Difference Equations and Its Applications

special session in

Fourth International Conference on Dynamical Systems and Differential Equations

Wilmington, NC, USA, May 24-27, 2002


Presentation: This symposium is concerned with the dynamics of Difference Equations and Differential Equations on time scale. It covers topics such as Asymptotic behaviors of solutions, Existence of periodic solutions, Perturbed systems, Control theory , Oscillation theory, etc….

List of Invited Speakers:


The nabla matrix exponential function on a time scale

Abstract: We will define the nabla matrix exponential function

$e^{A(t,t_0)}$ on a time scale and derive several of its properties. Also some results concerning Riccati equations will be given.


Positive Periodic Solutions Of Nonlinear Functional Difference Equations


In this paper, we apply a cone theoretic fixed point theorem and obtain sufficient conditions for the existence of multiple positive periodic solutions to the nonlinear functional difference equations


x(n+1) = a(n)x(n)\overset{+}{-}\lambda h(n) f(x(n-\tau(n))).


where $a(n), h(n), \lambda, f(x)$ and $\tau$ are positive and periodic of period $T$.


Accurate Estimates for the Solutions of Difference Equations


We will be concerned with a non-autonomous perturbed linear discrete

dynamical system. A well known result of Perron, which dates back \ 1929(see Ortega(1973) and LaSalle(1976)), states that this kind of systems are asymptotically stable if the matrix $A$ of the linear part is stable, i.e., the spectral radius of $A$ is less than one, and the perturbation satisfies

an asymptotic property. We can see that this kind of results are purelylocal results which gives no information about the size of the region of asymptotic stability nor the norm of the solutions. In this report, estimates for the norms of the solutions and the size of the regions of

stability of non-autonomous perturbed linear difference equations are derived. The methodology is based on the ''freezing'' method and on the recent estimates for the powers of a constant matrix. Finally, we will illustrate our main results by considering partial difference equations

which model reaction and diffusion processes.



Under consideration is a class of even ordered linear differential equations with Sturm-Liouville boundary conditions. The differential equation is, in fact, a general dynamic equation containing delta-derivatives whose solution is defined on a measure chain. For a pair of eigenvalue problems for this dynamic equation, we first verify the existence of a smallest possible eigenvalue and then establish a comparison between the smallest eigenvalues of each eigenvalue problem.


On the periodic nature of the solutions of the reciprocal difference equation with maximum


We prove that every positive solution of the difference equation

x[n]=max{A/x[n-1],B/x[n-2],C/x[n-3]} is eventually periodic of (not

necessarily prime) period T, which is explicitely determined in terms of the

coefficients A,B and C.


Positivity and Discrete Models for the Lotka-Volterra Equations



The first nontrivial mathematical model for predator-prey interactions was the Lotka-Volterra equations [1]. The two coupled, first-order ODE's has biological relevant solutions

in the first quadrant, i.e., x(0) > 0 and y(0) > 0, lead to x(t) > 0 and y(0) > 0. Further, there is a single non-negative fixed-point, around which all solutions periodically oscillate.

We consider the application of the nonstandard finite-difference techniques of Mickens [2] to formulate corresponding discrete time models of these equations. In particular, enforcement

of the positivity condition is made by the use of nonlocal discrete representations for both the linear and quadratic terms appearing in the Lotka-Volterra differential equation.

Our studies indicate that one must be careful in carrying out this procedure. Both linear stability analysis and numerical work will be used to illustrate our results.


[1] J. D. Murray 1989 Mathematical Biology (Springer-Verlag, Berlin); section 3.1.

[2] R. E. Mickens 1994 Nonstandard Finite Difference Models of Differential Equations (World Scientific, Singapore).


