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The equation axp-1Xn-k bxn-p±cXn-q° where the initial conditions are arbitrary positive real numbers, k, 1, p, q are non- negative integers and a, b, c are positive constants, is one of the difference equations whose solutions are associated with number sequences. Positive solutions of concrete Xn+1 = nE No, (1.1) Motivated by this line of investigations, here we show that the systems of differ- ence equations Xn+1 =f'(af (Pn-1)+bf (qn-2)), yn+1 =f'(af (rn-1)+bf (Sn-2)), (1.8) for n E No, where the sequences Pn, qn, In and Sn are some of the sequences xn and Yn, f : Df R is a "1 – 1" continuous function on its domain Df CR, the initial values x-j, y-j, je {0,1,2} are arbitrary real numbers and the parameters and a, b

The equation axp-1Xn-k bxn-p±cXn-q° where the initial conditions are arbitrary positive real numbers, k, 1, p, q are non- negative integers and a, b, c are positive constants, is one of the difference equations whose solutions are associated with number sequences. Positive solutions of concrete Xn+1 = nE No, (1.1) Motivated by this line of investigations, here we show that the systems of differ- ence equations Xn+1 =f'(af (Pn-1)+bf (qn-2)), yn+1 =f'(af (rn-1)+bf (Sn-2)), (1.8) for n E No, where the sequences Pn, qn, In and Sn are some of the sequences xn and Yn, f : Df R is a "1 – 1" continuous function on its domain Df CR, the initial values x-j, y-j, je {0,1,2} are arbitrary real numbers and the parameters and a, b

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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The equation axn-Xn-k Xn+1 = nE No, (1.1) bxn-p±cXn-q where the initial conditions are arbitrary positive real numbers, k, 1, p, q are non- negative integers and a, b, c are positive constants, is one of the difference equations whose solutions are associated with number sequences. Positive solutions of concrete Motivated by this line of investigations, here we show that the systems of differ- ence equations Xn+1 =f'(af (Pn-1)+bf (qn-2)), yn+1 =f'(af (rn-1)+bf (Sn-2)), (1.8) for n E No, where the sequences Pn, qn, Tn and Sn are some of the sequences xn and Yn, f : Df → R is a "1 – 1" continuous function on its domain Df CR, the initial values x-j, y-j, je {0,1,2} are arbitrary real numbers and the parameters and a, b
Transcribed Image Text:The equation axn-Xn-k Xn+1 = nE No, (1.1) bxn-p±cXn-q where the initial conditions are arbitrary positive real numbers, k, 1, p, q are non- negative integers and a, b, c are positive constants, is one of the difference equations whose solutions are associated with number sequences. Positive solutions of concrete Motivated by this line of investigations, here we show that the systems of differ- ence equations Xn+1 =f'(af (Pn-1)+bf (qn-2)), yn+1 =f'(af (rn-1)+bf (Sn-2)), (1.8) for n E No, where the sequences Pn, qn, Tn and Sn are some of the sequences xn and Yn, f : Df → R is a "1 – 1" continuous function on its domain Df CR, the initial values x-j, y-j, je {0,1,2} are arbitrary real numbers and the parameters and a, b
Lemma 4. Let {yn} be a positive solution of equation (1.1) with the corresponding sequence {z„} e So for n > N. Then: (i) (1– p)zn < yn < Zn for all n > N; (ii) {zn¢n} is increasing for all n > N. Proof. Assume that {yn} is a positive solution of equation (1.1) with the corres- ponding sequence {zn} E So. Then z, is positive, Zn 2 yn, and Yn = Zn – Pnyo(n) 2 (1– p)zn, n>N> no, so (i) is proved. It easy to see that z, E So implies lim b,(Az,)“ = 0; n00 otherwise we would eventually have Az, > 0 contradicting z, E So. Similarly, lim a„A(b,(Azn)“) = 0. A summation of equation (1.1) then yields 9szs+1 Zn+1 LIs. s=n s=n s=n Summing once more, we obtain as t=s s=n or Azn 2-Zn+1Qn. Hence, A(zn&n) = PnAzn + Zn+1Ao, > Zn+1(A0n – OnQn) =0 since {0,} is a solution of the difference equation (Ao, – Qnºn) = 0. Therefore, {z,9n} is increasing and this completes the proof of the lemma.
Transcribed Image Text:Lemma 4. Let {yn} be a positive solution of equation (1.1) with the corresponding sequence {z„} e So for n > N. Then: (i) (1– p)zn < yn < Zn for all n > N; (ii) {zn¢n} is increasing for all n > N. Proof. Assume that {yn} is a positive solution of equation (1.1) with the corres- ponding sequence {zn} E So. Then z, is positive, Zn 2 yn, and Yn = Zn – Pnyo(n) 2 (1– p)zn, n>N> no, so (i) is proved. It easy to see that z, E So implies lim b,(Az,)“ = 0; n00 otherwise we would eventually have Az, > 0 contradicting z, E So. Similarly, lim a„A(b,(Azn)“) = 0. A summation of equation (1.1) then yields 9szs+1 Zn+1 LIs. s=n s=n s=n Summing once more, we obtain as t=s s=n or Azn 2-Zn+1Qn. Hence, A(zn&n) = PnAzn + Zn+1Ao, > Zn+1(A0n – OnQn) =0 since {0,} is a solution of the difference equation (Ao, – Qnºn) = 0. Therefore, {z,9n} is increasing and this completes the proof of the lemma.
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