Applied Mathematics E-Notes, 1(2001), 18-23 \begin{matrix}c \\ ◯ \end{matrix} Available free at mirror sites of http://math2.math.nthu.edu.tw/∼amen/

Periodic Solutions of Abstract Difference Equations ^{∗}

Michael Gil’^{†}

Received 20 May 2000

Abstract We investigate whether a control mechanism can be introduced to support a periodic solution for an abstract difference equation modeling a diffusion problem. An existence theorem is proved and estimates of the norms of the periodic solution is also obtained.

1 Introduction

To motivate what follows, consider a large number of equally divided cells contained in a “very long” straight tube, each of which contains a certain solute dissolved in a unit volume of solvent. These cells are separated from each other by semipermeable membranes through which the solute may flow but not the solvent. Let us denote by u_{i}^{(t)} the amount of solute dissolved in the i-th cell and at different time periods t = 0, 1, 2, ... . Since each cell contains a unit volume of solvent, u_{i}^{(t)} also represents the concentration of solute in the i-th cell. During the time period t, if the concentration u_{i−1}^{(t)} is higher than u_{i}^{(t)}, solute will flow from the (i − 1)-th cell to the i-th cell. The amount of increase is u_{i}^{(t+1)} − u_{i}^{(t)}, and it is reasonable to postulate that the increase is proportional to the difference u_{i−1}^{(t)} − u_{i}^{(t)}. Similarly, solute will flow from the (i + 1)-th cell to the i-th cell if u_{i+1}^{(t)} > u_{i}^{(t)}. Thus, it is reasonable the total effect is

u_{i}^{(t+1)} − u_{i}^{(t)} = γ (u_{i−1}^{(t)} − u_{i}^{(t)}) + γ (u_{i+1}^{(t)} − u_{i}^{(t)}) ,

where γ is a proportionality constant. If the permeability of the membrane is time dependent, and if a time dependent control is introduced, it is plausible that the governing equation becomes

u_{i}^{(t+1)} − u_{i}^{(t)} = γ_{t} (u_{i+1}^{(t)} − 2u_{i}^{(t)} + u_{i−1}^{(t)}) + G (t, u_{i}^{(t)}) .

By writing {u_{i}^{(t)}}_{i=−∞}^{∞} as x_{t}, we may write the above equation in the form

x_{t+1} − x_{t} = γ_{t}Jx_{t} + g_{t}(x_{t}), t = 0, 1, 2, ...,

∗Mathematics Subject Classifications: 39A11

†Department of Mathematics, Ben Gurion University, P. O. Box 653, Beer Sheva 84105, Israel

where J is a doubly infinite matrix with diagonal elements equal to −2, and superdiagonal as well as subdiagonal elements equal to 1.

Existence of periodic solutions to difference equations similar to the one derived above have been studied by several authors, see e.g. [1-5]. Here, an important question arises naturally as to whether the control mechanism can maintain a periodic solution for the above equation. To this end, let C be the set of complex numbers and let X be a complex Banach space with a norm ∥·∥ . We will denote the identity operator defined on X by I and denote the closed ball with radius r by Ω_{r}, i.e. Ω_{r} = {v ∈ X| ∥v∥ ≤ r} , where 0 < r ≤ ∞. Consider sequences of the form {x_{k}}_{k=0}^{∞} in X which satisfies the perturbed difference equation

x_{k+1} = A_{k}x_{k} + F_{k}(x_{k}), k = 0, 1, 2, ...,  (1)

where {A_{k}}_{k=0}^{∞} is a periodic sequence of bounded operators defined on X such that

A_{k} = A_{k+T} , k ≥ 0,  (2)

and

I − A_{0}A_{1} ⋯ A_{T −1} is invertible,  (3)

and {F_{k}}_{k=0}^{∞} is a periodic sequence of functions from Ω_{r} into X such that

F_{k} = F_{k+T} , k ≥ 0,  (4)

and

∥F_{k}(x) − F_{k}(y)∥ ≤ q ∥x − y∥ , k = 0, 1, ..., T − 1; x, y ∈ Ω_{r}; q > 0.  (5)

In the special case when X = C^{n}, A_{k} = A for k ≥ 0 and ∥F_{k}(x)∥ ≤ α ∥x∥ + β for k ≥ 0 and x ∈ X, equation (1) has been studied [6], and a periodic solution is found under suitable conditions on A, α, and β. Here, we will also be interested with the existence of periodic solutions of our general abstract difference equation (1), as well as estimates of their norms.

2 Main Results

To accomplish our goals, let us set

U(k, j) = ∏_{i=j}^{k−1} A_{i}, 0 ≤ j < k ≤ T,

and set U(j, j) = I for j ≥ 0. Then it is easily checked that the unique solution of the equation

y_{k+1} = A_{k}y_{k} + f_{k}, f_{k} ∈ X, k = 0, 1, ...,

is given by

y_{k} = U(k, 0)y_{0} + ∑_{j=0}^{k−1} U(k, j)f_{j}, k = 1, 2, ... .

