Third and Fourth Order Piece-wise Defined Recursive Sequences

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1 International Mathematical Forum, Vol. 11, 016, no., HIKARI Ltd, Third and Fourth Order Piece-wise Defined Recursive Sequences Saleem Al-Ashhab Department of Mathematics Al-albayt University, Mafraq, Jordan Copyright c 015 Saleem Al-Ashhab. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We introduce a recursive sequence with a piecewise defined function as the right hand side. This function is discontinuous and includes nonlinear terms, i.e. one branch of this function contains the exponent three and four. We present results concerning the asymptotic behavior of the sequence. We specify conditions under which the sequence contains periodic subsequences. Mathematics Subject Classification: 11D5; 39A8; 39A99 Keywords: piecewise defined sequence, periodic sequnces, Cubic equations and Cardan s formula 1. Introduction Qena, Al-Ashhab and Guyker [6] considered the following difference equation: Q k = hq k 1, if Q k 1 v, Q k = hq k 1 j, if Q k 1 > v, for k = 1,,... where the initial value was set at Q 0 = 1. They proved the existence of periodic behavior for the sequence in some cases.

2 6 Saleem Al-Ashhab In [], [4] and [8] the work on this topic was extended and we find there more results concerning the nonlinear difference equation. We are going in this paper to study the cubic termss and terms of fourth order instead of quadratic terms. By doing this we will use some formulas related to cubic equations and equations of the fourth order. In [7] we find that the cubic equation y 3 + 3py + q = 0 possesses a solution of the form y = u + v, where u = 3 q + q + p 3, v = 3 q q + p 3 There exists a transformation from a general cubic equation to the previous form. Furhter, an equation of the form x 4 + bx 3 + cx + dx + e = 0 has the same roots of the equation x + b + A x + y + by d A where A = 8y + b 4c and y is any real root of the equation 8y 3 4cy + (bd 8e)y + e(4c b ) d = 0. We can take also the negative root as the value of A as is usually referred to Ferrari s method. But, it is enough for us here the described procedure.. Sequences with cubic exponent We generalize Qena s work in this section (see [5] and [6]). We replace the quadratic term by a term raised to an integer power. Later, we concentrate on the powers three and four. Definition 1 Let h, j, v and L 0 be positive real numbers. Let τ be an integer. We define the sequence L k as follows L k = hl τ k 1, if L k 1 v, L k = hl k 1 j, if L k 1 > v, for k = 1,,... In this section we will take τ = 3. We consider the following equation h 3 y 3 + (1 + h)(h n + y) + h = 0,

3 Third and fourth order piece-wise defined recursive sequences 63 where h, n 5. We use the notation G(i) = i j=0 τ j, T = 1 + h h 3, p = T 3, q = T hn + h. According to Cardan s Formula one of the solutions of the cubic equation is r = 3 q + R + q 3 R, R = q + p 3. We note that r < 0, r 3 + T r + T h n + h = 0. On the side the function f(x) = 3 x + a 3 x a, x 0 is decreasing for any real a > 0. Hence, we deduce that 3 R 3 R + q q < 3 q In particular we get r > 3 T hn + h... (1) Proposition 1: Let i > 0 be an integer. If we set n = G(i) + 1, j = h n + r, 3 4h n h v < h 0.75n 1, L 0 = 1, then we obtain the sequence L 0, L 1 = h,..., L i+5 = h,... Proof: According to definition and (1) L 0 = 1 v, L 1 = h v,..., L i = h G(i 1) 3 h n 1 v,

4 64 Saleem Al-Ashhab L i+1 = h G(i) = h n 1 > h 0.75n 1 > v, L i+ = h G(i)+1 j = r < 3 h n h v, L i+3 = hl 3 i+ = hr 3. According to definition r 3 = T r (T h n + h ) < T 3 T hn + h (T h n + h ). Hence, due to T < 0.5 we obtain r 3 < 3 T hn + h (T h n + h ). According to calculus the function f(x) = x 3 x is increasing for x > Since x x 3 4 x 3 > 0 for x > 4.5 we obtain for x > 4.5, 0 < α < 0.5 x + α 3 x + α > x 3 x > 4 x 3. Thus, by setting x = T h n, α = h we obtain L i+3 = hr 3 > 4 T 3 h 3n+4 > 4 h 3n > v, L i+4 = h( hr 3 ) j = h r 3 j = h T r + T h n+ + 1 h n r = r h + hn+3 + h n+ + 1 h n = h r + h 3 + h n+. h 3 h 3 Due to (1) we obtain h r > 3 4(T h n+6 + h 4 ) > 3 4(h n+3 + h n+4 + h 4 ) > 3 h n+4 > h n+. Since x x > x 1.9 for x > 10, we obtain by substituting x = h n+ h n+ + h r > h n+ h n+ > h 1.9(0.5n+1) = h 0.95n+1.9,

