Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 309098, 16 pages

http://dx.doi.org/10.1155/2013/309098

Research Article

Stability of a Generalized Trigonometric-Quadratic Functional Equation

Janyarak Tongsomporn¹ and Vichian Laohakosol²

¹School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand

²Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand

Received 28 May 2013; Accepted 25 September 2013

Academic Editor: Jaeyoung Chung

Copyright © 2013 Janyarak Tongsomporn and Vichian Laohakosol. 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.

A stability result for the generalized trigonometric-quadratic functional equation over the domain of a normed space and the range of a Banach space is derived. It states that under appropriate conditions on the function , the four functions , , , and are approximately equal to combinations of certain quadratic and additive functions.

1. Introduction

A generalized trigonometric functional equation is a functional equation of the form

which is so named because the two best known trigonometric functions, sine and cosine, are solutions of special cases of this equation. In [1], the authors investigated the stability of (1), where , , , and are nonzero functions from an abelian group to the complex field. Later in the same year, a generalized trigonometric functional equation with either the function or being bounded was investigated to complement the above earlier work where such function boundedness was not treated. A generalized quadratic functional equation is a functional equation of the form

which is so named because the quadratic function is a solution of one of its particular cases, namely,

and any solution of (3) is referred to as a quadratic function. The stability of the quadratic functional equation was first proved by Skof [2] in 1983 for functions from a normed space to a Banach space. In 1984, Cholewa [3] demonstrated that Skof’s theorem is also valid if a normed space is replaced by an abelian group. Later, Czerwik [4] extended Cholewa’s result by changing a control function and considering functions from a normed space to a Banach space. In another paper, [5], Czerwik investigated the stability of the “partially pexiderized” quadratic functional equation of the form . In 2000, Jung [6] considered (2), where, , , and are functions from a normed space to a Banach space . In the paper [7], the authors considered the stability of a combined generalized trigonometric-quadratic functional equation of the form

over the domain of an abelian group and the range , except the control function whose range is taken to be the nonnegative real numbers . There, the functions and are explicitly determined under certain restrictions on the remaining functions , , , and . It is natural to ask whether the restrictions on the functions , , , and can be removed and/or altered to obtain some stability result of (4). We answer this question affirmatively using Jung’s techniques elaborated in [6, 8].

2. Notation

We collect in this section the notation and terminology that will be kept standard throughout. A normed space is the domain and a Banach space is the range of all functions involved. Let . Define

Let be a given function satisfying the following three conditions (i) , (ii) , and (iii) there is an integer such that

For , define

Define a real sequence by

and let

Clearly, the functions , satisfy the conditions (i), (ii), and (iii).

3. Auxiliary Results

We start with an important lemma.

Lemma 1. If , , satisfies

then one has, for integer , and ,

Proof. We prove the lemma by induction on . To verify the case , we use induction on . Putting in (10), we obtain

that is, the assertion is true for . Assume that the statement is true for all . Replacing and in (10) by and , we get

and so

and we are done for . Assuming that the lemma is true for some , we have

verifying the lemma in general.

We come now to our first main proposition.

Proposition 2. Assume that satisfies

When for all , one further assumes that . Then there exists a unique quadratic function satisfying

provided that and for all , or

provided that and for all . Moreover, if is measurable or is continuous in for every fixed , then the function satisfies

Proof. From Lemma 1, we have

For with , from (20), we obtain

In (21), since the last term can be made as small as we wish by choosing sufficiently large, we deduce that is a Cauchy sequence in . Since is a Banach space, the limit

is well defined. Using and (20), the validity of the inequality (17) is easily verified. Next, replacing , in (16) by , , respectively, dividing the resulting inequality by , and using , , we get

which implies that is a quadratic function. If is another quadratic function satisfying the inequality (17), since and are quadratic functions, we have

It thus follows from (24), (17), , and that

that is, .

The other provision and is treated analogously by replacing by in (11) to obtain an upper bound for , which is then used to show that is a Cauchy sequence in so that the limit

is well defined and satisfies (18). Finally, the proof that (19) holds when is measurable or is continuous is the same as that in [4].

We now proceed to our second main proposition.

Proposition 3. If satisfies

then there exists a unique additive function such that either

provided that for , one has and

or

provided that for , one has and

Moreover, if is continuous in for every fixed , then is linear.

