# On the convergence of solutions to nabla dynamic equations on time scales.

1. Introduction

Many scientific disciplines, for instance in physics, chemistry, biology, and economics, are described by ordinary differential equations (ODEs). Thus, finding solutions of ODEs is important both in theory and practice. However, almost ODEs can not be solved analytically. Therefore, it is necessary to find a numeric approximation to the solutions in science and engineering. The Euler methods is very well-known because it is simple and useful to perform this, see [4, 9, 11, 18]. For solving the stiff initial value problem

(1.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

at each step [[t.sub.m-1], [t.sub.m]], the implicit Euler approximation of (1.1) is

(1.2) [x.sub.m] [x.sub.m-1] + hf ([t.sub.m], [x.sub.m])

where [t.sub.m] = [t.sub.m-1] + h and [x.sub.m] is the approximative value of x(t) at t = [t.sub.m]. The quantity [e.sub.m] := [absolute value of (x([t.sub.m]) - [x.sub.m])] is called the error of this method after m - 1 time steps which characterizes the difference between the approximative solution and the exact solution. The interested problem is how the error [e.sub.m] can be estimasted when the mesh step h tends to zero. We have known that [e.sub.m] tends to zero as h tends to zero with some added assumptions on f. Further, it has been shown that the implicit Euler method is more stable than explicit one (see [9, 11, 14]).

On the other hand, in recent years, the theory of the analysis on time scales has received a lot of attentions, see [1, 2, 7, 12, 13, 15, 16] in order to unify the continuous and discrete analysis. By using the notation of the analysis on time scales, equations (1.1) and (1.2) can be rewritten under the form

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with the time t belongs to the time scales T = R or [T.sub.h] = hZ. Thus, using the implicit Euler method means we consider equation (1.1) on the time [T.sub.h], which is "close" to T = R in some sense by view of analysis on time scale. Then the problem of the error estimation above can be restated as follows: How do the solutions of (1.3) on [T.sub.h] converge to the solution of (1.3) on T = R as the mesh step h tends to zero? In case of positive answer, what is the convergent rate of the error [e.sub.m]?

Following this idea in a more general context, in this article, we will consider the behavior of solutions of equation (1.3) on time scales [{[T.sub.n]}.sup.[infinity].sub.n=1] when [T.sub.n] tends to T by the Hausdorff distance. Under assumption that f (t,x) satisfies the Lipschitz condition in the variable x, we will prove that

(1.4) [x.sub.n](t) [right arrow] x(t) as [T.sub.n] [right arrow] T,

where [{[x.sub.n](t)}.sup.[infinity].sub.n=1], x(t) are solutions of equation (1.3) on time scales [{[T.sub.n]}.sup.[infinity].sub.n=1], T, respectively. Moreover, if f satisfies the Lipschitz condition in both variables t and x then the convergent rate of solutions is estimated as a same degree as the Hausdorff distance between [T.sub.n] and T, i.e.,

(1.5) [parallel]xn(t) - x(t)[parallel] [less than or equal to] [C.sub.2][d.sub.H](T, [T.sub.n]), for all t [member of] T [intersection] [T.sub.n] : [t.sub.0] [less than or equal to] t [less than or equal to] T.

By using these results, we obtain the convergence of the implicit Euler method as a consequence. It can be considered as a new and general approach to the convergence problems of the approximative solutions.

This paper is organized as follows. Section 2 summarizes some preliminary results on time scales. In Section 3, we study the convergence of solutions of nabla dynamic equations on time scales. The main results of the paper are derived here. In Section 4, we give some illustrating examples and show the convergence of the implicit Euler method. The last section deals with some conclusions.

2. Preliminaries

Let T be a closed subset of R, endowed with the topology inherited from the standard topology on R. Let [sigma](t) = inf{s [member of] T : s > t}, [mu](t) = [sigma](t) - t and [rho](t) = sup{s [member of] T : s < t}, v(t) = t - [rho](t) (supplemented by sup 0 = inf T, inf 0 = sup T). A point t [member of] T is said to be right-dense if [sigma](t) = t, right-scattered if [sigma](t) > t, left-dense if [rho](t) = t, left-scattered if [rho](t) < t and isolated if t is simultaneously right-scattered and left-scattered.

A function f defined on T is called rd-continuous provided it is continuous at right-dense points in T and its left-sided limits exist (finite) at left dense points. Similarly, f is Id-continuous if it is continuous at every left-dense point and if the right-sided limit exists in every right-dense point. It is easy to see that a function is continuous if and only if it is both rd-continuous and ld-continuous. A function f from T to R is regressive (respectively positively regressive) if 1 - v(t)f (t) [not equal to] 0 (respectively 1 - v(t)f (t) > 0) for every t [member of] T.

Definition 2.1 (Nabla Derivative). A function f : T [right arrow] [R.sup.d] is called nabla differentiable at t if there exists a vector [f.sup.[nabla]] (t) such that for all [epsilon] > 0

If ([rho](t)) - f (s) - [f.sup.[nabla]] (t)([rho](t) - s)[parallel] [less than or equal to] [epsilon][absolute value of ([rho](t) - s)]

for all s [member of] (t - [delta], t + [delta]) [intersection] T and for some [delta] > 0. The vector [f.sup.[nabla]] (t) is called the nabla derivative of f at t.

If T = R then the nabla derivative is f'(t) from continuous calculus; if T = Z then the nabla derivative is the backward difference, [nabla]f (t) = f (t) - f (t - 1), from discrete calculus.

Let f be a ld-continuous function and a, b [member of] T. Then, the Riemann integral [[integral].sup.b.sub.a] f(s)[nabla]s exists (see, e.g., [6, 7, 10]). In case b [not member of] T, writing [[integral].sup.b.sub.a] f(s)[nabla]s means [[integral].sup.b.sub.a] [nabla]s where [bar.b] = maxjt < b : t [member of] T}.

Consider the dynamic equation on the time scale T

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where f : T x [R.sup.d] [right arrow] [R.sup.d]. If f is ld-continuous and satisfies the Lipschitz condition in the variable x with a positively regressive Lipschitz coefficient then the problem (2.1) has a unique solution. For the existence, uniqueness and extensibility of solution of equation (2.1) we refer to [5, 7].

For any regressive ld-continuous functions p(x) from T to R, the solution of the dynamic equation [x.sup.[nabla]] = p(t)x, with the initial condition x(s) = 1, defines a so-called exponential function. We denote this exponential function by [[??].sub.p](T; t, s). For the properties of exponential function [[??].sub.p] T; t, s) the interested reader can see [1] and [7]. To simplify notations, we write [[??].sub.p](T; t, s) for [[??].sub.p](t,s) if there is no confusion. It is known that for any positively regressive number a, we have the estimate

(2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [C.sub.0] is a constant depending on the bounds of v (see [1, 2, 7]).

It is easy to see that if f (t,x) is a continuous function in (t,x) then x(t) is a solution to (2.1) if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 2.2 (Gronwall-Bellman lemma, see [7, 10]). Let x(t) be a continuous function and k > 0, [x.sub.0] [member of] R. Assume that x(t) satisfies the inequality

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where k is positively regressive. Then, the relation

(2.4) x(t) [less than or equal to] [x.sub.0][[??].sub.k](t, [t.sub.0]) for all t [member of] T, t [greater than or equal to] [t.sub.0]

holds.

Fix [t.sub.0] [member of] R. Let T = T([t.sub.0]) be the set of all time scales with bounded graininess such that [t.sub.0] [member of] T for all T [member of] T. We endow T with the Hausdorff distance, that is Hausdorff distance between two time scales [T.sub.1] and [T.sub.2] defined by

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For properties of the Hausdorff distance, we refer interested readers to [3, 8, 17].

3. Convergence of solutions

In this section, we consider the dynamic equation (2.1) on the sequence [{[T.sub.n]}.sub.n[member of]N] of time scales satisfying:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

by the Hausdorff distance and [t.sub.0] [member of] [T.sub.n] for any n [member of] N. We define the time scale

(3.1) [??] = [bar.[[union].sub.n[member of]N] [T.sub.n] [union] T].

