Showing papers in "Journal of Difference Equations and Applications in 2002"
TL;DR: This paper gives an introduction to nonstandard finite difference methods useful for the construction of discrete models of differential equations when numerical solutions are required.
Abstract: This paper gives an introduction to nonstandard finite difference methods useful for the construction of discrete models of differential equations when numerical solutions are required. While the general rules for such schemes are not precisely known at the present time, several important criterion have been found. We provide an explanation of their significance and apply them to several model ordinary and partial differential equations. The paper ends with a discussion of several outstanding problems in this area and other related issues.
238 citations
TL;DR: In this paper, the existence of multiple positive solutions to the nonlinear second-order three-point boundary value problem on time scales was established, using the Leggett-Williams fixed-point theorem.
Abstract: In the first part of the paper, we establish the existence of multiple positive solutions to the nonlinear second-order three-point boundary value problem on time scales, u ▵▿ (t)+f(t,u(t))=0, u(0)=0, 𝛂u(𝛈)=u(T) for t∈[0,T]⊂╥, where ╥ is a time scale, 𝛂>0, η∈(0,p(T)⊂╥, and 𝛂η
140 citations
TL;DR: In this article, extensions of certain averaging techniques for the second-order scalar differential equation (r ( t ) x j ( t )) j + q ( t) x ( t ), on a time scale (measure chain) T were presented.
Abstract: We are interested in extensions of certain averaging techniques for the second-order scalar differential equation ( r ( t ) x j ( t )) j + q ( t ) x ( t )=0, on a time scale (measure chain) T . These results include some earlier criteria for the difference equations case.
117 citations
TL;DR: In this paper, the Beverton-Holt equation is modified for population dynamics, where the constant carrying capacity of a population is replaced by a periodic sequence of positive carrying capacities.
Abstract: has a unique positive equilibrium K and all solutions with x0 . 0 approach K as t !1: This equation (known as the Beverton–Holt equation) arises in applications to population dynamics, and in that context K is the “carrying capacity” and r is the “inherent growth rate”. A modification of this equation that arises in the study of populations living in a periodically (seasonally) fluctuating environment replaces the constant carrying capacity K by a periodic sequence Kt of positive carrying capacities.
110 citations
TL;DR: In this paper, it was shown that every nonzero solution x of (∗) satisfies the same asymptotic relation as the fundamental solutions described earlier by Perron, i.e., ϱ = lim supn→∞ |x(n)| is equal to the modulus of one of the roots of the characteristics equation χ k + q 1χ k−1++ qk=0.
Abstract: Consider the scalar kth order linear difference equation: x(n + k) + pi(n)x(n + k - 1) + … + pk(n)x(n) = 0 where the limits qi=limn→∞Pi(n) (i=1,…,k) are finite. In this paper, we confirm the conjecture formulated recently by Elaydi. Namely, every nonzero solution x of (∗) satisfies the same asymptotic relation as the fundamental solutions described earlier by Perron, ie., ϱ= lim supn→∞ |x(n)| is equal to the modulus of one of the roots of the characteristics equation χ k + q 1χ k−1+…+qk=0. This result is a consequence of a more general theorem concerning the Poincare difference system x(n+1)=[A+B(n]x(n), where A and B(n) (n=0,1,…) are square matrices such that ‖B(n)‖ →0 as n → ∞. As another corollary, we obtain a new limit relation for the solutions of (∗).
108 citations
TL;DR: In this article, the fixed point theorem of cone expansion and compression of norm type is generalized by replacing the norms with two functionals satisfying certain conditions to produce a fixed-point theorem of c.
Abstract: The fixed point theorem of cone expansion and compression of norm type is generalized by replacing the norms with two functionals satisfying certain conditions to produce a fixed point theorem of c...
62 citations
TL;DR: In this paper, the concepts of Riemann's delta and nabla integrals on time scales were introduced and investigated, and the main theorems of the integral calculus on time scale were proved.
