Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Theorem 7 Let 0 < p < 1. Then every solutions of Eq. (8) is bounded and persist such that 1- p" n- 1< Yn < 1- P where C1 = p (y0 +). Proof. Let {yn} be a positive solution of Eq.(8) and p > 0. Then, we have from Eq.(8) Yo 1+p 2 > 1, Y1 Y1 1+P2 Yi-m > 1. Y2 Thus we obtain by induction yn > 1 for n > 1. Now we consider the other side. We have from Eq.(8) Yn+1 = 1+p2 Yn <1+ pyn. (12) Yn-m where According to Theorem 3, there exist a sequence yn S Un, n = 0, 1, · .., {un} satisfies (13) Un+1 = 1+ pun, n > 1, such that us = Ys,Us+1 = Ys+1, 8 E {-m, -m + 1, ·..},n > s. Therefore the solution of the difference equation (13) is 1- pr + p2-C1 1- p (14) Un =

Theorem 7 Let 0 < p < 1. Then every solutions of Eq. (8) is bounded and persist such that 1- p" n- 1< Yn < 1- P where C1 = p (y0 +). Proof. Let {yn} be a positive solution of Eq.(8) and p > 0. Then, we have from Eq.(8) Yo 1+p 2 > 1, Y1 Y1 1+P2 Yi-m > 1. Y2 Thus we obtain by induction yn > 1 for n > 1. Now we consider the other side. We have from Eq.(8) Yn+1 = 1+p2 Yn <1+ pyn. (12) Yn-m where According to Theorem 3, there exist a sequence yn S Un, n = 0, 1, · .., {un} satisfies (13) Un+1 = 1+ pun, n > 1, such that us = Ys,Us+1 = Ys+1, 8 E {-m, -m + 1, ·..},n > s. Therefore the solution of the difference equation (13) is 1- pr + p2-C1 1- p (14) Un =

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
See similar textbooks
icon
Related questions
Question

Show me the steps of determine green and inf is here

Theorem 7 Let 0 < p < 1. Then every solutions of Eq.(8) is bounded and persist such that 1- pr 1< yn < +p"-!Cı +p?-C1 where C1 = p (v0 +). Proof. Let {yn} be a positive solution of Eq.(8) and p > 0. Then, we have from Eq.(8) Yo > 1, 1+Pym Y1 Y1 1+P2 Y1-m > 1. Y2 Thus we obtain by induction yn > 1 for n > 1. Now we consider the other side. We have from Eq.(8) Yn Yn+1 = 1+p <1+ pyn. (12) 2 Yn-m According to Theorem 3, there exist a sequence yn < Un, n = 0, 1, .., where {un} satisfies (13) Un+1 = 1+ pun, n > 1, such that us = Ys,Us+1 = Ys+1, 8 € {-m, -m + 1, ..:},n > s. Therefore the solution of the difference equation (13) is 1- p" Un + p?-C1 (14) 1 - p where C1 = p( Yo+ Moreover we obtain from (12) and (14), p- Yn+1 - Un+1 <p(yn – Un) where n > s and p E (0, 1). So, we get yn < Um, n > s as desired. I
Transcribed Image Text:Theorem 7 Let 0 < p < 1. Then every solutions of Eq.(8) is bounded and persist such that 1- pr 1< yn < +p"-!Cı +p?-C1 where C1 = p (v0 +). Proof. Let {yn} be a positive solution of Eq.(8) and p > 0. Then, we have from Eq.(8) Yo > 1, 1+Pym Y1 Y1 1+P2 Y1-m > 1. Y2 Thus we obtain by induction yn > 1 for n > 1. Now we consider the other side. We have from Eq.(8) Yn Yn+1 = 1+p <1+ pyn. (12) 2 Yn-m According to Theorem 3, there exist a sequence yn < Un, n = 0, 1, .., where {un} satisfies (13) Un+1 = 1+ pun, n > 1, such that us = Ys,Us+1 = Ys+1, 8 € {-m, -m + 1, ..:},n > s. Therefore the solution of the difference equation (13) is 1- p" Un + p?-C1 (14) 1 - p where C1 = p( Yo+ Moreover we obtain from (12) and (14), p- Yn+1 - Un+1 <p(yn – Un) where n > s and p E (0, 1). So, we get yn < Um, n > s as desired. I
Theorem 3 (See [8]) Let n e N, and g (n, u, v) be a decreasing function in u and v for any fixed n. Suppose that for n > no, the inequalities Yn+1 <g (n, Yn, Yn-1) < Un+1 hold. Then Yno-1 < Uno-1, Yno < Uno implies that Yn < Un, n 2 no. Consider the scalar kth-order linear difference equation x (n + k) + P1(n)x (n + k – 1) + .. + Pk(n)x (n) = 0, (4) %3| where k is a positive integer and pi : Z+ → C for i = 1,... , k. Assume that qi = lim p:(n), i = 1, , к, (5) exist in C. Consider the limiting equation of (4): x (n + k) + q1x (n + k – 1) +... + qkx (n) = 0. (6) Yn Yn+1 = 1+p (8) yn-m ...hone m. B hendle the dieTone uotion I0) The unigu
Transcribed Image Text:Theorem 3 (See [8]) Let n e N, and g (n, u, v) be a decreasing function in u and v for any fixed n. Suppose that for n > no, the inequalities Yn+1 <g (n, Yn, Yn-1) < Un+1 hold. Then Yno-1 < Uno-1, Yno < Uno implies that Yn < Un, n 2 no. Consider the scalar kth-order linear difference equation x (n + k) + P1(n)x (n + k – 1) + .. + Pk(n)x (n) = 0, (4) %3| where k is a positive integer and pi : Z+ → C for i = 1,... , k. Assume that qi = lim p:(n), i = 1, , к, (5) exist in C. Consider the limiting equation of (4): x (n + k) + q1x (n + k – 1) +... + qkx (n) = 0. (6) Yn Yn+1 = 1+p (8) yn-m ...hone m. B hendle the dieTone uotion I0) The unigu
Expert Solution
Step 1

We shall clear the doubts in next step. According to the company we shall answer first three subparts only. For other kindly post again by mentioning properly.

steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS