Abstract. In this paper we study nonlinear control systems with multiple delays on controls and states. To obtain asymptotic stability, we impose Hölder-type assumptions on the perturbing function, and show a Gronwall-type inequality for difference equations with delay. We prove that a nonlinear control system can be stabilized if its linear control system can be stabilized. Some examples are i...

We derive explicit stability conditions for time-dependent difference equations with several delays in C n (the set of n complex vectors) and estimates for the size of the solutions. The growth rates obtained here are not necessarily decay rates. 1. Introduction. Stability of systems of difference equations with delays has been discussed by many authors, for example, see Gil' and Cheng [6], Zha...

In this paper, a class of impulsive difference equations with time-varying delays is considered. By establishing an impulsive delay difference inequality, some sufficient conditions ensuring all solutions either oscillate or tend to zero are obtained. An example is given to illustrate the feasibility of the obtained result. AMS Subject Classifications: 34K15, 34K10.

This technical note deals with a modified algebraic Riccati equation (MARE) and its corresponding inequality and difference equation, which arise in modified optimal control and filtering problems and are introduced into the cooperative control problems recently. The stabilizing property of the solution to MARE is presented. Then, the uniqueness is proved for the almost stabilizing and positive...

e study of dynamic equations on time scales goes back to the 1989 Ph.D. thesis of Hilger [1, 2] and is currently an area of mathematics receiving considerable attention [3– 7]. Although the basic aim of the theory of time scales is to unify the study of differential and difference equations in one and the same subject, it also extends these classical domains to hybrid and in-between cases. A g...

In this note our aim is to point out that certain inequalities for modified Bessel functions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equivalent to the corresponding Turán type inequalities for these functions. Moreover, we present some new Turán type inequalities for the aforementioned functions and we show that their product is decreasing as a funct...

In 1964 A. Garsia gave a stunningly brief proof of a useful maximal inequality of E. Hopf. The proof has become a textbook standard, but the inequality and its proof are widely regarded as mysterious. Here we suggest a straightforward first step analysis that may dispel some of the mystery. The development requires little more than the notion of a random variable, and, the inequality may be int...

Various generalizations of the Gronwall inequality 1, 2 are fundamental tools in the study of existence, uniqueness, boundedness, stability, invariant manifolds, and other qualitative properties of solutions of differential equations and integral equation. There are a lot of papers investigating them such as 3–8 . Along with the development of the theory of integral inequalities and the theory ...

In this article, existence of multiple positive T -periodic solutions for the first order delay difference equation of the form ∆x(n) = a(n)g(x(n))x(n) − λf(n, x(n− τ(n))) has been studied. Leggett-Williams multiple fixed point theorem has been employed to prove the results, which are established considering different cases on functions g and f .

We explore the relationship between inequality and demand structure in an endogenous growth model where consumers expand consumption along a hierarchy of needs and desires. The consumption hierarchy is captured by non-homothetic preferences implying that the shape of the demand curves for various goods depends on the distribution of income. This setting enables us to study a mechanism that so f...