5 Third and fourth order piece-wise defined recursive sequences 65 L i+4 > h0.95n h 3 h 3 > h 0.95n 1.1 > h 0.75n 1 > v. Finally, according to definition and the properties of r we obtain L i+5 = h( h r 3 ) (1 + h)j = h 3 r 3 (1 + h)(h n + r) = h. We illustrate the proposition with the following example: We set h = i =. We have then n = G() + 1 = 14, 33 v < 74. The equation 8x 3 + 3( 14 + x) + = 0 has the solution Hence, we have j = = We obtain the sequence: L 1 =, L = 4 = 16, L 3 = 13 = 819 > v, L 4 = 14 j = < v, L 5 = ( ) 3 = 175 > v, L 6 = = 8184, L 7 = =, Sequences with fourth order terms In this section we will take τ = 4. We consider the following 4 th order equation h 3 (x h n ) 4 (1 + h)x h = 0. Using the substitution z = x h n, we transform it into z h h 3 z h + hn (1 + h) h 3 = 0 This equation will be solved symbolically exact by applying a procedure based on Cardan s formula: We move on to the cubic equation 8y h + hn (1 + h) y ( 1 + h ) = 0 h 3 h 3 One real solution of the cubic equation is y = 3 q + R + 3 q R

6 66 Saleem Al-Ashhab where p = h + hn (1 + h), q = 1 3h 3 16 (1 + h ), R = q + p 3. h 3 Now, we have to solve the quadratic equation where A = 8y. Its discriminant is x A x + (y h + 1 h 3 A ) = 0 = A h 3 4y + 4h h 3 A = y 3 h 1 h 3 y Now, > 0 iff (h y) 3 < h + 1 iff y 3 < 1 (1 + h T ) iff y < 3 h 3. On the other hand, we know that As we already know y = 3 ( T 4 ) + R + ( T 3 4 ) R. 3 a + x + 3 a x < 3 a, x > 0. Specifically, by taking a = ( T 4 ) we conclude then that is positive. We denote by r the greater root of the quadratic equation. In other words, we set r = A +. Now, r is a positive real solution of the 4 4th order equation, i.e. r h h 3 r h + hn (1 + h) h 3 = 0, or r 4 T r T h n 1 h = 0 We deduce that r 4 = T (r + h n ) + 1 h > T (r + hn ).

7 Third and fourth order piece-wise defined recursive sequences 67 But T > h. Hence, we conclude that r 4 > r + hn h > h n. Proposition : Let i > 0 be an integer. If we set n = G(i) + 1, j = h n + r, h G(i 1) v < h G(i), L 0 = 1, then we obtain the sequence L 0, L 1 = h,..., L i+5 = h,... Proof: As in the proof of propostion 1 we obtain L 0 = 1, L 1 = h, L = h 5,..., L i+1 = h n 1 > v, L i+ = hl i+1 j = r < v, L i+3 = hr 4 > h n 1 > v, L i+4 = hl i+3 j = h r 4 j = h r 4 h n r. But, we know that Thus, r 4 r h hn h = r h 3 + h + hn h 3. L i+4 = r + h + hn h > h n 1 > v, L i+5 = hl i+4 j = h 3 r 4 (1+h)(r+h n ) = h. We illustrate the proposition with the following example: We set h =, i = 1. In This case we have j = 66.38, since is a root for 8(64 x) 4 3x =. For any v < 3 we obtain the sequence: L 0 = 1, L 1 =, L = 3, L 3 =.38, L 4 = 50, L5 = > 3, L6 =. 4. Conclusion In this paper we proved two similar results. It is still open if we can generalize this result for any integer value of τ. Also, another open question is: what will happen when v lies outside the specified intervals? We expect to obtain a divergent behavior as it was the case in other papers. Also, what are the

8 68 Saleem Al-Ashhab conditions under wich the sequence will return after more steps to first term, like for example L i+6 = h? Acknowledgements. Saleem Al-Ashhab thanks the Al-albayt university for support, since this paper was prepared during the year of sabbatical leave offered by the university. References [1] S. Al-Ashhab, J. Guyker, Piecewise Defined Recursive Sequences with Applications in Matrix Theory, Journal of Mathematical and Computational Sciences, (01), no. 4, [] S. Al-Ashhab, R. Sabra, Nonlinear Piecewise-Defined and Cubic Difference Equations, 4th Annual International Conference on Computational Mathematics, Computational Geometry and Statistics CMCGS cmcgs15.30 [3] E. M. Elsayed, Dynamics of a Rational Recursive Sequence, International Journal of Difference Equations, 4 (009), no., [4] A. Hamadneh, On a Study of Some Difference Equations, Ms. C. thesis, AL al-bayt University, 013, The thesis appeared as: A Study of some Difference Equations including Bifurcation Analysis in the book ISBN by. lap-publishing.com [5] M. Qena, A Study on The Fixed and Periodic Points of Certain Discrete Dynamical Systems Equations, M. S. Thesis, AL al-bayt University, 01, The thesis appeared as New concepts in sequences, with the project number 51030, and ISBN by lap-publishing.com [6] M. Qena, S. Al-Ashhab, J. Guyker, Nonlinear Piecewise Defined Difference Equations, International Mathematical Forum, 7 (01), no. 1, [7] I. Bronstein, K. Semendjajew, Taschenbuch der Mathematik, Teubner verlagsgesellschaft, Leipzig, [8] Ramadan Sabra, Saleem Al-Ashhab, Bifurcations of Quadratic Piece-wise Defined Recursive Sequences, International Mathematical Forum, 10 (015), no. 6, [9] Ramadan Sabra, Saleem Al-Ashhab, Nonlinear Piecewise-Defined Differ-

9 Third and fourth order piece-wise defined recursive sequences 69 ence Equations with Reciprocal and Cubic Terms, GSTF Journal of Mathematics, Statistics and Operations Research JMSOR, 3 (015) Received: October 10, 015; Published: December, 015