Assume that f is continuous on [??] and satisfies the Lipschitz condition in the variable x, that is there exists a constant k > 0 such that

(3.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where k is positively regressive. By these assumptions, the initial value problems (IVPs)

(3.3) [x.sup.[nabla].sub.n] (t) = f(t, [x.sub.n](t)), t [member of] [T.sub.n], [x.sub.n] ([t.sub.0]) = [x.sub.0], n = 1, 2, ...

and

(3.4) [x.sup.[nabla]](t) = f (t,x(t)), t [member of] T, x([t.sub.0])= [x.sub.0],

have a unique solution [x.sub.n](t) defined on [T.sub.n] (respectively solution x(t) defined on T). It is clear that the solutions of the IVPs (3.3) and (3.4) are given by

(3.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(3.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

respectively, where [integral] f[[nabla].sub.n]s denotes the integral on time scale [T.sub.n].

The following lemma gives the uniformly bounded property of solutions of the IVPs (3.3) and (3.4) on different time scales.

Lemma 3.1. Let [x.sub.S](t) be the solution to the dynamic equation

[x.sup.[nabla]](t) = f (t,x(t)), t [member of] S, x([t.sub.0]) = [x.sub.0].

Then, for any T > [t.sub.0] one has

(3.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Let S [member of] T; S [subset] [??]. For any t [member of] S, we have

(3.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By virtue of continuity of f on [??], one has [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover, since f satisfies the Lipschitz condition (3.2),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By using the Gronwall-Bellman lemma, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [[??].sub.k](S; t, [t.sub.0]) is exponential function defined on S. Thus, by (2.2), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The proof is complete.

Let n [member of] N, we denote by [[rho].sub.n] the backward jump operator on the time scale [T.sub.n]. For any t [member of] T, there exists a unique [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], such that either s = t or t [member of] ([[rho].sub.n](s), s). It is easy to check that the function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is ld-continuous on T. Also, there exists [t.sup.*.sub.n] = [t.sup.*.sub.n](t) [member of] [T.sub.n] satisfying

(3.9) [absolute value of (t - [t.sup.*.sub.n])] = d(t, [T.sub.n]) = inf{[absolute value of (t - s)] : s [member of] [T.sub.n]}.

We choose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Define

(3.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(3.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Assume that [T.sub.n] [subset] T. Then, by the definition of Riemann integral on time scales, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for any t [member of] [T.sub.n] (see, e.g. [6, 7]).

Since [d.sub.H](T, [T.sub.n]) [right arrow] 0 as n [right arrow] [infinity], we can assume that [t.sup.*.sub.n](t) < T + 1 when t [less than or equal to] T.

By Lemma 3.1, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and hence let

M = sup{[parallel]f(t,x)[parallel] : [t.sub.0] [less than or equal to] t [less than or equal to] T + 1, [parallel]x[parallel] < A}.

Now, we need the following lemmas for proving the convergence of the solution sequence {[x.sub.n](t)} of the IVPs (3.3) when [T.sub.n] tends to T.

Lemma 3.2. Let [x.sub.n](t), n = 1, 2, ... be solutions to the IVPs (3.3) and x(t) be the solution to the IVP (3.4). Assume that [T.sub.n] [subset] T. Then,

(3.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and

(3.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [t.sup.*.sub.n] is defined by (3.9) and

(3.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. For any t [member of] [T.sub.n], t [less than or equal to] T we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By virtue of Lipschitz condition

[parallel] f(s,x(s)) - f(s, [x.sub.n](s))[parallel] [less than or equal to] k[parallel]x(s) - [x.sub.n](s)[parallel],

it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By using Gronwall-Bellman lemma, we obtain (3.12). If t [member of] T, t [less than or equal to] T then

[parallel]x(t) - [x.sub.n] ([t.sup.*.sub.n])[parallel] [less than or equal to][parallel]x(t) - x([t.sup.*.sub.n])[parallel] + [parallel]x([t.sup.*.sub.n]) - [x.sub.n] ([t.sup.*.sub.n])[parallel]

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Further,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Summing up, (3.13) holds. The proof is complete.

