George L. Karakostas

Stable steady state of some population models

Applying an analytical method and several limiting equations arguments, some sufficient conditions are provided for the existence of a unique positive equilibriumK for the delay differential equationx=−γx+D(xt), which is the general form of many population models. The results are concerned with the global attractivity, uniform stability, and uniform asymptotic stability ofK. Application of the results to some known population models, which shows the effectiveness of the methods applied here, is also presented.

The dynamics of some discrete population models

IN THIS paper we are interested in providing sufficient conditions guaranteeing the fact that all positive solutions of the discrete delay difference equation A II+1 = AA, + F(A,_,) (E) converge as n + 00. In (E), A is a constant real number with 0 0 sufficiently small. Thus, formally, equation (E) comes from x(t + h) x(t) h = -yx(t) + D(x(t r)) for small h. If we set ~,z(f) := x(ht), then the preceding equation for t = n and r/h = m becomes yh@ + 1) = (1 ?‘h)Y, (n) + hD(Y,(n m)), which is of the form (E), where A := 1 yh and F(u) := hD(u), u 2 0. Clearly h > 0 is small if and only if A E (0, 1). 1069 @h)

Sufficient conditions for the existence of nonnegative solutions of a nonlocal boundary value problem

Abstract In this paper, we provide sufficient conditions for the existence of nonnegative solutions of a nonlocal boundary value problem for a second-order ordinary differential equation. By applying Krasnoselskiis fixed-point theorem in a cone, first we prove the existence of solutions of an auxiliary BVP formulated by truncating the response function. Then the Arzela-Ascoli Theorem is used to take C 1 limits of sequences of such solutions.

Asymptotic 2–periodic difference equations with diagonally self–invertible responses

We provide sufficient conditions which guarantee that all positive solutions of a nonlinear difference equation of third order is asymptotically periodic 2. An estimate of the width of such solutions in terms of the initial values is also provided

On the Recursive Sequence

The boundedness, global attractivity, oscillatory and asymptotic periodicity of the nonnegative solutions of the difference equation in the title is investigated, where all the coefficients are nonnegative real numbers. The paper is motivated by an open problem proposed by Ladas [Open problems and conjectures, J. Differ. Equations Appl., 5 (1999), 211–215]. E-mail: sstevic@ptt.yu; sstevo@matf.bg.ac.yuThe boundedness, global attractivity, oscillatory and asymptotic periodicity of the nonnegative solutions of the difference equation in the title is investigated, where all the coefficients are nonnegative real numbers. The paper is motivated by an open problem proposed by Ladas [Open problems and conjectures, J. Differ. Equations Appl., 5 (1999), 211–215]. E-mail: sstevic@ptt.yu; sstevo@matf.bg.ac.yu

Asymptotic Behavior of the Solutions of the Difference Equation x_{n + 1} = x_{n}^{2} f(x n-1)

The difference equation where f is a nonincreasing real valued function, is discussed and asymptotic properties of the (nonnegative) solutions are provided.

Multiple positive solutions for a functional second-order boundary value problem

By using the Krasnoselskii fixed point theorem on cones in Banach spaces some existence results of positive solutions of a boundary value problem concerning a second-order functional differential equation are given.

Triple solutions for a nonlocal functional boundary value problem by leggett–williams theorem

The existence of triple solutions for a second-order nonlocal functional boundary value problem is proved by using a fixed-point theorem on cones in Banach spaces due to Leggett and Williams. The obtained results are new even in the ordinary case for three-point boundary value problems discussed quite recently in [Xiaoming He and Weigao Ge (2002). Triple solutions for second-order three-point boundary value problems. J. Math. Anal. Appl., 268, 256–265].

Slowly Varying Solutions of the Difference Equation x_{n + 1} = f(x_{n}, \ldots ,x_{n-k}) + g(n,x_{n},x_{n-1}, \ldots ,x_{n-k})

We provide conditions which guarantee that all bounded solutions of the difference equation are slowly varying in the sense that

ON A DIFFERENCE EQUATION WITH MIN-MAX RESPONSE

We investigate the global behavior of the (positive) solutions of the difference equation xn