We consider nonlinear difference equations of unbounded order of the form , where are suitable functions. We establish sufficient conditions for the boundedness and the convergence of as . Some of these conditions are interesting mainly for studying stability of numerical methods for Volterra integral equations.
1. Introduction
We consider the following nonlinear Volterra discrete
equation of nonconvolution type: The existence problem for
solution of Volterra discrete equations arises in the nonlinear implicit case.
For linear implicit equations and nonlinear explicit equations, the problem is
easily solved. Recently, some local and global existence theorems for Volterra
discrete equations in the general case are given in [1, 2].
From now on, we assume that there exists a strictly
increasing function such that Note that (1.2) implies
that The above difference equation
can be considered as the discrete counterpart of the Volterra integral equation
whose importance in the applications is well known (see, e.g., [3, 4]), and arises also in the application of numerical
methods to Volterra integral and integrodifferential equations. The theory of
the qualitative behavior of this type of nonlinear difference equation is very
important, in particular for the study of numerical stability of such methods
(see, e.g., [5–11] and the references
therein).
In this paper, we study some sufficient conditions for
the boundedness of the solutions (if they exist) of (1.1), subject to (1.2),
and their asymptotic behavior as . In particular, in Section 2 we investigate the asymptotic behavior when is upper bounded by a linear function. The
case of nonnegative coefficients is investigated in Section 3 and, with
additional monotonicity assumptions, in Section 4.
2. Case of
Assume that, in (1.1) with (1.2), the
following additional hypotheses hold: Observe that the second
part of (2.1) is true if in (1.2) .
The following lemma can be
easily proved.
Lemma 2.1. If for all .
Here and in the sequel we assume a sum with a
negative superscript to be zero. By using (2.1) and Lemma 2.1, from (1.1), we have that and we
set This inequality will be useful
in order to find sufficient conditions for the boundedness of and for its convergence to zero as tends to infinity.
Theorem 2.2. Consider (1.1) with (1.2)
and (2.1), if there exists a positive constant such that for some positive integer ,
then is bounded and Moreover, if then .
Proof. Let us consider (2.3), by using (2.5), we have
that In particular, assume that the
third part of (2.5) holds, then and thus, Hence, the following
inequalities hold for each : For this reason, from which we obtain
that Thus, is bounded and satisfies (2.6).
Assume that and put and .
Then, since is bounded and the third of (2.5) holds, we
have that and .
Let's take any and consider a continuous function on .
Then, by ,
there exists a constant such that For the above ,
there exists a positive integer such that and ,
for any .
By assumption (2.7), we have that for ,
there exists a positive integer such that Then, for and ,
we have that which is a contradiction with
the definition. Hence, and we obtain .
Note that the third part
of (2.5) is equivalent to .
The theorem above gives some conditions on the
coefficients of (1.1) for the boundedness of which supplement the results in [12, Theorem 2.1]. Moreover, it
worths while to compare our result with the ones in [9, Theorem 3.1] and [5, Theorem 4.1]. In order to do
that, we assume , and then is given. In this case, following the line of
the proof of Theorem 2.2, we can still show that vanishes as provided that (2.5) and the second
part of (2.7) hold. Observe that this represents an
additional result with respect to [9, Theorem 3.1] and [5, Theorem 4.1] which, involving the sum of the coefficients on the second index, enlarges the set of conditions
for to be bounded and convergent to zero. As an
example, for equation (3.2) in [9] or the sufficient condition
in [5]
is not satisfied, however (2.5) is fulfilled. Moreover, it is easy to
see that, in the convolution case ,
the third of (2.5) coincides with the known one [5, 10] and the second
part of (2.7) is implied by (2.5).
Theorem 2.2 turns out to be quite useful in the linear
case when (1.1) represents the linearized equation for the global error of a
numerical method applied to a Volterra integral equation. In this case, represents the local truncation error of the
method at the step .
Thus, if is bounded for all and if (2.5) holds, then the error is bounded and the bound is given in (2.6).
The following theorem provides some sufficient conditions
on the coefficients of (1.1) for the summability of ,
which turn out to be less restrictive of those stated by [13, Theorem 2.8].
Theorem 2.3. For (1.1) with (1.2), assume
(2.1). If then ,
and consequently, .
Proof. By (2.3), Therefore, by (2.19), we have
that and then, .
In the case (1.1) is linear, the following theorem is easily
proved.
Theorem 2.4. For the linear equation (2.22),
assume ,
and for
(i)Suppose that Then, .
In particular, if there exists a positive integer such that then is bounded and Moreover, if then .(ii)If then ,
and consequently, .