Thus the periodic boundary value problem

y_{k+1} = A_{k}y_{k} + f_{k}, f_{k} ∈ X, k = 0, 1, ..., T,
y_{0} = y_{T}

has a solution provided

y_{0} = y_{T} = U(T, 0)y_{0} + ∑_{j=0}^{T −1} U(T, j)f_{j},

or

y_{0} = (I − U(T, 0))^{−1} ∑_{j=0}^{T −1} U(T, j)f_{j},

and in such a case, this solution is given by

y_{k} = U(k, 0) (I − U(T, 0))^{−1} ∑_{j=0}^{T −1} U(T, j)f_{j} + ∑_{j=0}^{k−1} U(k, j)f_{j}, 0 ≤ k ≤ T,  (6)

and its maximum norm satisfies

max_{0≤k≤T −1} ∥y_{k}∥ ≤ γ_{T} max_{0≤k≤T −1} ∥f_{k}∥ ,  (7)

where

γ_{T} = max_{0≤k≤T −1} ∑_{j=0}^{T −1} {∥U(k, 0)(I − U(T, 0))^{−1}U(T, j)∥ + ∥U(k, j)∥} .  (8)

THEOREM 1. Under the conditions (2)-(5), if γ_{T} (qr + l_{T}) < r < ∞, where l_{T} = max_{0≤k≤T −1} ∥F_{k}(0)∥ , then the periodic boundary value problem

x_{k+1} = A_{k}x_{k} + F_{k}(x_{k}), k = 0, 1, ..., T − 1,  (9)
x_{0} = x_{T} ,  (10)

has a unique solution.

PROOF. Let Φ be the Cartesian product X^{T} . When equipped with the maximum norm defined by

∥(x_{0}, ..., x_{T −1})∥_{Φ} = max_{0≤k≤T −1} ∥x_{k}∥ , (x_{0}, ..., x_{T −1}) ∈ Φ,

Φ becomes a Banach space. Let

G = {x ∈ Φ| ∥x∥_{Φ} ≤ r} .

Furthermore, let Ψ : G → Φ be defined as follows: for each x = (x_{0}, ..., x_{T −1}) ∈ G, define (Ψx)_{0} = 0 and

(Ψx)_{k} = U(k, 0) (I − U(T, 0))^{−1} ∑_{j=0}^{T −1} U(T, j)F_{j}(x_{j}) + ∑_{j=0}^{k−1} U(k, j)F_{j}(x_{j})

for 0 ≤ k ≤ T − 1. Then for each x ∈ G,

∥Ψx∥_{Φ} ≤ γ_{T} max_{0≤k≤T −1} ∥F_{k}(x_{k})∥ ≤ γ_{T} max_{0≤k≤T −1} {q ∥x_{k}∥ + l_{T} } ≤ γ_{T} (qr + l_{T}) < r,

which implies Ψx ∈ G. Furthermore,

∥Ψx − Ψy∥_{Φ} ≤ γ_{T} q ∥x − y∥_{Φ} , x, y ∈ G.

Since γ_{T} (qr + l_{T}) < r implies 0 ≤ γ_{T} l_{T} < r(1 − γ_{T} q), we see that Ψ is a contraction mapping on G. By Banach’s fixed point theorem, Ψ has a unique fixed point u in G. It is easily seen that u is a solution of (9)-(10). The proof is complete.

We remark that the unique solution asserted in the above theorem satisfies

max_{0≤k≤T −1} ∥x_{k}∥ ≤ \frac{γ_{T} l_{T}}{1 − qγ_{T}} .  (11)

Indeed, if {x_{k}}_{k=0}^{T} is such a solution, then in view of (7), (8) and (5), we will have

max_{0≤k≤T −1} ∥x_{k}∥ ≤ γ_{T} max_{0≤k≤T −1} ∥F_{k}(x_{k})∥ max_{0≤k≤T −1} ∥x_{k}∥ ≤ γ_{T} max_{0≤k≤T −1} {q ∥x_{k}∥ + l_{T} } ,

which implies (11).

There are at least two important variations of Theorem 1. First of all, if r = ∞, we may replace the assumption γ_{T} (qr + l_{T}) < r < ∞ by γ_{T} l_{T} < 1 in the above theorem: Under the conditions (2)-(5) where r = ∞, if γ_{T} l_{T} < 1, then the periodic boundary problem (9)-(10) has a unique solution {x_{k}}_{k=0}^{T} which satisfies (11). Second, if we assume that γ_{T} l_{T} > 0, then the condition γ_{T} (qr + l_{T}) < r in the above Theorem can be replaced by the condition γ_{T} (qr + l_{T}) ≤ r : Under the conditions (2)-(5) and γ_{T} l_{T} > 0, if γ_{T} (qr + l_{T}) ≤ r, where l_{T} = max_{1≤k≤T −1} ∥F_{k}(0)∥ , then the periodic boundary value problem (9)-(10) has a unique solution {x_{k}}_{k=0}^{T} which satisfies (11). The proofs of these statements are not much different from that of Theorem 1 and hence omitted.

