New Results On Composition-Delay Equations With Asymptotically Periodic Solutions

Richard T. Guy, Kenneth S. Berenhaut

Dept. of Mathematics, Wake Forest University, Winston-Salem, NC Abstract The purpose of this thesis is to convey several new results in the field of piecewise difference equations, paying particular attention to higher order equations with asymptotically periodic solutions. A study of solutions to the equation

y_{n} = min{y_{n−k_{1}} − y_{n−m_{1}}, y_{n−k_{2}} − y_{n−m_{2}}}

is presented. Related results are then obtained for a class of difference equations satisfying certain symmetry and monotonicity conditions. In particular we consider equations of the form

y_{n} = min{f(y_{n−k_{1}}, y_{n−m_{1}}), f(y_{n−k_{2}}, y_{n−m_{2}})},

where f(u, v) = h(u, v)/v for h symmetric in u and v, and f satisfies monotonicity conditions. The results are then extended to the form

y_{n} = min{f(y_{n−k_{1}}, y_{n−m_{1}}), f(y_{n−k_{2}}, y_{n−m_{2}}), …, f(y_{n−k_{L}}, y_{n−m_{L}})}.

Introduction

Definition 1 (Difference Equation) An n^{th} order difference equation is an equation of the form

y_{i} = f(y_{i−1}, …, y_{i−n})  (1)

where f is some function (often continuous) mapping Y^{n} → Y.

Definition 2 A set {y_{i}} is a solution to a difference equation given by f if (??) holds.

• Integer initial values

Examples:

• Linear equations: y_{n} = y_{n−1} + y_{n−2} (Fibonaccci, Lucas numbers)

• Collatz Conjecture:

a_{k} = \{ \begin{matrix}\frac{a_{k−1}}{2} & if a_{k−1} is even \\ 3a_{k−1} + 1 & if a_{k−1} is odd \end{matrix}  (2)

** Goal: Develop tools to understand behavior of non-linear, non-differentiable difference equations. **

The Equation y_{n} = min{y_{n−k_{1}} − y_{n−m_{1}}, y_{n−k_{2}} − y_{n−m_{2}}}

Theorem 1 Suppose k, m, j ≥ 1 and consider solutions to the equation y_{n} = y_{n−k} − y_{n−m}, n ≥ 0.

1.There exists a prime period 6j solution if and only if (k, m) ∈ {(j, 2j), (5j, 4j)} mod 6j. 2.If gcd(k, m) = 1 then all solutions are eventually periodic if and only if (k, m) = (1, 2). Furthermore, if (k, m) = (1, 2) then all

non-trivial solutions must be strictly periodic with period six. 3.All solutions are either asymptotically periodic or satisfy

lim sup_{n→∞} \frac{|y_{n}|}{n} = ∞.  (3)

Theorem 2 Suppose k_{1} = m_{1} + k_{2} and gcd(k_{1}, m_{1}, k_{2}, m_{2}) = 1.

1.There exists a non-trivial solution which has period p = m_{1} + m_{2}. 2.All solutions are asymptotically periodic with period p = m_{1} + m_{2} if and only if k_{2}|m_{2}. 3.If k_{2} ∤ m_{2} then there exists a non-trivial period k_{2} solution. 4.If gcd(k_{2}, m_{1} + m_{2}) > 1, then there exists an unbounded integer solution which satisfies

lim sup_{n→∞} \frac{|y_{n}|}{n} = \frac{1}{lcm(k_{2}, m_{1} + m_{2})}.  (4)

5.Any unbounded integer solution to the equation must satisfy

lim sup_{n→∞} \frac{|y_{n}|}{n} < ∞.  (5)

Other Results and Conjectures: The “absolute value case” y_{n} = min{y_{n−k_{1}} − y_{n−m_{2}}, y_{k_{2}} − y_{m_{2}}} was completely solved.

