M. Federson

Impulsive retarded differential equations in Banach spaces via Bochner–Lebesgue and Henstock integrals

ẋ(t)=f(t; xt); xt0 = ; (1.1) where is a given continuous function from [ − r; 0]; r? 0, to a Banach space X and f is a continuous map from an open set ⊂ R × C([ − r; 0]; X ) to X . Given a continuous function x : [t0 − r; t0 + a] → X; a? 0; t0 ∈R, we follow [11] to deAne xt : [−r; 0] → X by xt( )= x(t+ ); ∈ [−r; 0]; t ∈ [t0; t0+a]. The initial value problem (1.1) is equivalent to the integral equation

Topological dynamics of retarded functional differential equations

Abstract We prove that a local flow can be constructed for a general class of nonautonomous retarded functional differential equations (RFDE). This is an extension to a result of Artstein (J. Differential Equations 23 (1977) 216) and fits in the classical theory of R. Miller and G. Sell. The main tool in this paper are generalized ordinary differential equations according to Kurzweil (Czech. Math. J. 7 (82) (1957) 418). In obtaining our results, we must prove the space of RFDEs can be embedded in a space of generalized ordinary differential equations. In opposition to the technical hypotheses of Oliva and Vorel (Bol. Soc. Mat. Mexicana 11 (1996) 40), this auxiliary result, as we present, is advantageous in the sense that our assumptions have an explanatory character. Applications based on topological dynamics techniques follow naturally from our results. As an illustration of this fact we show how to achieve in this setting a theorem on continuous dependence on initial data of solutions of RFDEs.

Oscillation by impulses for a second-order delay differential equation

We consider a certain second-order nonlinear delay differential equation and prove that the all solutions oscillate when proper impulse controls are imposed. An example is given.

Existence and impulsive stability for second order retarded differential equations

We consider certain impulsive second order delay differential equations and give conditions for the existence of solutions. Moreover we prove that the non-impulsive equations can be stabilized by the imposition of proper impulse controls generalizing recent results by Li and Weng. We also comment on some possible applications and give examples.

Oscillation for a second-order neutral differential equation with impulses

We consider a certain type of second-order neutral delay differential systems and we establish two results concerning the oscillation of solutions after the system undergoes controlled abrupt perturbations (called impulses). As a matter of fact, some particular non-impulsive cases of the system are oscillatory already. Thus, we are interested in finding adequate impulse controls under which our system remains oscillatory.

Continuous dependence for impulsive functional dynamic equations involving variable time scales

Using a known correspondence between the solutions of impulsive measure functional differential equations and the solutions of impulsive functional dynamic equations on time scales, we prove that the limit of solutions of impulsive functional dynamic equations over a convergent sequence of time scales converges to a solution of an impulsive functional dynamic equation over the limiting time scale.

The monotone convergence theorem for multidimensional abstract Kurzweil vector integrals

AbstractWe prove two versions of the Monotone Convergence Theorem for the vector integral of Kurzweil,

Linear Integral Equations with Discontinuous Kernels and the Representation of Operators on Regulated Functions on Time Scales

\smallint _R {\text{d}}\alpha {\text{(}}t{\text{)}}\;f(t)