International Journal of Analysis
Volume 2015 (2015), Article ID 894625, 4 pages
http://dx.doi.org/10.1155/2015/894625

Research Article

An Integral Mean Value Theorem concerning Two Continuous Functions and Its Stability

Monea Mihai

University Politehnica of Bucharest, Splaiul Independenţei 313, 060042 Bucharest, Romania

Received 5 July 2015; Revised 12 September 2015; Accepted 15 September 2015

Academic Editor: Yaozhong Hu

Copyright © 2015 Monea Mihai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The aim of this paper is to investigate an integral mean value theorem proposed by one of the references of this paper. Unfortunately, the proof contains a gap. First, we present a counterexample which shows that this theorem fails in this form. Then, we present two improved versions of this theorem. The stability of the mean point arising from the second result concludes this paper.

1. Introduction

The mean value theorems represent some of the most useful mathematical analysis tools. The first known result is due to Lagrange (1736–1813). In the years that followed, more mathematicians investigated this subject. As consequences of this fact, now we can find similar results, more generalizations, or extensions. Sahoo and Riedel’s book [1] presents a large collection of old and new mean value theorems. The readers can consult [2], [3], [4], or [5] to find some recent results. Reference [6] leads this subject to a new direction.

In [7], the following theorem was presented.

Theorem 1. Let and be two nonnegative continuous functions on the interval and . There exists such that

This result is very generous. Moreover, the authors obtained some interesting consequences. Unfortunately, the proof contains a gap. The aim of this paper is to present a counterexample which shows that the result from the previous theorem is not necessarily valid under this hypothesis. Afterwards, we present some conditions for which equality (1) holds. Finally, we include a stability result.

2. A Counterexample for Theorem 1

Let us consider the functions defined, for any , by and . These functions are continuous and nonnegative. We have

We choose . Then

for any . Hence

Then, relation (1) fails.

3. Two Integral Mean Value Theorems

In this section we present two valid versions of Theorem 1, which were suggested by their original proofs.

Theorem 2. Let and be two nonnegative continuous functions on the interval . One supposes that there exists such that , , , and . Then, for any , there exists such that equality (1) holds.

Proof. First, for every , we have Hence, is nonnegative; then

and we obtain

Equality in any side of (6) holds if is constant or Similarly, we obtain

We obtain equality in any side of (7) if or is constant.
Let Let us consider the continuous function defined, for any , by

Then

From (6) and (7), we obtain In the same mode, we have Then, there exists between and such that We obtain (1).
It remains to show that If , then the conclusion is clear. We suppose that Using (6) and (7), we conclude that and are constant functions. We obtain , for any Then, we can replace with any point from A similar conclusion is obtained if Now, the proof is complete.

As consequences of the previous theorem, we obtain the following result.

Theorem 3. Let and be two nonnegative continuous functions on the interval and monotone of the same type. Let Then there exists such that equality (1) holds.

Proof. We suppose that and are increasing functions. We have , , , and The conclusion follows by applying the previous theorem.

4. A Stability Result

The parents of the stability concept are considered to be the mathematicians Ulam and Hyers (see [8–10]). This notion is associated with the functional equations, the differential equations, or the linear recurrences. Starting with [11], a new direction was created. It is about the stability of the point arising from the mean value theorems. Reference [12] or [13] is relevant.

We want to conclude this paper with a new stability result. It is about the Hyers-Ulam stability of the mean point defined by Theorem 3. We apply a similar reasoning to Theorem from [13].

Theorem 4. Let Let and be two nonnegative continuous functions on the interval and monotone of the same type. One assumes that there exists a unique which is satisfying equality (1). Then, for any , there exists with the following property: for any continuous functions such that and , for any , there exists such that and

Proof. We assume that the functions and are increasing. We define the function by

for any We have Moreover,

The uniqueness of goes to Similar arguments show that Then , for any , and , for any
Now, let For any continuous functions such that and , for any , we define the function by

for any Further,

and we obtain

for any In the same mode, we obtain

for any Moreover, we have

so

By using (15), (16), and (18), we obtain

for any
Now, let and such that Similarly, let such that Then We consider such that

We consider such that

Further, , so Similarly, we have , so Then, there exists such that From , we obtain The equality goes to

and the proof is complete.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work has been funded by the Sectoral Operational Programme Human Resources Development 2007–2013 of the Ministry of European Funds through the Financial Agreement POSDRU 187/1.5/S/155420.

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