Lemma 3.3. Assume that [T.sub.n] [subset] T. For each [epsilon] and T [member of] T, there exists [theta] = [theta] ([epsilon], T) such that if [d.sub.H](T, [T.sub.n]) < [theta] then

(3.15) [[delta].sup.(n).sub.T] [less than or equal to] (T - [t.sub.0]) [epsilon] + [2M(T- [t.sub.0])/[theta]][d.sub.H](T, [T.sub.n]),

where [[delta].sup.n.sub.T] is defined by (3.14).

Proof. Since f is continuous, f is uniformly continuous on [[t.sub.0], T] x B(0, A) where B(0, A) is the ball with the center 0 and radius A. Therefore, for each [epsilon], there exists [delta] = [delta]([epsilon]) such that if [absolute value of ([t.sub.1] - [t.sub.2])] + [parallel][x.sub.1] - [x.sub.2][parallel] < [delta] then

[parallel]f([t.sub.1], [x.sub.1]) - f ([t.sub.2], [x.sub.2])[parallel] [less than or equal to] [epsilon] on [[t.sub.0], T] x B(0, A).

We choose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This implies that if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then

[parallel]f(t, x(t)) - [f.sub.n](t, [[??].sub.n](t))[parallel] < [epsilon].

We see that the number of values s [member of] [T.sub.n] satisfying [t.sub.0] [less than or equal to] s [less than or equal to] T and

{t [member of] T : [[rho].sub.n](s) < t < s, s - t [greater than or equal to] [theta]} [not equal to] 0,

is less than or equal to [T-[t.sub.0]/[theta]]. Assume that these values are [s.sub.1] < [s.sub.2] < ... < [s.sub.r] with r [less than or equal to] [T-[t.sub.0]/[theta]] In case [d.sub.H](T,[T.sub.n]) < [theta], we see that if t [member of] T such that [[rho].sub.n]([s.sub.i]) < t < [s.sub.i], [s.sub.i] - t [greater than or equal to] [theta] then

t - [[rho].sub.n]([s.sub.i]) = d(t, [T.sub.n]) [less than or equal to] [d.sub.H](T, [T.sub.n]).

Let

[[tau].sub.i] = max {t [member of] T : [[rho].sub.n]([s.sub.i]) < t < [s.sub.i], [s.sub.i] - t [greater than or equal to] [theta]}]i = [bar.1, r].

It is clear [[tau].sub.i] - [[rho].sub.n]([s.sub.i]) [less than or equal to] [d.sub.H](T, [T.sub.n]). Further,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [parallel]f(s,x(s)) - [f.sub.n](s, [[??].sub.n](s))[parallel] < [epsilon] for all s [member of] [[t.sub.0], [[rho].sub.n]([s.sub.1])] [union] ([[tau].sub.i], [[rho].sub.n]([s.sub.i+1])] [union] ([[tau].sub.r], T]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, for i = 1, 2,..., r we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The proof is complete.

We are now derive the convergence theorem for the IVPs (3.3) and (3.4).

Theorem 3.4. Let the sequence of time scales [{[T.sub.n]}.sup.[infinity].sub.n=1] satisfy [lim.sub.n[right arrow][infinity]] [T.sub.n] = T. Let [x.sub.n](t), n = 1, 2, ... be the solutions to the IVPs (3.3) and x(t) be the solution to the IVP (3.4). Then, for any T > to we have

(3.16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [t.sup.*.sub.n] is defined by (3.9).

Proof. Firstly, we assume that [T.sub.n] [subset] T for all n [member of] N. From Lemma 3.2, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for any [t.sub.0] [less than or equal to] t [less than or equal to] T. By Lemma 3.3, we get [lim.sub.n[right arrow][infinity]] [[delta].sup.(n).sub.T+1] = 0. Therefore, (3.16) holds.

In the general case, we put

[[??].sub.n] = [T.sub.n] [union] T.

Then, it is easy to see that

(3.17) [d.sub.H](T, [T.sub.n]) = max{[d.sub.H]([[??].sub.n], T), [d.sub.H]([[??].sub.n], [T.sub.n])}.

Let [[??].sub.n](t) be the solution to equation (2.1) on time scale [[??].sub.n]. For t [member of] T, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since T [subset] [[??].sub.n] and [T.sub.n] [subset] [[??].sub.n], we can apply Lemma 3.2 to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

(3.18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By Lemma 3.1, Lemma 3.3 and equality (3.17), we imply that [[??].sup.(n1).sub.T] - 0[[??].sup.(n2).sub.T+1] [right arrow] 0 [infinity]. Thus, (3.16) holds. The proof is complete.

