces. The obtained results are generalizations for infinite-dimensional case of some well-known results of Elaydi and Hajek on exponential trichotomy of differential systems.

Abstract:
The paper emphasizes the properties of exponential dichotomy and exponential trichotomy for skew-evolution semiflows in Banach spaces, by means of evolution semiflows and evolution cocycles. The approach is from uniform point of view. Some characterizations which generalize classic results are also provided.

Abstract:
We are interested in an open problem concerning the integral characterizations of the uniform exponential trichotomy of skew-product flows. We introduce a new admissibility concept which relies on a double solvability of an associated integral equation and prove that this provides several interesting asymptotic properties. The main results will establish the connections between this new admissibility concept and the existence of the most general case of exponential trichotomy. We obtain for the first time necessary and sufficient characterizations for the uniform exponential trichotomy of skew-product flows in infinite-dimensional spaces, using integral equations. Our techniques also provide a nice link between the asymptotic methods in the theory of difference equations, the qualitative theory of dynamical systems in continuous time, and certain related control problems.

Abstract:
We investigate the exponential stability of the zero solution to a system of dynamic equations on time scales. We do this by defining appropriate Lyapunov-type functions and then formulate certain inequalities on these functions. Several examples are given.