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We investigate the global asymptotic stability of the solutions of X n+1 = βX n−l +γX n−k A+X n−k for n = 1, 2,. . ., where l and k are positive integers such that l = k. The parameters are positive real numbers and the initial conditions are arbitrary nonnegative real numbers. We find necessary and sufficient conditions for the global asymptotic s...
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We investigate the global asymptotic stability of the solutions of {equation presented}v where l and k are positive integers such that l ≠ k. The parameters are positive real numbers and the initial conditions are arbitrary nonnegative real numbers. We find necessary and sufficient conditions for the global asymptotic stability of the zero equilibr...
Citations
... The nonlinear difference equations of higher orders come quite naturally in modelling various systems we come across such as ecology, physiology, physics, engineering, economics, probability theory, genetics, psychology and resource management [4], [5], [6]. It is very interesting to investigate the behaviour of solutions of a higher-order rational difference equation and to discuss the local asymptotic stability of its equilibrium points [7], [8], [9]. Studying non-linear rational difference equations of higher orders is not though easy but very interesting with enriched phenomena [10]. ...
The asymptotic dynamics of the classes of rational difference equations (RDEs) of third order defined over the positive real-line as $$\displaystyle{x_{n+1}=\frac{x_{n}}{ax_n+bx_{n-1}+cx_{n-2}}}, \displaystyle{x_{n+1}=\frac{x_{n-1}}{ax_n+bx_{n-1}+cx_{n-2}}}, \displaystyle{x_{n+1}=\frac{x_{n-2}}{ax_n+bx_{n-1}+cx_{n-2}}}$$ and $$\displaystyle{x_{n+1}=\frac{ax_n+bx_{n-1}+cx_{n-2}}{x_{n}}}, \displaystyle{x_{n+1}=\frac{ax_n+bx_{n-1}+cx_{n-2}}{x_{n-1}}}, \displaystyle{x_{n+1}=\frac{ax_n+bx_{n-1}+cx_{n-2}}{x_{n-2}}}$$ is investigated computationally with theoretical discussions and examples. It is noted that all the parameters $a, b, c$ and the initial values $x_{-2}, x_{-1}$ and $x_0$ are all positive real numbers such that the denominator is always positive. Several periodic solutions with high periods of the RDEs as well as their inter-intra dynamical behaviours are studied.
... Other related work on rational difference equations see in refs. [30,31,32,36]. ...
In this paper, we have investigated a nonlinear rational difference equation of higher order. Our concentration is on invariant intervals, periodic
character, the character of semicycles and global asymptotic stability of all positive solutions of
xn+1=α+βxn−kA+Bxn−k,n=0,1,...,
where the parameters α, β and A, B and the initial conditions xk, ..., x1, x0 are positive real numbers k={1,2,3,...}. It is shown that the equilibrium point is globally asymptotically stable under the condition \ β≤A, and the unique positive solution is also globally asymptotically stable under the conditions β ≤ A ≤ β. It is shown that there does not exists any periodic solution for any positive parameters but it is shown computationally that there are periodic solutions with low as well as high periods.