Abstract: In this paper we introduce and investigate the concepts of Riemann's delta and nabla integrals on time scales. Main theorems of the integral calculus on time scales are proved.
61 citations
TL;DR: In this article, the authors show that the equation in the title with nonnegative parameters and nonnegative initial conditions exhibits a trichotomy character concerning periodicity, convergence, and boundedness which depends on whether the parameter n is equal, less, or greater than the sum of the parameters g and A.
Abstract: We show that the equation in the title with nonnegative parameters and nonnegative initial conditions exhibits a trichotomy character concerning periodicity, convergence, and boundedness which depends on whether the parameter n is equal, less, or greater than the sum of the parameters g and A .
58 citations
TL;DR: In this article, it was shown that in connected metric spaces n - cycles are not globally attracting (where n S 2 ) and applied this result to a two species discrete-time Lotka-Volterra competition model with stocking.
Abstract: In this article, we prove that in connected metric spaces n - cycles are not globally attracting (where n S 2 ). We apply this result to a two species discrete-time Lotka-Volterra competition model with stocking. In particular, we show that an n - cycle cannot be the ultimate life-history of evolution of all population sizes. This solves Yakubu's conjecture but the question on the structure of the boundary of the basins of attractions of the locally stable n - cycles is still open.
52 citations
TL;DR: In this article, a double fixed-point theorem is applied to obtain the existence of at least two positive solutions of a right focal boundary value problem for a second order impulsive dynamic equation on a time scale.
Abstract: A double fixed-point theorem is applied to obtain the existence of at least two positive solutions of a right focal boundary value problem for a second order impulsive dynamic equation on a time scale.
46 citations
TL;DR: Using Lyapunov type of function a comparison theorem for the fuzzy difference equation is obtained in terms of ordinary difference equations, which is used as a tool to study the stability results of the fuzzy different equations.
Abstract: The notion of fuzzy difference equation is introduced. Using Lyapunov type of function a comparison theorem for the fuzzy difference equation is obtained in terms of ordinary difference equations, which is used as a tool to study the stability results of the fuzzy difference equations.
TL;DR: In this article, the authors derived second order necessary and sufficient optimality conditions for a discrete optimal control problem with one variable endpoint and the other fixed, and with equality control constraints.
Abstract: In this work, we derive second order necessary and sufficient optimality conditions for a discrete optimal control problem with one variable endpoint and the other fixed, and with equality control constraints. In particular, the positivity of the second variation, which is a discrete quadratic functional with appropriate boundary conditions, is characterized in terms of the nonexistence of intervals conjugate to 0, the existence of a certain conjoined basis of the associated linear Hamiltonian difference system, or the existence of a symmetric solution to the implicit and explicit Riccati matrix equations. Some results require a certain minimal normality assumption, and are derived using the sensitivity analysis technique.
TL;DR: In this paper, the authors present a survey and refinement of recent results in the discrete optimal control theory, connecting the discrete control problems with the theory of conjugate intervals, Hamiltonian systems, and Riccati equations.
Abstract: In this paper, we present a survey and refinement of our recent results in the discrete optimal control theory. For a general nonlinear discrete optimal control problem (P) , second order necessary and sufficient optimality conditions are derived via the nonnegativity ( I S 0) and positivity ( I >0) of the discrete quadratic functional I corresponding to its second variation. Thus, we fill the gap in the discrete-time theory by connecting the discrete control problems with the theory of conjugate intervals, Hamiltonian systems, and Riccati equations. Necessary conditions for I S 0 are formulated in terms of the positivity of certain partial discrete quadratic functionals, the nonexistence of conjugate intervals, the existence of conjoined bases of the associated linear Hamiltonian system, and the existence of solutions to Riccati matrix equations. Natural strengthening of each of these conditions yields a characterization of the positivity of I and hence, sufficiency criteria for the original problem (P) ...
TL;DR: In this article, it was shown that the z -limit sets are closed intervals consisting of fixed points of f X only, under certain conditions these closed intervals contain exactly one point each.