Proof. By (2.22), we obtain that Then, we have
that Thus, analogously to the proofs
of Theorems 2.2 and 2.3, we obtain the conclusion of this theorem.
3. Nonnegative Coefficients
In this
section, we focus on the solutions of (1.1) with (1.2) and Such discrete equations are
useful, above all, in the investigations on the behavior of the solution of
some numerical methods when used to solve nonlinear heat flow in a material
with memory (see [14]
and the bibliography therein). Let us start with the following lemma, which
describes some aspects of the solution of (1.1)-(1.2) with (3.1) when has a sign eventually constant for all .
The utility of this lemma is not in itself, but as an instrument to prove some
of the next theorems (see Theorems 3.4, 4.1 and 4.3).
Lemma 3.1. Let be the solution of (1.1) and assume that
(i)and for each ;(ii)there exists such that (resp., ) for any ,
then where and is a positive constant.
Moreover,
assume that one of the following conditions holds:
(iii1),(iii2)and there exists a strictly increasing
function on such that and (resp., ), then .
Furthermore, if, in addition to ,
there exists a positive constant such that (resp., ), then .
Proof. Since and is a continuous function, then is bounded. Assume that there exists a
nonnegative integer such that for any (the analysis of the case for all is analogous). Then, by the fact that, for the
main hypothesis (1.2), whenever , we have Hence, the first part of the
lemma is proved. Consider now the two cases and separately.
Case (iii1): of course implies .
Case (iii2): put .
Assume that ,
and let be a subsequence of such that .
Then, one can prove that .
By (1.1) and assumptions, we have Therefore, which is a contradiction because and .
Hence, we have .
In addition, suppose that there exists a positive
constant such that .
Then, we have that Thus, from ,
we conclude that .
The proof is completely analogous when there exists a
nonnegative integer such that for any .
Remark 3.2. Observe that in the linear case (2.22), the last conditions of Lemma
3.1 are satisfied whenever and .
Hereafter, we investigate on the boundedness of the
solution of (1.1)-(1.2) when
Lemma 3.3. Let be the solution of (1.1) with (1.2) and (3.7),
and assume that then is bounded.
Proof. Let be the bound for and .
Let us write as the sum of the following two
contributions: where and .
Therefore, since (1.2), (3.1), (3.7), and (3.8) hold, we have
that Thus, is bounded and the proof is complete.
As an example we consider the equation in this case and .
Hence, Another example is given by the
explicit equation Here and ,
hence From Figure 1 it is clear that
the bounds established by Lemma 3.3 (represented by dotted lines) may be quite
sharp. We are able to
prove the following result.
Figure 1: Plot of (
3.13) and its
bounds given by (
3.14).
Theorem 3.4. Assume that is continuous on , for all .
Then .
Proof. Let and assume .
Since we are in the hypotheses of Lemma 3.3, is bounded and then .
For any fixed ,
consider a continuous function on .
Then, by ,
there exists a constant such that For the above ,
there exists a positive integer such that and ,
for any .
By Assumption (3.16), we have that for ,
there exists a positive integer such that Then, for and ,
we have that Let us rewrite (1.1) in the
following form: where ,
for and ,
for and .
Thus, and, since is an increasing function, we have that, for
all , Since we are in the hypothesis
that the coefficients are nonnegative, it follows
that In conclusion,
from and by using (3.20), (3.19), and
(3.17), the following inequality holds: This result contradicts the definition. Hence, ,
so are eventually nonnegative. Since it is easy
to see that we are in the hypotheses of Lemma 3.1 (case (iii1)), then .
Remark 3.5. Once again, in the convolution
case, the first part of (3.15) implies the second
one of (3.16).
For the special case ,
we establish the following sufficient condition from Theorem 3.4.
Theorem 3.6. Suppose that and assume that hold, then the solution of (1.1) tends to zero as tends to infinity.
Proof. Put . Since for ,
hence is a strictly monotone decreasing function in .
Now, we will prove that ,
for .
Let for .
Then we have that By recalling that and ,
we have .
Since the function is increasing for ,
there results .
Thus, .
Hence, we have that ,
for .
Thus, for with ,
the second of (3.15) is true and, by Theorem 3.4, we have .
Remark 3.7. From the proof of Theorem 3.6 it is
clear that the second part of (3.15) is satisfied by . By playing with ,
this allows us to consider a wide variety of functions which satisfy (1.2). For instance, in the
cases and the stained areas in Figure 2 represent the
admissible regions for the functions
respectively (the solid lines show, as an example, the graphs of and ).