The constant γ_{T} defined by (8) is difficult to evaluate. To simplify matters, let us set Q_{j,j} = 1 for j ≥ 0,

Q_{k,j} = ∏_{i=j}^{k−1} ∥A_{i}∥ , 0 ≤ j < k ≤ T,

and

Q_{k,j} = \frac{1}{Q_{j,k}} , 0 ≤ k < j ≤ T.

Then ∥U(k, j)∥ ≤ Q_{k,j} for 0 ≤ k ≤ j ≤ T − 1 and Q_{k,j}Q_{j,i} = Q_{k,i} for i ≤ j ≤ k. Furthermore, if Q_{T,0} < 1, then

γ_{T} ≤ max_{0≤k≤T −1} ∑_{j=0}^{T −1} {Q_{k,0}(1 − Q_{T,0})^{−1}Q_{T,j} + Q_{k,j}}

≤ max_{0≤k≤T −1} ∑_{j=0}^{T −1} Q_{k,0} {(1 − Q_{T,0})^{−1}Q_{T,0} + 1} Q_{0,j}

≤ \frac{1}{1 − Q_{T,0}} max_{0≤k≤T −1} Q_{k,0} ∑_{j=0}^{T −1} \frac{1}{Q_{j,0}} .

THEOREM 2. Assume that Q_{T,0} < 1. Let l_{T} = max_{1≤k≤T −1} ∥F_{k}(0)∥ , and

ρ_{T} = \frac{1}{1 − Q_{T,0}} max_{0≤k≤T −1} Q_{k,0} ∑_{j=0}^{T −1} \frac{1}{Q_{j,0}} .

Under the conditions (2)-(5), if ρ_{T} (qr + l_{T}) < r, then the boundary value problem (9)-(10) has a unique solution {u_{k}}_{k=0}^{T}. Moreover, the inequality

max_{0≤k≤T −1} ∥u_{k}∥ ≤ \frac{ρ_{T} l_{T}}{1 − qρ_{T}}

is valid.

The proof is similar to that of Theorem 1 and thus omitted.

3 An Example

We now turn to our diffusion problem. Let us find a control such that our problem can be written in the form

u_{i}^{(t+1)} − u_{i}^{(t)} = a_{t}u_{i+1}^{(t)} − b_{t}u_{i}^{(t)} + c_{t}u_{i−1}^{(t)} + G (t, u_{i}^{(t)}) ,

where i = 0, ±1, ..., and t = 0, 1, ... . The above partial difference equation can be written in the form

u_{t+1} = A_{t}u_{t} + F_{t}(u_{t}), t = 0, 1, 2, ...,

where A_{t} is a doubly infinite matrix with diagonal elements equal to 1 + b_{t}, and superdiagonal and subdiagonal elements equal to a_{t} and c_{t}, respectively. Suppose {a_{t}} , {b_{t}} and {c_{t}} are T -periodic real sequences, then the matrix sequence {A_{t}} is also T - periodic. Take X to be the space of doubly infinite bounded sequences endowed with the supremum norm. Then

∥A_{t}∥ = |1 − b_{t}| + a_{t} + c_{t}.

Therefore,

Q_{k,j} = ∏_{i=j}^{k−1} (|1 − b_{i}| + a_{i} + c_{i}) .

Suppose further that Q_{T,0} < 1 (which is satisfied, for example, when a_{t} + c_{t} < b_{t} < 1 for all t.) Then the quantity ρ_{T} can be calculated in a straightforward manner, and Theorem 2 can then be applied.

References

[1] A. Halanay, Periodic and almost periodic solutions of systems of finite difference equations, Arch. Rat. Mech. Anal., 12(1963), 134-149.

[2] R. P. Agarwal and J. Popenda, Periodic solutions of first order linear difference equations, Math. Comput. Modelling, 22(1)(1995), 11-19.

[3] T. Y. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly, 82(1975), 985-992.

[4] G. P. Pelyukh, On the existence of periodic solutions of discrete difference equations (Russian), Uzbek. Mat. Zh., 3(1995), 88-90.

[5] Kh. Turaev, On the existence and uniqueness of periodic solutions of a class of nonlinear difference equations (Russian), Uzbek. Mat. Zh., 2(1994), 52-54.

[6] M. Gil’ and S. S. Cheng, Periodic solutions of a perturbed difference equation, preprint.