Sentence examples for using the exponential dichotomy from inspiring English sources

Using the exponential dichotomy techniques and a fixed point strategy with stochastic analysis theory, we establish the existence of p-mean piecewise pseudo almost periodic mild solutions for a class of impulsive nonautonomous partial stochastic evolution equations in Hilbert spaces.

For the model, we have given some sufficient conditions ensuring the existence and global exponential stability of almost periodic solutions by using the exponential dichotomy of linear dynamic equations on time scales, Banach's fixed point theorem and the theory of calculus on time scales.

More precisely, by using the exponential dichotomy techniques, stochastic analysis theory, and Leray-Schauder nonlinear alternative combined with the new composition theorem, we discussed the existence and exponential stability of p-mean piecewise pseudo almost periodic mild solutions for these equations.

Then, using this new composition theorem together with the Leray-Schauder nonlinear alternative and the exponential dichotomy techniques with stochastic analysis theory, we get new existence and exponential stability results.

We then apply our results to study the robustness of the exponential dichotomy of evolution families on a half-line under small perturbations.

This shows that the exponential dichotomy on time scales is more general and unifies the notions of exponential dichotomies on the continuous and discrete case.

Complete characterizations for the exponential dichotomy of cocycles are also obtained.

Finally, our results are applied at the study of the exponential dichotomy of -semigroups.

The theory of the exponential dichotomy and its applications are widely developed.

Several new sufficient conditions are obtained ensuring the existence and exponential stability of the almost periodic solution for this model based on inequality analysis technique and combing the exponential dichotomy with fixed point theorem.

Specifically, if and are two Banach function spaces with the property that either or, then the admissibility of the pair implies the existence of the exponential dichotomy.