Proof. First, we show that

Putting in (27), we obtain

This shows that (32) holds for . Assume that (32) is valid for ; that is,

Replacing by in (27) and using (34), we have

verifying the case , and so the assertion (32) holds for all and all .

Next, we use induction on to show that

The case follows from (32). Assuming that this holds for , putting , and then substituting for in (36), we obtain

verifying (36) generally. Rewriting (36), we get

Using (38), , and the inequality

and proceeding as in the proof of Proposition 2, we deduce that is a Cauchy sequence. Thus, the limit

is well defined. To show that is additive, we substitute and for and in (27), respectively, to get

From and

it follows that

Using this last inequality, dividing both sides of (41) by and letting , we obtain .

According to (38), we have

To show that is unique, let be another additive mapping satisfying (44). Thus,

Using and

we see that the last term becomes arbitrarily small for sufficiently large , implying . Finally, the same proof as in [9] shows that is linear if is a Banach space and is continuous in for every fixed .

The other provision and

is treated analogously by replacing by in (36) to obtain

which is then used to show that is a Cauchy sequence in for all , so that the limit

is a well-defined additive function. Using

we can show that is unique. Finally, that is linear if is a Banach space and is continuous in for every fixed is proved by the same technique as that in [9].

4. Main Result

Theorem 4. Adopting the notation of Section 2, if the functions satisfy

then there exist a quadratic function and two additive functions such that, for all , one has

where

Moreover, if and are continuous in for each , then the two additive functions , are linear, while the quadratic function satisfies

Proof. Putting in (51), we obtain

For , setting

we get . Subtracting (51) and (55), we have

Substituting for and for in (57), we get

Combining (57) and (58), we obtain

Let

Note that , , , are the even parts of the functions , , , , respectively, and that

From (59), we obtain

Substituting in (62), we get

Putting in (62) and replacing by , we get

Putting in (62), we have

Substituting in (62), we get

From (64) and (63), we have

From (65) and (66) and replacing by , we obtain

From (62), (67), and (68), we get

Substituting in (69), we get

From (69) and (70), we obtain

and so

By Proposition 2, there is a unique quadratic function such that

If is continuous in for each , then the quadratic function satisfies . From (68), (70), and (73), we have

From (70) and (73), we get

From (67) and (73), we obtain

Next, we consider the odd part of functions. Subtracting (57) and (58), we obtain

For , let

Clearly, , , , are the odd parts of , , , , respectively. Note that

From (77), we get

Substituting in (80), we get

Putting in (80) and replacing by , we get

Letting in (80), we have

Substituting in (80), we get

From (81) and (82), we obtain

From (83) and (85), we have

From (84) and (86), we get

From (87) and (88), we have

In view of (80), (85), (86), and (89), we have

Substituting by in (90), we obtain

From (90) and (91), we have

Putting and in (92), we obtain

By Proposition 3, there exists a unique additive function such that

Moreover, if is continuous in for every fixed , we have that is a linear function. Putting and in (93), we get

Again by Proposition 3, there exists a unique additive function such that

Furthermore, if is continuous in for all , then is also linear. From (85), (95), and (97), it follows that

The theorem now follows from the inequalities (73), (74), (75), (76), (95), (97), and (98).

References

J. Tongsomporn, V. Laohakosol, C. Hengkrawit, and P. Udomkavanich, “Stability of a generalized trigonometric functional equation,” Journal of Computational and Applied Mathematics, vol. 234, no. 5, pp. 1448–1457, 2010.

F. Skof, “Local properties and approximation of operators,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113–129, 1983.

P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76–86, 1984.

S. Czerwik, “On the stability of the quadratic mapping in normed spaces,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 62, pp. 59–64, 1992.

S. Czerwik, “The stability of the quadratic functional equation,” in Stability of Mappings of Hyers-Ulam Type, Th. M. Rassias and J. Tabor, Eds., pp. 81–91, Hadronic Press, Palm Harbor, Fla, USA, 1994.

S. M. Jung, “Stability of the quadratic equation of Pexider type,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 70, pp. 175–190, 2000.

C. Hengkrawit, V. Laohakosol, and J. Tongsomporn, “A stability result for a generalized trigonometric-quadratic functional equation with one unbounded function,” WSEAS Transactions on Mathematics, vol. 10, no. 8, pp. 279–290, 2011.

S. M. Jung, “On the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 204, no. 1, pp. 221–226, 1996.

T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.