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Xn+1 = f (n, Xn-k), n = 0, 1, · · ..

Xn+1 = f (n, Xn-k), n = 0, 1, · · ..

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Theorem 2 ([12], p. 18) Let f : [a, b]k where k is a positive integer, and where [a, b] is an interval of real numbers and consider the following difference equation [a, b] be a continuous function, псе Xn+1 = = f (xn, · ·· , ¤n-k), n = 0, 1, · . . . (3) Suppose that f satisfies the following conditions: Yn Yn+1 = 1++p (8) Yn-m B .u. hendle the inen og uetion (0) The unigu
Transcribed Image Text:Theorem 2 ([12], p. 18) Let f : [a, b]k where k is a positive integer, and where [a, b] is an interval of real numbers and consider the following difference equation [a, b] be a continuous function, псе Xn+1 = = f (xn, · ·· , ¤n-k), n = 0, 1, · . . . (3) Suppose that f satisfies the following conditions: Yn Yn+1 = 1++p (8) Yn-m B .u. hendle the inen og uetion (0) The unigu
Theorem 9 Let 0 < p < . Then the equilibrium point y of Eq.(8) is globally asymptotically stable. Proof. Firstly, we consider the following function Yn f (u, v) = f (yn; Yn-m) = 1+p¬2 Yn-m The function f (u, v) is nondecreasing in u and nonincreasing in v. Let (m, M) is a solution of the system f(m, M) and M f (M, т). т — Hence we obtain that M ,M = 1+p; m2. m т 3 1+р- Therefore we have m = M. According to Theorem 2, every solution of Eq.(8) converges to x, as desired.
Transcribed Image Text:Theorem 9 Let 0 < p < . Then the equilibrium point y of Eq.(8) is globally asymptotically stable. Proof. Firstly, we consider the following function Yn f (u, v) = f (yn; Yn-m) = 1+p¬2 Yn-m The function f (u, v) is nondecreasing in u and nonincreasing in v. Let (m, M) is a solution of the system f(m, M) and M f (M, т). т — Hence we obtain that M ,M = 1+p; m2. m т 3 1+р- Therefore we have m = M. According to Theorem 2, every solution of Eq.(8) converges to x, as desired.
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