Abstract: The paper investigates z -limit sets for discrete-time dynamical systems of the form x n +1 = f n +1 ( x n ), n S 0, with each f n mapping an interval I of R into itself. For autonomous systems, i.e. f n = f for all n , and f continuous on I =[ a , b ], the case that all z -limit sets consist of one point only is characterized by several equivalent conditions, one being that f has no 2-periodic points. The non-autonomous case assumes that the functions f n converge uniformly to a continuous function f X that has no 2-periodic points. It is shown that the z -limit sets are closed intervals consisting of fixed points of f X only. Under certain conditions these closed intervals contain exactly one point each. This allows a treatment of certain discrete-time dynamical systems in R n .
TL;DR: In this article, the asymptotic behavior of solutions of difference equations of the form x n + 1 = x n p f (x n m k 1, x n m m k 2,
, xn m k r ), n = 0,1,
where p is a positive constant and k 1,
, k r are (fixed) nonnegative integers.
Abstract: In this paper, we shall study the asymptotic behavior of solutions of difference equations of the form x n +1 = x n p f ( x n m k 1 , x n m k 2 ,
, x n m k r ), n =0,1,
, where p is a positive constant and k 1 ,
, k r are (fixed) nonnegative integers. In particular, permanence and global attractivity will be discussed.
TL;DR: In this article, the freezing method for ordinary differential systems is extended to difference systems, and new stability criteria and solution estimates for linear difference systems are derived by virtue of that method.
Abstract: The freezing method for ordinary differential systems is extended to difference systems. By virtue of that method, new stability criteria and solution estimates for linear difference systems are derived.
TL;DR: In this paper, the authors explore a basic partial differential equation, search for solutions, and find conditions which generate solutions of a given type. But their work with partial differential equations on time scales is still beginning.
Abstract: Work with partial differential equations on time scales is just beginning. In this paper we explore a basic partial differential equation, search for solutions, and find conditions which generate solutions of a given type.
TL;DR: In this article, the Darboux method is extended to discrete dynamical systems, providing a way of searching invariants for them and several examples of application, such as finding first integrals for polynomial ODEs.
Abstract: There is a method for searching first integrals for polynomial ordinary differential equations (usually called Darboux method) based on the knowledge of several of their invariant algebraic hypersurfaces. We extend this method to discrete dynamical systems, providing a way of searching invariants for them and we give several examples of application.
TL;DR: In this article, the authors consider positive solutions of the difference equation x n =max A x n m k, B x nm m, n =0,1,
, where A, B are any positive real numbers and k, m are any integers.
Abstract: We consider positive solutions of the following difference equation x n =max A x n m k , B x n m m , n =0,1,
, where A , B are any positive real numbers and k , m are any positive integers We prove that every positive solution is eventually periodic and determine the period in terms of the parameters A , B , k , and m
TL;DR: In this paper, the problem of asymptotic behavior between weighted bounded solutions of a system of homogeneous linear functional difference equations and its perturbation under non-classical dichotomic properties was studied.
Abstract: We study the problem of asymptotic behavior between weighted bounded solutions of a system of homogeneous linear functional difference equations and its perturbation under non-classical dichotomic properties and also we obtain some results about approximation. We apply our results to Volterra difference systems with infinite delay.
TL;DR: In this article, the existence and attractivity of a 2 k -periodic orbit with respect to the contractive map principle was investigated. But the existence of a stable periodic orbit in the case of a step function was not shown.
Abstract: We consider the discrete-time system x ( n )= g x ( n m 1)+ f ( y ( n m k )), y ( n )= g y ( n m 1)+ f ( x ( n m k )), n ] N describing the dynamic interaction of two identical neurons, where g ] (0,1) is the internal decay rate, f is the signal transmission function and k is the signal transmission delay. We construct explicitly an attractive 2 k -periodic orbit in the case where f is a step function (McCulloch-Pitts Model). For the general nonlinear signal transmission functions, we use a perturbation argument and sharp estimates and apply the contractive map principle to obtain the existence and attractivity of a 2 k -periodic orbit. This is contrast to the continuous case (a delay differential system) where no stable periodic orbit can occur due to the monotonicity of the associated semiflow.
