A new analytical method for the linearization of dynamic equation on measure chains

In this paper, by introducing the concept of topological equivalence on measure chain, we investigate the relationship between the linear system xΔ=A(t)x and the nonlinear system xΔ=A(t)x+f(t,x). Some sufficient conditions are obtained to guarantee the existence of a equivalent function H(t,x) sending the (c,d)-quasibounded solutions of nonlinear system xΔ=A(t)x+f(t,x) onto those of linear system xΔ=A(t)x. Our results generalize the Palmer's linearization theorem in [K.J. Palmer, A generalization of Hartman's linearization theorem, J. Math. Anal. Appl. 41 (1973) 753-758] to dynamic equation measure chains. In the present paper, we give a new analytical method to study the topological equivalence problem on measure chains. As we will see, due to the completely different method to investigate the topological equivalence problem, we have a considerably different result from that in the pioneering work of Hilger [S. Hilger, Generalized theorem of Hartman-Grobman on measure chains, J. Aust. Math. Soc. Ser. A 60 (2) (1996) 157-191]. Moreover, we prove that equivalent function H(t,x) is also ω-periodic when the systems are ω-periodic. Hilger [S. Hilger, Generalized theorem of Hartman-Grobman on measure chains, J. Aust. Math. Soc. Ser. A 60 (2) (1996) 157-191] never considered this important property of the equivalent function H(t,x).     

On functions behaving like additive functions

Let 0 < 1. In the paper we consider the following inequality: |f(x + y) – f(x) – f(y)| min{|f(x + y)|, |f(x) + f(y)|}, wheref: R R. Solutions and continuous solutions of this inequality are investigated. They have similar properties as additive functions, e.g. if the solution is bounded above (below) on a set of positive inner Lebesgue measure then it is continuous. Some sufficient condition for this inequality is also given.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems

In this paper, we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We develop a comprehensive Floquet theory including Lyapunov transformations and their various stability preserving properties, a unified Floquet theorem which establishes a canonical Floquet decomposition on time scales in terms of the generalized exponential function, and use these results to study homogeneous as well as nonhomogeneous periodic problems. Furthermore, we explore the connection between Floquet multipliers and Floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales. Finally, we show this unified Floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated (but time varying) system?s poles in the complex plane. We include several examples to show the utility of this theory.     

Qualitative behavior of global solutions to some nonlinear fourth order differential equations

We study global solutions to a fourth order semilinear ordinary differential equation. We determine sufficient conditions on the nonlinearity that ensure global continuation of the solutions. Furthermore, we discuss their qualitative behaviors such as oscillations and boundedness.     

Oscillation of second-order half-linear delay dynamic equations with damping on time scales

By using the generalized Riccati transformation and the inequality technique, we establish a few new oscillation criterions for certain second-order half-linear delay dynamic equations with damping on a time scale. Our results extend and improve some known results, but also unify the oscillation of the second-order half-linear delay differential equation with damping and the second-order half-linear delay difference equation with damping.     

Interval criteria for oscillation of nonlinear second-order dynamic equations on time scales

Interval oscillation criteria are established for a second-order nonlinear dynamic equation on time scales by utilizing a generalized Riccati technique and an integral averaging technique. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.     

Asymptotic properties of solutions of certain third-order dynamic equations

In this paper, the well known oscillation criteria due to Hille and Nehari for second-order linear differential equations will be generalized and extended to the third-order nonlinear dynamic equation

(r₂(t)((r₁(t)x^Δ(t))^Δ)^γ)^Δ+q(t)f(x(t))=0     

Asymptotic preconditioning of linear homogeneous systems of differential equations

We consider the asymptotic behavior of solutions of a linear differential system x^′=A(t)x, where A is continuous on an interval ([a,∞). We are interested in the situation where the system may not have a desirable asymptotic property such as stability, strict stability, uniform stability, or linear asymptotic equilibrium, but its solutions can be written as x=Pu, where P is continuously differentiable on [a,∞) and u is a solution of a system u^′=B(t)u that has the property in question. In this case we say that P preconditions the given system for the property in question.     