For estimating the convergent rate, we need the following lemma.

Lemma 3.5. Assume that [T.sub.n] [subset] T. Then, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. With the value [theta] = [d.sub.H](T, [T.sub.n]), we follow a similar way as in the proof of Lemma 3.3 to construct the sequence [s.sub.1], [s.sub.2], ..., [s.sub.r] and the sequence [[tau].sub.1], [[tau].sub.2], ..., [[tau].sub.r] satisfying

[[rho].sub.n] ([s.sub.1]) < [[tau].sub.1] < [s.sub.1] < ... < [[rho].sub.n]([s.sub.r]) < [[tau].sub.r] < [s.sub.r].

Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The proof is complete.

Assume further that f (t,x) satisfies the Lipschitz condition in both variables t and x, that is

(3.19) [parallel]f(t,x) - f(s,y)[parallel] [less than or equal to] k([absolute value of (t - s)] + [parallel] x - y[parallel]), for all s, t [member of] T and x, y [member of] [R.sup.d].

We now estimate the convergent rate of approximation.

Theorem 3.6. Assume that assumption (3.19) is satisfied. Let [x.sub.n] (t), n = 1, 2, ... be solutions of the IVPs (3.3) and x(t) be the solution of the IVP (3.4). If t [member of] T : to [less than or equal to] t < T then

(3.20) [parallel]x(t) - [x.sub.n] ([t.sup.*.sub.n])[parallel] [less than or equal to] [C.sub.1][d.sub.H] (T, [T.sub.n]),

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined by (3.9). Moreover, if t [member of] T [intersection] [T.sub.n] : [t.sub.0] [less than or equal to] t < T then

[parallel]x(t) - [x.sub.n](t)[parallel] [less than or equal to] [C.sub.2][d.sub.H](T, [T.sub.n]),

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Let

[[??].sub.n] = [T.sub.n] [union] T,

and [[??].sub.n](t) be the solution of equation (2.1) on the time scale [[??].sub.n]. It is showed in

Theorem 3.4, for t [member of] T : to [less than or equal to] t [less than or equal to] T, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [[??].sup.(n1).sub.t], [[??].sup.(n2).sub.t] are given by (3.18). Note that if t [member of] T [intersection] [T.sub.n] : [t.sub.0] [less than or equal to] t [less than or equal to] T then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since f(t,x) satisfies the Lipschitz condition (3.19),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, by Lemma 3.5, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly, we imply that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Similarly, if t [member of] T [intersection] [T.sub.n] : [t.sub.0] [less than or equal to] t < T then

[parallel]x(t) - [x.sub.n](t)[parallel] [less than or equal to] [C.sub.2][d.sub.H] (T, [T.sub.n]),

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The proof is complete.

4. Examples

Example 4.1. Consider the IVP

(4.1) x' = f (t,x), [t.sub.0] [less than or equal to] t [less than or equal to] T, x([t.sub.0]) = [x.sub.0].

In numerical analysis, approximations to the solution x(t) of (4.1) will be generated at various values, called mesh points, in the interval [[t.sub.0], T]. For a positive integer n, we select a subdivision of the interval [[t.sub.0], T]

(4.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Associating with (4.2), we study a difference equation, called the implicit Euler method [9, 11, 14], as follows

(4.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then T and [T.sub.n] are time scales.

This leads to that we can rewrite (4.1) and (4.3) as follows

[x.sup.[nabla]](t) = f (t,x(t)), t [member of] T, x([t.sub.0]) = [x.sub.0],

and

[x.sup.[nabla]] ([theta]) = f ([tau], [x.sub.n]([tau])), [tau] [member of] [T.sub.n], [x.sub.n]([t.sub.0]) = [x.sub.0],

respectively. In this case, it is easy to see that

(4.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Suppose that f (t, x) is continuous and satisfies Lipschitz condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By Theorem 3.4 and (4.4), we see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] uniformly in [[t.sub.0], T]. Hence, we obtain the well-known result for the convergence of implicit Euler method in numerical analysis [9, 11, 14].