Existence of Periodic Solutions of Nonlinear Discrete Second Order Equations


\documentclass[12pt]{article} \normalsize \renewcommand{\baselinestretch}{1.5} \begin{document} {\centerline{\large\bf Existence of Periodic Solutions} \centerline{\large\bf of Nonlinear Discrete Second Order Equations} \bigskip In this paper we study the existence of $T$-periodic solutions of equations of the form \[ \begin{array}{lll} x(t+2)+bx(t+1)+cx(t)+f(t, x(t))=h(t) \end{array} \] \noindent where $f$ is nonlinear, smooth, $f(t+T,x)=f(t,x)$ for each $(t,x)$ and $h(t+T)=h(t)$ for all $t$. Under the assumption of the existence of a $T$-periodic solution for a specific forcing term $h$, we aim to find conditions on the nonlinear term $f$ which will allow us to establish the existence of $T$-periodic solutions to a system of the form \[ \begin{array}{lll} x(t+2)+bx(t+1)+cx(t)+f(t, x(t))=h(t)+g(t) \end{array} \] \noindent where $g$ is $T$-periodic and ``relatively large." We formulate our problem as an operator equation in a space of $T$-periodic sequences and use a homotopy argument that eventually connects the existence of $T$-periodic solutions of the discrete equation to a differential equation on a sequence space. \end{document}


Exponential Stability In NonLinear Difference Equations


We employ non-negative definite Lyapunov functionals to obtain conditions that guarantee exponential stability and uniform exponential stability of the zero solution of the nonlinear discrete system

$$x(n+1) = f(n, x(n)), x(n_{0}) = x_{0}, for n \geq n_{0}.$$

The theory is illustrated with several examples.


Stability Properties of Linear Volterra Discrete Systems With Nonlinear Perturbation


We consider a Volterra discrete system with nonlinear perturbation

$$x(n+1)= A(n)x(n) + \sum^{n}_{s=0}B(n,s)x(s) + g(n,x(n))$$

and obtain necessary and sufficient conditions for stability properties of the zero solution employing the resolvent equation coupled with the variation of parameters formula.


Singular Conjugate Boundary Value Problems on a Time Scale


Let $\T_1$ be a time scale symmetric about $1/2$. Let $1/2 \in \T$ be dense and define $\T = \T_1 \bigcap [0,1]$. The conjugate nonlinear boundary value problem,


-u^{\D\D}(t) = a(t)f(u(t)), t \in \T\setminus\{0,1\}\\

u(0) = u(1) = 0,


where $a(t)$ is singular at $t = 1/2$ and $f$ satisfies certain

growth conditions, is shown to have infinitely many solutions

using Krasnosel'ski\u{\i}'s fixed point theorem.




Oscillatory Properties of Third Order Neutral Delay Differential Equations


The authors consider the third order neutral delay differential equation

$$ \left(a(t)\left(b(t)\left(y(t) + py(t-\tau)\right)'\right)'\right)'

+ q(t)f(y(t-\sigma)) = 0, \eqno(*) $$

where $a(t) > 0$, $b(t) > 0$, $q(t) \ge 0$, $0 \le p <1$, $\tau > 0$, and

$\sigma > 0$. Criteria for the oscillation of all solutions of ($\ast$) are

obtained. Examples illustrating the results are included.



Equations with partial derivatives and differential equations used for simulating acausal pulses in mathematical physics


Some phenomena in physics (such as the phenomenon of photonic echo) appears for an external observer as non-causal pulses suddenly emerging from an active medium (prepared by some other optical pulses). Such a pulse is very hard to be simulated without using physical quantities corresponding to the internal state of a great number of atoms. The only mathematical possibility of simulating such pulses without using a great number of variables consists in the use of test-functions. It is shown that such functions can be put in correspondence with acausal pulses in physics. This study shows that the wave-equation considered on the length interval (0, 1) (an open set), starting at the initial moment of time from null initial conditions, can possess as possible solution a test-function represented by a propagating direct wave coming from outside the length-interval. For explaining the reason why such acausal pulses do not appear in real circumstances some methods from statistical physics. are used. While at the zero moment of time all derivatives of the amplitude of the "real" string are equal to zero, it is shown that we may consider the zero moment of time as a bifurcation point


The Method of Quasilinearization and a three-point Boundary Value Problem



The method of quasilinearization generates a monotone iteration scheme whose iterates converge quadratically to a unique solution of the problem at hand. In this paper, we apply the method to two families of three-point boundary value problems for second order ordinary differential equations; linear boundary conditions and nonlinear boundary conditions are addressed independently. For linear boundary conditions, an appropriate Green's function is constructed. For nonlinear boundary conditions, we show that these nonlinearities can be addressed similarly to the nonlinearities in the differential equation.