1. Introduction
The stability problem for functional equations or differential equations began with the well-known question of Ulam [
1]:
Let and be a group and a metric group with a metric , respectively. Given , does there exist a such that if a function satisfies the inequality for all , then there exists a homomorphism with for all ?
In short, the Ulam’s question states as follows: Under what conditions does there exist an additive function near an approximately additive function?
In 1941, Hyers [
2] gave a partial solution to the question of Ulam under the assumption that relevant functions are defined on Banach spaces.
Theorem 1 (Hyers [
2])
. Given , assume that is a function between Banach spaces such thatfor all . Then the limitexists for each and is the unique additive function such thatfor any . Based on Theorem 1, we say that the Cauchy additive functional equation,
, has (or satisfies) the Hyers-Ulam stability or it is stable in the sense of Hyers and Ulam. Since then, the stability problems for several functional equations and differential equations have been extensively investigated by many mathematicians (see [
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15] and the references therein).
Let
be a function with continuous partial derivatives
and
and let
L be the line segment joining the distinct points
and
in
. If a point
on the line segment
L satisfies the equation
then
is called a two-dimensional Lagrange’s mean value point of
f in
L.
In
Section 3, we prove the Hyers-Ulam stability of two-dimensional Lagrange’s mean value points. The main result of this paper is an extension and a generalization of the previous work [
16] (Theorem 2.2) (or see Theorem 4 below). Moreover, we introduce an efficient method for applying our main theorem (Theorem 6) in the practical use.
2. Preliminaries
The concept of Hyers-Ulam stability can be applied to the case of other mathematical objects. Ulam and Hyers [
17] seem to be the first mathematicians who applied the concept of Hyers-Ulam stability to differential expressions.
Theorem 2 (Ulam and Hyers [
17])
. Assume that is n times differentiable in a neighborhood N of a point and and changes sign at . For any , there corresponds a such that for every function which is n times differentiable in N and satisfies for all , there exists a point such that and . Given an
, if we choose a sufficiently small
, Theorem 2 would certainly be true but the choice of sufficiently small
imposes a constraint on practical use of this theorem. Indeed, we are interested in choosing the
as large as possible. Therefore, we introduce an algorithm for the choice of
strongly based on the proof of ([
17], Theorem 1).
Remark 1. Assume that ε and be the quantities given in Theorem 2. The following steps provide us with an efficient method for choosing the δ:
ε is chosen less than the radius of N;
we choose , and α such that , , , and , where and ;
we choose the δ as large as possible with .
In 1958, Flett [
18] proved a variant of Lagrange’s mean value theorem: If a function
is continuously differentiable and
, then there exists a point
such that
.
A similar problem as the Ulam’s question can be formulated for the mean value points:
If a function f has a mean value point η and g is a function quite near to f, does g have a mean value point near η?
Indeed, Das, Riedel and Sahoo [
19] examined the stability problem for Flett’s mean value points. Unfortunately, there were some errors in the proof of the main theorem of [
19]. In 2009, Lee, Xu and Ye [
20] proved the Hyers-Ulam stability of the Sahoo-Riedel’s points. We remind that for a differentiable function
, a point
is called a Sahoo-Riedel’s point of
f in
provided
satisfies
. Moreover, they obtained the following theorem concerning the stability of Flett’s points as a corollary.
Theorem 3 (Lee et al. [
20])
. Assume that are differentiable functions and η is a Sahoo-Riedel’s point of f in . If f is twice differentiable at η andthen for any and any neighborhood of η, there exists a with the property that for every h satisfying for and , there exists a Sahoo-Riedel’s point of h with . Thereafter, Găvrută, Jung and Li [
16] have proved the stability of the Lagrange’s mean value point which is a point
of a differentiable function
satisfying
.
Theorem 4 (Găvrută et al. [
16])
. Let be real numbers satisfying . Assume that is a twice continuously differentiable function and η is the unique Lagrange’s mean value point of f in an open interval and moreover that . Suppose is a differentiable function. Then, for any given , there exists a with the property that if for all , then there is a Lagrange’s mean value point of g with . We can extend the intimate concept of Lagrange’s mean value points to the two-dimensional cases. We will now introduce the two-dimensional Lagrange’s mean value theorem (see [
21] (Theorem 4.1)). Courant introduced the following theorem in his book [
22] (or see Sahoo and Riedel [
21] (Theorem 4.1)).
Theorem 5 (Courant [
22])
. For every function with continuous partial derivatives and and for all distinct points and in , there exists an intermediate point on the line segment L joining the points and such that . The geometrical interpretation of Theorem 5 is that the difference between values of the function at points and is equal to the differential at an intermediate point on the line segment joining those two points.
3. Main Theorem
Assume that
is a function with continuous partial derivatives
and
and
L is the line segment joining the distinct points
and
in
. We remind that an intermediate point
on
L is called a two-dimensional Lagrange’s mean value point of
f in
L provided that the point
satisfies the equation
By making use of Theorem 2, we will prove our main theorem concerning the Hyers-Ulam stability of the two-dimensional Lagrange’s mean value points.