TL;DR: In this article, the solvability of initial value problems and MPBVPs for linear implicit non-autonomous systems of difference equations was studied in the context of non-convex systems.
Abstract: This paper deals with the solvability of initial-value problems (IVPs) and multipoint boundary-value problems (MPBVPs) for linear implicit non-autonomous systems of difference equations.
TL;DR: In this paper, the difference equation x n + 1 = ~ i = 0 k f i x n m i p i, n = 0,1,2,
, where k is a positive integer.
Abstract: In this note we improve Theorem 2 in Ref. [3] , about the difference equation x n +1 = ~ i =0 k f i x n m i p i , n =0,1,2,
, where k is a positive integer, f i , p i ] (0, X ) for i =0,
, k , and the initial conditions x m k , x m k +1 ,
, x 0 are arbitrary positive numbers.
TL;DR: In this article, the authors consider a Volterra discrete system with nonlinear perturbation and obtain necessary and sufficient conditions for stability properties of the zero solution employing the resolvent equation coupled with the variation of parameters formula.
Abstract: We consider a Volterra discrete system with nonlinear perturbation x ( n +1)= A ( n ) x ( n )+ ~ s =0 n 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.
TL;DR: In this paper, the authors considered the problem of functional differential equations with almost periodic time dependence and gave conditions under which such equations with such time dependence will have unique almost periodic solutions.
Abstract: The study of functional differential equations with piecewise constant arguments usually results in a study of certain related difference equations. In this paper we consider certain neutral functional differential equations of this type and the associated difference equations. We give conditions under which such equations with almost periodic time dependence will have unique almost periodic solutions, and for certain autonomous cases, we obtain certain stability results and also conditions for chaotic behavior of solutions. We are particularly concerned with such equations which are partially discretized versions of non-forced Duffing equations.
TL;DR: In this article, the authors define R -semiconjugate maps as a natural means of relating a map F of R m to a mapping { of the interval via a link map H.
Abstract: R -semiconjugate maps are defined as a natural means of relating a map F of R m to a mapping { of the interval via a link map H . Invariants are seen to be special types of semiconjugate links where { is the identity. Basic relationships between the dynamical behaviors of { and F are established, and conditions under which a link map H is a Liapunov function are obtained. Examples and applications involving concepts from stability to chaos are discussed.
TL;DR: In this paper, the second order Sturm-Liouville difference equation (SLDE) problem is considered in which the parameter that is part of the SLDE also appears linearly in the boundary conditions.
Abstract: This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions....
TL;DR: In this paper, sufficient conditions for local controllability of time discrete dynamical systems into steady states were given for an emission reduction model, and the results were applied to an emission-reduction model.
Abstract: Sufficient conditions are given for local controllability of time discrete dynamical systems into steady states. The results are applied to an emission reduction model.
TL;DR: In this paper, a model of a strip of cardiac tissue consisting of a one-dimensional chain of cardiac units is derived in the form of a non-linear partial difference equation.
Abstract: A model of a strip of cardiac tissue consisting of a one-dimensional chain of cardiac units is derived in the form of a non-linear partial difference equation. Perturbation analysis is performed on this equation, and it is shown that regular perturbations are inadequate due to the appearance of secular terms. A singular perturbation procedure known as the method of multiple scales is shown to provide good agreement with numerical simulation except in the neighborhood of a singularity of the slow flow. The perturbation analysis is supplemented by a local numerical simulation near this singularity. The resulting analysis is shown to predict a "spatial bifurcation" phenomenon in which parts of the chain may be oscillating in period-2 motion while other parts may be oscillating in higher periodic motion or even chaotic motion.