A Global Convergence Result for a Higher Order Difference Equation
Let f(z1,…,zk)∈C(Ik,I) be a given function, where I is (bounded or unbounded) subinterval of ℝ, and k∈ℕ. Assume that f(y1,y2,…,yk)≥f(y2,…,yk,y1) if y1≥max{y2, …,yk}, f(y1,y2,…,yk)≤f(y2,…,yk,y1) if y1≤min{y2,…,yk}, and f is non- decreasing in the last variable zk. We then prove that every bounded s...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2007-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2007/91292 |
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| Summary: | Let f(z1,…,zk)∈C(Ik,I) be a given function, where I is (bounded or unbounded) subinterval of ℝ,
and k∈ℕ. Assume that f(y1,y2,…,yk)≥f(y2,…,yk,y1) if y1≥max{y2, …,yk},
f(y1,y2,…,yk)≤f(y2,…,yk,y1) if y1≤min{y2,…,yk}, and f is non- decreasing in the last variable zk.
We then prove that every bounded solution of an autonomous
difference equation of order k, namely, xn=f(xn−1,…,xn−k), n=0,1,2,…, with initial values x−k,…,x−1∈I, is convergent, and every unbounded solution tends either to +∞ or to −∞. |
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| ISSN: | 1026-0226 1607-887X |