Global behavior of nonlinear difference equations of higher order with applications
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Therefore the result follows. Theorem 3. The proof is by induction. Corollary 3. Then using Theorem 3. From 2. Then the last inequality implies that. This implies that. Hence we have. Now suppose that. Example 5. The author, therefore, acknowledge with thanks DSR technical and financial support. Agarwal, Difference Equations and Inequalities, vol. Kocic and G. Ladas, "Global attractivity in a second-order nonlinear difference equation," Journal of Mathematical Analysis and Applications, vol.
Kulenovic, G. Ladas, and N. Camouzis and G. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, vol. Kulenovic and G. Aloqeili, "Dynamics of a rational difference equation," Applied Mathematics and Computation, vol. Hamza and R. Khalaf-Allah, "Global behavior of a higher order difference equation," Journal of Mathematics and Statistics, vol. Al-Shabi and R.
Abo-Zeid, "Global asymptotic stability of a higher order difference equation," Applied Mathematical Sciences, vol. Khalaf-Allah, "Asymptotic behavior and periodic nature of two difference equations," Ukrainian Mathematical Journal, vol.
Abo-Zeid and C. However, users may print, download, or email articles for individual use. CC BY. Similar topics of scientific paper in Mathematics , author of scholarly article — R. Introduction and Preliminaries Although some difference equations look very simple, it is extremely difficult to understand thoroughly the global behaviors of their solutions.
An equilibrium point for 1. Consider the following linear difference equations:. Furthermore, from system 2 and 6 we obtain that.
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From 6 and 7 , it follows that. Let us consider fourth-dimensional discrete dynamical system of the following form:. To construct the corresponding linearized form of system 2 we consider the following transformation:. The following theorem shows the existence and uniqueness of positive equilibrium point of system 2. Then, we obtain that. Furthermore, assume that condition 16 is satisfied; then one has. The proof is therefore completed. Arguing as in [ 2 ], we have following result for global behavior of 2.
Then, it is easy to see that f x , y is increasing in x and decreasing in y. Moreover, g x , y is decreasing in x and increasing in y. Furthermore, from 31 and 32 , we obtain. Under the conditions of Theorems 7 and 9 the unique positive equilibrium of 2 is globally asymptotically stable.
Stability analysis of a system of second‐order difference equations
In this section, we will determine the rate of convergence of a solution that converges to the unique positive equilibrium point of the system 2. The following result gives the rate of convergence of solutions of a system of difference equations:. Suppose that condition 35 holds. To find the error terms, one has from the system 2.
Using Proposition 11 , one has following result. In this section, we study the behavior of unbounded solutions of system 2. Consider system 2.
Global Behavior of a Nonlinear Difference Equation with Applications
Furthermore, from system 2 it follows that. Moreover, from system 2 we have. Then, it is easy to see that solution of 51 is given by.
On the contrary, suppose that the system 2 has a distinctive prime period-two solutions. Then, from system 2 , one has. After some tedious calculations from 54 , we obtain. From 55 , it follows that.
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Hence, the proof is completed. Then, system 2 can be written as. Moreover, in Figure 1 , the plot of x n is shown in Figure 1 a , the plot of y n is shown in Figure 1 b , and an attractor of the system 59 is shown in Figure 1 c. Moreover, in Figure 2 , the plot of x n is shown in Figure 2 a , the plot of y n is shown in Figure 2 b , and an attractor of the system 60 is shown in Figure 2 c. Moreover, in Figure 3 , the plot of x n is shown in Figure 3 a , the plot of y n is shown in Figure 3 b , and an attractor of the system 61 is shown in Figure 3 c.
In literature, several articles are related to qualitative behavior of competitive system of planar rational difference equations [ 20 ]. It is very interesting mathematical problem to study the dynamics of competitive systems in higher dimension. This work is related to qualitative behavior of competitive system of second-order rational difference equations.
Behavior of a Competitive System of Second-Order Difference Equations
We have investigated the existence and uniqueness of positive steady state of system 2. Under certain parametric conditions the boundedness and persistence of positive solutions is proved. Moreover, we have shown that unique positive equilibrium point of system 2 is locally as well as globally asymptotically stable.
Furthermore, rate of convergence of positive solutions of 2 which converge to its unique positive equilibrium point is demonstrated. Finally, existence of unbounded solutions and periodicity nature of positive solutions of this competitive system are given. The authors thank the main editor and anonymous referees for their valuable comments and suggestions leading to improvement of this paper.
For the first author, this work was partially supported by the Higher Education Commission of Pakistan. The authors declare that they have no conflict of interests regarding the publication of this paper. National Center for Biotechnology Information , U. Journal List ScientificWorldJournal v.
Published online May Ibrahim , 2 , 3 and K. Khan 4.
Linear homogeneous equations
Author information Article notes Copyright and License information Disclaimer. Din: moc. Din et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Introduction Systems of nonlinear difference equations of higher order are of paramount importance in applications. Boundedness and Persistence The following theorem shows the boundedness and persistence of every positive solution of system 2. Proof — The proof follows by induction.
Lemma 10 — Under the conditions of Theorems 7 and 9 the unique positive equilibrium of 2 is globally asymptotically stable. Rate of Convergence In this section, we will determine the rate of convergence of a solution that converges to the unique positive equilibrium point of the system 2.
Proposition 11 Perron's Theorem, [ 19 ] — Suppose that condition 35 holds. Proposition 12 see [ 19 ] — Suppose that condition 35 holds. Existence of Unbounded Solutions of 2 In this section, we study the behavior of unbounded solutions of system 2.