Periodic solutions for dynamic equations on time scales

Massera type criteria are established for the existence of periodic solutions of linear and nonlinear dynamic equations on time scales. Some interesting properties of the exponential function on time scales are presented. Furthermore, a sufficient condition guaranteeing the boundedness of the solutions of the equation is presented.     

Limit behavior of solutions of ordinary linear differential equations

A classification of classes of equivalent linear differential equations with respect to -limit sets of their canonical representations is introduced. Some consequences of this classification with respect to the oscillatory behavior of solution spaces are presented.     

Two counterexamples related to the Kreiss matrix theorem

The modern version of the Kreiss matrix theorem states that A ⁿ esK (for alln0) ifA is ans ×s matrix satisfying a resolvent condition with respect to the unit disk with constantK. In this paper we show for any fixedK + 1 that the upper boundesK is sharp in the sense that a linear dependence on the dimensions is the best possible. An analogous result is obtained for the continuous version of the Kreiss matrix theorem, which states that exp(tA) esK (for allt0) ifA is ans ×s matrix satisfying a resolvent condition with respect to the left half plane with constantK.The research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences (K.N.A.W.) and was carried out at the Department of Mathematics and Computer Science, University of Leiden, The Netherlands.     

An asymptotic analysis of positive solutions of generalized Thomas-Fermi differential equations — The sub-half-linear case

The existence and the asymptotics behavior for the large value of the variable of the positive solutions of generalized Thomas-Fermi equation presented in this article are proved. It is assumed that coefficient q(t) belongs to the class of regularly varying functions in the sense of Karamata. Properties of these functions and the Schauder-Tychonoff fixed point theorem are the main tools for the proofs.     

On monotonic solutions of systems of nonlinear second order differential equations

Bounded, monotonic, and asymptotic properties of solutions for a class of second order nonlinear differential equations are discussed. Some necessary and sufficient conditions for boundedness of all solutions are established. Moreover, all solutions are classified into four disjoint subsets which are characterized in terms of the convergence and divergence of several integrals. The obtained results extend and improve many known results of various authors.     

Local variability of non-smooth functions

With the help of Müntz powers a formula of the Taylor type for non-smooth functions is presented. The approximation provides a local study for the variability of some curves which do not have a derivative. The approach includes the classical case but, at the same time, other non-analytical and non-differentiable mappings. In the first place, a Müntz curve representing the local variability of a function is defined. The coefficient and exponent of the model allow a numerical characterization of the relative extremes and differentiability of the map. The introduction of exponents of higher order provides a generalization of the Taylor’s formula including some cases of non-differentiability. In the last part, a series expansion of non necessarily integer powers representing the function is presented. Several properties of convergence, continuity, integrability and density are studied.     

Basic theory of fractional differential equations

In this paper, the basic theory for the initial value problem of fractional differential equations involving Riemann–Liouville differential operators is discussed employing the classical approach. The theory of inequalities, local existence, extremal solutions, comparison result and global existence of solutions are considered.     

On the dynamics of a higher order rational difference equations

The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the difference equation

$y_{n+1}=a{y}_n+b{y}_{n?t}+c{y}_{n?l}+\frac{d{y}_{n?k}+e{y}_{n?s}}{\alpha y_{n?k}+\beta y_{n?s}},n=0,1,\dots ,$

with positive parameters and non-negative initial conditions. Numerical examples to the difference equation are given to explain our results.     

The monotonic property and stability of solutions of fractional differential equations

In this paper we improve on the monotone property of Lemma 1.7.3 in Lakshmikantham et al. (2009) [5] for the case g(t,u)=λu with a nonnegative real number λ. We also investigate the Mittag-Leffler stability of solutions of fractional differential equations by using the fractional comparison principle.     

Exponential stability preservation in discrete-time analogues of artificial neural networks with distributed delays

This paper demonstrates that there is a discrete-time analogue which does not require any restriction on the size of the time-step in order to preserve the exponential stability of an artificial neural network with distributed delays. The analysis exploits an appropriate Lyapunov sequence and a discrete-time system of Halanay inequalities, and also either a Young inequality or a geometric-arithmetic mean inequality, to derive several sufficient conditions on the network parameters for the exponential stability of the analogue. The sufficiency conditions are independent of the time-step, and they correspond to those that establish the exponential stability of the continuous-time network.