Theorem 6. Assume that L is the line segment joining two distinct points and and that is a twice continuously partial differentiable function. Suppose is the unique two-dimensional Lagrange’s mean value point of f in L and the point satisfieswhere and . Then, for any given , there corresponds a such that if a partial differentiable function satisfies for all , then there is a two-dimensional Lagrange’s mean value point of g in L with . Proof. We know that the coordinates of each point on the line segment
L are given by
for some
. We define an auxiliary function
by
and calculate its derivative as
We define another auxiliary function
by
for all
. Obviously,
is twice continuously differentiable and
. Hence, by the Rolle’s theorem, there exists an
with
which implies that
is the unique two-dimensional Lagrange’s mean value point of
f in
L.
Furthermore, in view of (
1), we get
and
is continuous. Hence, there exists a neighborhood
of
such that either
for all
or
for all
. Since
,
changes sign at
.
We now translate Theorem 2 into the statement (
2) below by substituting as we see in the following table. (The function
H will be chosen later.)
Regarding the table above, Theorem 2 states that
For any given
, let
where
is chosen such that the statement in (
2) holds true. Let
be a partial differentiable function satisfying
for all
. If we define a differentiable function
by
for all
, then it holds that
for all
.
Hence, by the statement in (
2) with
, there exists a point
such that
and
. We note that
implies that
where
is the point on the line segment
L. Indeed,
is a two-dimensional Lagrange’s mean value point of
g in
L. Moreover, it holds that
Hence, the point is a two-dimensional Lagrange’s mean value point of g in L with , which completes the proof. □
We are now interested in choosing an appropriate
in Theorem 6 because the magnitude of
seems to be important for the practical use of this theorem. We only need to apply the algorithm in Remark 1 and refer to the statement in (
2) for the following algorithm.
Remark 2. For the notations r, , , and , we refer the proof of Theorem 6 and we introduce an efficient algorithm for choosing the δ:
by considering , we choose such that ;
we choose , and α such that , , , and ;
we choose the as large as possible with ;
we determine δ by .
The following two corollaries show that our main result (Theorem 6) is a generalization and an improvement of [
16] (Theorem 2.2). If we put
in Theorem 6, then the condition (
1) is reduced to
. Hence, we get the following corollary.
Corollary 1. Assume that L is the line segment joining two distinct points and and that is a twice continuously partial differentiable function. Suppose is the unique two-dimensional Lagrange’s mean value point of f in L and . Then, for any given , there exists a with the property that if a partial differentiable function satisfies for all , then there is a two-dimensional Lagrange’s mean value point of g in L with . In particular, throughout the statement of this corollary.
If we put
in Theorem 6, then the condition (
1) is reduced to
. Thus, we obtain the following corollary.
Corollary 2. Assume that L is the line segment joining two distinct points and and that is a twice continuously partial differentiable function. Suppose is the unique two-dimensional Lagrange’s mean value point of f in L and . Then, for any given , there exists a with the property that if a partial differentiable function satisfies for all , then there is a two-dimensional Lagrange’s mean value point of g in L with . In particular, throughout the statement of this corollary.
4. Example
Assume that
L is the set of all points on the line segment joining the points
and
and that
is a twice continuously partial differentiable function defined by
Then we have
,
,
, and
. In addition,
is the unique two-dimensional Lagrange’s mean value point of
f in
L and
i.e.,
f satisfies the condition (
1).
Consulting the proof of Theorem 6, we now define . It then follows from the equation that . Moreover, . Thus, we conclude that for all and for all , i.e., changes sign at and we can choose an arbitrary (we see in the proof of Theorem 6 and we take Remark 2 into account).
We now follow Remark 2
and choose
,
, and
with
,
, and
such that
If we set
,
, and
, then we get
Hence, by considering (
3) and Remark 2
, we can choose
which is consistent with
According to Theorem 6 and Remark 2 , for any given , if a partial differentiable function satisfies for all , then there exists a two-dimensional Lagrange’s mean value point of g in L with .
5. Discussions
In this paper, we prove the Hyers-Ulam stability of two-dimensional Lagrange’s mean value point:
Assume that L denote the line segment joining two points in the plane and that is a twice continuously partial differentiable function. Moreover, suppose is the unique two-dimensional Lagrange’s mean value point of f in L and the condition is fulfilled. Then, for any given , there exists a with the property that if a partial differentiable function satisfies for all , then there exists a two-dimensional Lagrange’s mean value point of g in L with .
The main theorem of this paper is an extension and an improvement of a previous work [
16] (Theorem 2.2). Indeed, even Corollary 1 or 2 is a generalization and an improvement of [
16] (Theorem 2.2).
Moreover, we introduce an algorithm for determining an appropriate constant depending on f and only. The larger is chosen, the more efficiently we apply our main theorem in practical use. This method helps us to choose a ‘large’ such that if a partial differentiable function g satisfies inequality for all , then there exists a two-dimensional Lagrange’s mean value point of g with .