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Now, we can see that the function g(u, v, w) increasing in u, v and w. Let (m, M) be a solution of the system M = g(M, M, M) and m = g(m, m, m). Then from Eq. (1), we see that aM am М — ВМ +аМ + Вт + am + m = bM + c' bm + c thus b(la – B)M² – c(1 - a - B)M b(la – B)m? – c(1 - a - B)m аМ, am.

Now, we can see that the function g(u, v, w) increasing in u, v and w. Let (m, M) be a solution of the system M = g(M, M, M) and m = g(m, m, m). Then from Eq. (1), we see that aM am М — ВМ +аМ + Вт + am + m = bM + c' bm + c thus b(la – B)M² – c(1 - a - B)M b(la – B)m? – c(1 - a - B)m аМ, am.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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THEOREM 3.2. The equilibrium point T1 is a global attractor of difference equation (1) if B+a < 1. (6) Proof: Suppose that ( and n are real numbers and assume that g : 5,n]3 → 5,nl is a function defined by g(u, υ , w ) βυ αυ + aw bw+c Then ag(u, v, w) dv дg(и, v, w) = B, ас = a and g(u, v, w) (bw+c)² · du Now, we can see that the function g(u, v, w) increasing in u, v and w. Let (m, M) be a solution of the system M 9(М, М, М) and m — g(m, m, m). Then from Eq. (1), we see that aM am М — ВМ + аM + m = Bm + am + bM + c' bm + c thus b(la – B)M² – c(1 – a – B)M 6(la – B)m? – c(1- a - B)m аМ, am.
Transcribed Image Text:THEOREM 3.2. The equilibrium point T1 is a global attractor of difference equation (1) if B+a < 1. (6) Proof: Suppose that ( and n are real numbers and assume that g : 5,n]3 → 5,nl is a function defined by g(u, υ , w ) βυ αυ + aw bw+c Then ag(u, v, w) dv дg(и, v, w) = B, ас = a and g(u, v, w) (bw+c)² · du Now, we can see that the function g(u, v, w) increasing in u, v and w. Let (m, M) be a solution of the system M 9(М, М, М) and m — g(m, m, m). Then from Eq. (1), we see that aM am М — ВМ + аM + m = Bm + am + bM + c' bm + c thus b(la – B)M² – c(1 – a – B)M 6(la – B)m? – c(1- a - B)m аМ, am.
Our goal is to obtain some qualitative behavior of the positive solutions of the difference equation aIn-t In+1 = Bxn-it axn-k + n = 0, 1, (1) bxn-t+c' where the parameters B, a, a, b and c are positive real numbers and the initial conditions x-s, -s+1,..., x-1, xo are positive real numbers where s = max{l, k, t}.
Transcribed Image Text:Our goal is to obtain some qualitative behavior of the positive solutions of the difference equation aIn-t In+1 = Bxn-it axn-k + n = 0, 1, (1) bxn-t+c' where the parameters B, a, a, b and c are positive real numbers and the initial conditions x-s, -s+1,..., x-1, xo are positive real numbers where s = max{l, k, t}.
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