14,465 research outputs found

    Asymptotic behavior of a competitive system of linear fractional difference equations

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    We investigate the global asymptotic behavior of solutions of the system of difference equations xn+1 = (a+x n )/(b+y n ), yn+1 = (d+y n )/(e+x n ), n = 0,1,..., where the parameters a, b, d, and e are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. In certain range of parameters, we prove the existence of the global stable manifold of the unique positive equilibrium of this system which is the graph of an increasing curve. We show that the stable manifold of this system separates the positive quadrant of initial conditions into basins of attraction of two types of asymptotic behavior. In the case where a = d and b = e, we find an explicit equation for the stable manifold to be y = x

    Dynamics of a rational system of difference equations in the plane

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    We consider a rational system of first order difference equations in the plane with four parameters such that all fractions have a common denominator. We study, for the different values of the parameters, the global and local properties of the system. In particular, we discuss the boundedness and the asymptotic behavior of the solutions, the existence of periodic solutions and the stability of equilibria

    Asymptotic solutions of forced nonlinear second order differential equations and their extensions

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    Using a modified version of Schauder's fixed point theorem, measures of non-compactness and classical techniques, we provide new general results on the asymptotic behavior and the non-oscillation of second order scalar nonlinear differential equations on a half-axis. In addition, we extend the methods and present new similar results for integral equations and Volterra-Stieltjes integral equations, a framework whose benefits include the unification of second order difference and differential equations. In so doing, we enlarge the class of nonlinearities and in some cases remove the distinction between superlinear, sublinear, and linear differential equations that is normally found in the literature. An update of papers, past and present, in the theory of Volterra-Stieltjes integral equations is also presented

    On competitive discrete systems in the plane. I. Invariant Manifolds

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    Let TT be a C1C^{1} competitive map on a rectangular region R⊂R2R\subset \mathbb{R}^{2}. The main results of this paper give conditions which guarantee the existence of an invariant curve CC, which is the graph of a continuous increasing function, emanating from a fixed point zˉ\bar{z}. We show that CC is a subset of the basin of attraction of zˉ\bar{z} and that the set consisting of the endpoints of the curve CC in the interior of RR is forward invariant. The main results can be used to give an accurate picture of the basins of attraction for many competitive maps. We then apply the main results of this paper along with other techniques to determine a near complete picture of the qualitative behavior for the following two rational systems in the plane. xn+1=α1A1+yn,yn+1=γ2ynxn,n=0,1,...,x_{n+1}=\frac{\alpha_{1}}{A_{1}+y_{n}},\quad y_{n+1}=\frac{\gamma_{2}y_{n}}{x_{n}},\quad n=0,1,..., with α1,A1,γ2>0\alpha_1,A_{1},\gamma_{2}>0 and arbitrary nonnegative initial conditions so that the denominator is never zero. xn+1=α1A1+yn,yn+1=ynA2+xn,n=0,1,...,x_{n+1}=\frac{\alpha_{1}}{A_{1}+y_{n}},\quad y_{n+1}=\frac{y_{n}}{A_{2}+x_{n}},\quad n=0,1,..., with α1,A1,A2>0\alpha_1,A_{1},A_{2}>0 and arbitrary nonnegative initial conditions.Comment: arXiv admin note: text overlap with arXiv:0905.1772 by other author
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