Conjecture 1 Suppose gcd{k_{1}, k_{2}, m_{1}, m_{2}} = 1 and k_{1} + k_{2} = m_{1} = m_{2} = m. Then, all solutions to y_{n} = min{y_{n−k_{1}} − y_{n−m}, y_{n−k_{2}} − y_{n−m}} are strictly periodic, and moreover there exists a Q > 1 such that for all q > Q there exists some solution with prime period 2q.

Majority of work discussed in K. S. Berenhaut, R. T. Guy, “Periodicity and boundedness for the integer solutions to a minimum-delay difference equation,” in press. (Journal of Difference Equations and Applications)

Extension to y_{n} = min{f(y_{n−k_{1}}, y_{n−m_{1}}), …, f(y_{n−k_{L}}, y_{n−m_{L}})}

Theorem 3 Suppose that {y_{i}} satisfies y_{n} = f(y_{n−k}, y_{n−m}) with f(u, v) = h(u, v)/v for some function h where

(a) h ∈ C((0, ∞)^{2}, (a, ∞)) is symmetric in u and v and increasing in u,

(b) the function f is decreasing in v,

(c) for all v > a there exist C_{v} and D_{v} such that

lim_{u→a^{+}}f(u, v)/u = C_{v} > 1 and lim_{u→∞}f(u, v)/u = D_{v} < 1  (6)

Then {y_{i}} is periodic with not necessarily prime period 2.

Theorem 4 Set K = {k_{1}, k_{2}, …, k_{L}}, M = {m_{1}, m_{2}, … m_{L}} and ρ = gcd{k|k ∈ K}, and suppose the following hold.

• The function f satisfies the requirements of the theorem above.

• There exists a T ≥ 0, such that for each m ∈ M, there exists an m^{∗} ∈ M satisfying

m + m^{∗} = T mod ρ

• There exists a Q ≥ 0 and an m ∈ M such that

m + QT = 0 mod ρ.

Then every positive solution of

y_{n} = min{f(y_{n−k_{1}}, y_{n−m_{1}}), …, f(y_{n−k_{L}}, y_{n−m_{L}})}

converges to a unique equilibrium.

Tables and Figures

\begin{matrix}13 & 30 & 15 & 21 & 3 & 13 & 9 & 20 & 25 & 27 & −14 \\ −10 & 10 & 6 & −17 & −23 & 1 & −14 & −42 & −50 & 15 & −14 \\ −52 & −56 & 6 & 9 & −53 & −42 & −14 & 20 & −68 & −28 & −14 \\ 0 & −74 & −37 & 9 & −23 & −60 & −57 & 20 & −11 & −46 & −57 \\ 0 & 20 & −55 & −34 & −23 & 23 & −75 & −23 & −11 & −12 & −75 \\ −43 & 20 & −31 & −52 & −66 & 23 & −8 & −41 & −54 & −12 & 35 \\ −61 & −23 & −31 & 19 & −84 & −15 & −8 & −23 & −72 & −50 & 35 \\ −43 & −41 & −69 & 19 & −28 & −33 & −46 & −23 & 22 & −68 & −3 \\ −43 & 20 & −87 & 19 & −22 & −21 & −64 & −23 & 42 & −64 & −21 \\ −43 & 20 & −83 & 1 & −22 & −21 & −60 & −41 & 42 & −64 & −17 \\ −61 & 20 & −65 & 1 & −40 & −21 & −24 & −41 & 24 & −64 & 37 \\ −61 & 20 & −65 & 1 & −40 & −21 & −24 & −41 & 24 & −64 & 37 \end{matrix}

• K. S. Berenhaut, R. T. Guy, “Symmetric Functions and Difference Equations with Asymptotically Period-Two Solutions,” in press. (International Journal of Difference Equations)

• K. S. Berenhaut, R. T. Guy and C. L. Barrett, “Globally asymptotic behavior for minimum difference equations,” in preparation, to be submitted to the Journal of Difference Equations and Applications.