Figure 2: Plots of some admissible regions
for
according to Theorem
3.6.
4. Monotonic Nonnegative Coefficients
In this
section, for (1.1), first, we consider the case that We provide the following theorem
which generalizes [15, Theorem 2.1] to the nonlinear case.
Theorem 4.1. In addition to condition (4.1), suppose that Then, any solution of (1.1) satisfies (resp., ), .
Moreover, if ,
for all and there exists a strictly increasing
function on such that and (resp., ), then .
In addition, if there exists a positive constant such that (resp., ), then .
Proof. We prove the theorem in the case ,
the proof for is perfectly symmetric. Then by (1.1), we have
that hence, for the properties of described in (1.2), it has to be .
Proceeding by induction, suppose that .
By (1.1), and hence, by adding the two
relations and taking into account that, for the second
part of (4.1), ,
we have that So we have that and, from (1.1), .
Thus, we are in the hypotheses of Lemma 3.1 part and, hence, we get the thesis.
Observe that when (1.1) is linear, the last condition
of Theorem 4.1 is satisfied by choosing and .
In this case, the hypotheses of Theorem 4.1 include, as particular cases, those
of [15, Theorem 2.1]. In particular we note that, as Theorem 4.1 prove the
summability of the solution ,
it is interesting when applied to the equation satisfied by the fundamental
matrix of a Volterra difference equation (see, e.g., [15, equation (1.4)]). Namely, in
[15] it is underlined
that such a result can be employed in the study of the stability of some
numerical methods.
A simple application of Theorem 4.1 in the linear case
is given by the following example.
Example 4.2. Let us consider the difference
equation whose solution is given
by Then, for and ,
all the conditions in Theorem 4.1 are satisfied with ,
which implies and . Observe that in this case the bound coincides with the exact value of the sum of the series.
Next, we provide the following theorem whose proof is
a direct extension of the proof of Crisci et al. [6, Theorem 2.1], which gives a
priori bound for the solution of (1.1) depending on the forcing terms .
Theorem 4.3. In addition to the conditions (4.1), assume that Then, any solution of (1.1) is bounded and satisfies Moreover, suppose that and there exists a strictly increasing function on such that and (resp., ), then .
In addition, if there exists a positive constant such that (resp., ), then .
Proof. Consider the two possible subcases: (a) ,
and (b) .
(a) Assume .
If ,
then by (1.1), we get and .
Hence, (4.9) holds if is oscillatory about .
Let and denote by the time moment of the first passage of the
solution through the zero, that is, The time moment of the following
passage through the zero of the solution after is denoted by ,
that is, In a similar way, we introduce
the indexes as follows:
(1) Consider that ,
hence .
Then, from (1.1), we have that where every summation of the
type involves only positive ,
while the negative ones. Now observe that, for ,
by using (4.1) and the fact that ,
we have ,
furthermore , because of (4.12), hence both and are less than or equal to zero, thus .
By using these considerations it is easy to see that the following inequality
holds: With analogues considerations we
get By adding each side of (4.13) and
taking into account (4.14), (4.15), it comes out that By using the monotonicity of stated by (4.1) and the main hypothesis (1.2),
taking into account (4.12), we have that
(2) Consider that ,
hence .
Proceeding as above, we have Hence, from (1) and (2), we
obtain (4.9). Part (b) of the proof is essentially mirror-like of part (a) and
leads once again to (4.9). Thus, any solution of (1.1) is bounded and satisfies (4.9).
Moreover, suppose that and there exists a strictly increasing
function on such that and (resp., ).
If there exists a nonnegative integer such that (resp., ) for any ,
since and for all ,
we are in the hypotheses of Lemma 3.1 part (iii2) and we obtain .
On the contrary, if such an index does not exist, let and consider the extract subsequence of all the positive values in .
Assume that ,
then Taking into account (4.12), there
exists an index such that ,
then plays the role of in (4.13) and, analogously to (4.16), we
have Hence, since ,
for all ,
we have that and so, since only positive
quantities are involved, we get Passing to the as ,
we have that Taking into account that all the involved in the summations above form the
extract of the positive values in ,
we get ,
which is a contradiction with (4.19), so .
An analogous proof on the extract subsequence of all negative values of leads to .
The same happens when .
Hence, in conclusion, we obtain that .
In addition, suppose that there exists a positive constant such that .
Then, by (4.22) and the fact that is strictly positive, we conclude that .
Similarly, we obtain that .
Hence, .
Acknowledgment
This research was partially supported by Waseda University grant for special research Projects 2006B–167 and Scientific Research (c), no. 19540229 of Japan Society for the Promotion of Science.