Assume further that f satisfies the Lipschitz condition in both variables with constant k. That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By Theorem 3.6, we get an estimation of the convergent rate as well as an error bound for the implicit Euler method as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 4.2 (Approximation of solutions to logistic equations on [R.sub.+]). Let T = [0, [infinity]). We now consider plant population models. Let x(t) be the number of plants of one particular kind at time t [member of] T in a certain area. By experiments we know that x(t) grows according to the logistic equation

(4.5) [x.sup.[nabla]](t) = x(t) [1 - 4x(t)], t [member of] T and x(0) = 1 > 0.

Suppose that we are unable to observe the values of x(t) but [x.sub.n](t) with [x.sub.n](t) to be the number of plants of one particular kind at time t [member of] [T.sub.n] in a certain area, subjecting to the equation

(4.6) [x.sup.[nabla].sub.n] (t) = [x.sub.n](t) [1 - 4[x.sub.n](t)], t [member of] [T.sub.n] and [x.sub.n](0) = 1 for all n [member of] N,

where [T.sub.n] is a time scale given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This means that we lack the observation, for some reasons, at the times in the intervals (2k/n, 2k+1/n). It is easy to see that [d.sub.H]([T.sub.n],T) = 1/2n. Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We have x(t) = [e.sup.t]/1+4([e.sup.t] -1); t [member of] T and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for all k = 1, 2, ..., n [member of] N. Note that if f satisties the local Lipschitz condition and the solution sequence {[x.sub.n](t)} is bounded then Theorem 3.4 also holds. Therefore, by this theorem, we imply that [x.sub.n](t) [right arrow] x(t) as n [right arrow] to.

The discrete graph of solutions [x.sub.n](t) and x(t) on the interval [0,1] is shown in Figure 1.

[FIGURE 1 OMITTED]

Example 4.3 (Approximation of solutions to logistic equation on Cantor set). Let K be be the Cantor set in [0,1]. Following the construction of this Cantor set, we define [K.sub.0] = [0,1]. We obtain [K.sub.1] by removing the "middle third" of [K.sub.0], i.e., the open interval (1/3, 2/3) from [K.sub.0]. [K.sub.2] is obtained by removing two "middle thirds of [K.sub.1], i.e., the two open intervals and (1/9, 2,9) from (7/9, 8/9) Proceeding in this manner we obtain a sequence of time scales [([K.sub.n]).sub.n[member of]N]. The Cantor set is defined

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let ([T.sub.n]) be a sequence of time scales, where [T.sub.n] = [K.sub.n] [union] ([K.sub.n] +1) and T = K [union] (K + 1). Consider the dynamic equation (4.5) with x(0) = 1. It is known that we are unable to give an explicit formula for solutions as well as a numerical solution to equation (4.5). However, we can use Theorem 3.4 to approximate these solutions.

We illustrate this approximation by Figure 2. It is seen that the graph on [T.sub.4] of the equation (4.5) (the green line) is similar to one on [T.sub.0] = [0, 2] (the red line).

[FIGURE 2 OMITTED]

5. Conclusion

In this paper, we have proved the convergence of solutions of nabla dynamic equations [x.sup.[nabla]](t) = f (t,x) on time scales [{[T.sub.n]}.sup.[infinity].sub.n=1] when this sequence converges to the time scale T. The convergent rate of solutions is estimated when f satisfies the Lipschitz condition in both variables.

Acknowledgments This work was supported financially by Vietnam National Foundation for Science and Technology Development (NAFOSTED) 101.03-2014.58.

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NGUYEN THU H (a), NGUYEN HUU DU (b), LE CONG LOI (b), AND DO DUC THUAN (c)

(a) Department of Basic Science, Electric Power University, 235 Hoang Quoc Viet Str., Hanoi, Vietnam, ntha2009@yahoo.com

(b) Department of Mathematics, Mechanics and Informatics, Vietnam National University, 334 Nguyen Trai Str., Hanoi, Vietnam, dunh@vnu.edu.vn, loilc@vnu.edu.vn

(c) School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet Str., Hanoi, Vietnam, ducthuank7@gmail.com