Stability of L1 contractions

The notion of an exponential contraction is only one among many possible rates of contraction of a nonautonomous system, while for an autonomous system all contractions are exponential. We consider the notion of an L1 contraction that includes exponential contractions as a very particular case and that is naturally adapted to the variation-of-parameters formula. Both for discrete and continuous time, we show that under very general assumptions the notion of an L1 contraction persists under sufficiently small linear and nonlinear perturbations, also maintaining the type of stability. As a natural development, we establish a version of the Grobman–Hartman theorem for nonlinear perturbations of an L1 contraction.

Nonlinear Variation of Parameters Formula for Impulsive Differential Equations with Initial Time Difference and Application

This paper establishes variation of parameters formula for impulsive differential equations with initial time difference. As an application, one of the results is used to investigate stability properties of solutions.

Necessary and sufficient conditions for uniform stability of Volterra integro-dynamic equations using new resolvent equation

We consider the system of Volterra integro-dynamic equations and obtain necessary and sufficient conditions for the uniform stability of the zero solution employing the resolvent equation coupled with the variation of parameters formula. The resolvent equation that we use for the study of stability will have to be developed since it is unknown for time scales. At the end of the paper, we furnish an example in which we deploy an appropriate Lyapunov functional. In addition to generalization, the results of this paper provides improvements for its counterparts in integro-differential and integro-difference equations which are the most important particular cases of our equation.

Boundary value problems associated with first order rectangular: Kronecker product systems

In this paper first, we establish a general solution of the non-linear Kronecker product system (P(Q)(t)y'(t)+(R(S)(t)y(t) = f(t, y(t)) with the help of variation of parameters formula. Finally, we prove existence and uniqueness results for the non-linear Kronecker product system satisfying general boundary condition Uy = ?, by using Schauder-Tychonov's and Brouwer's fixed point theorems.

Kernel-resolvent relations for an integral equation

We consider a scalar integral equation where |G(t,z)| ≤ ϕ(t)|z|, C is convex, and . Related to this is the linear equation and the resolvent equation . A Liapunov functional is constructed which gives qualitative results about all three equations. We have two goals. First, we are interested in conditions under which properties of C are transferred into properties of the resolvent R which is used in the variation-of-parameters formula. We establish conditions on C and functions b so that as t→∞ and is in L2[0, ∞] implies that as t→∞ and is in L2[0, ∞]. Such results are fundamental in proving that the solution z satisfies z(t) →a(t) as t→∞ and that . This is in final form and no other version will be submitted.

PRINCIPAL MATRIX SOLUTIONS AND VARIATION OF PARAMETERS FOR VOLTERRA INTEGRO-DYNAMIC EQUATIONS ON TIME SCALES

We introduce the principal matrix solution Z(t, s) of the linear Volterra-type vector integro-dynamic equation and prove that it is the unique matrix solution of We also show that the solution of is unique and given by the variation of parameters formula

Stability analysis of nonlinear Lyapunov systems associated with an nth order system of matrix differential equations

This paper introduces the notion of Lipschitz stability for nonlinear nth order matrix Lyapunov differential systems and gives sufficient conditions for Lipschitz stability. We develop variation of parameters formula for the solution of the nonhomogeneous nonlinear nth order matrix Lyapunov differential system. We study observability and controllability of a special system of nth order nonlinear Lyapunov systems.