Abstract and Applied Analysis

Volume 2013 (2013), Article ID 612576, 4 pages

http://dx.doi.org/10.1155/2013/612576

Research Article

A Fixed Point Approach to the Stability of an Integral Equation Related to the Wave Equation

Soon-Mo Jung

Mathematics Section, College of Science and Technology, Hongik University, Sejong 339-701, Republic of Korea

Received 24 July 2013; Accepted 16 October 2013

Academic Editor: Hichem Ben-El-Mechaiekh

Copyright © 2013 Soon-Mo Jung. 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.

We will apply the fixed point method for proving the generalized Hyers-Ulam stability of the integral equation which is strongly related to the wave equation.

1. Introduction

In 1940, Ulam [1] gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms:

 Let be a group and let be a metric group with the metric . Given , does there exist a such that if a function satisfies the inequality for all , then there exists a homomorphism with for all ?

The case of approximately additive functions was solved by Hyers [2] under the assumption that and are the Banach spaces. Indeed, he proved that each solution of the inequality , for all and , can be approximated by an exact solution, say an additive function. In this case, the Cauchy additive functional equation, , is said to have the Hyers-Ulam stability.

Rassias [3] attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows:

and proved the Hyers theorem. That is, Rassias proved the generalized Hyers-Ulam stability (or Hyers-Ulam-Rassias stability) of the Cauchy additive functional equation. (Aoki [4] has provided a proof of a special case of Rassias’ theorem just for the stability of the additive function. Aoki did not prove the stability of the linear function, which was implied by Rassias’ theorem.) Since then, the stability of several functional equations has been extensively investigated [5–12].

The terminologies generalized Hyers-Ulam stability, Hyers-Ulam-Rassias stability, and Hyers-Ulam stability can also be applied to the case of other functional equations, differential equations, and various integral equations.

Let and be fixed real numbers with . For any differentiable function , the function defined as

is a solution of the wave equation

as we see

from which we know that satisfies the wave equation (3).

Conversely, we know that every solution of the wave equation (3) can be expressed by

where are arbitrary twice differentiable functions. If these and satisfy

for all , then expressed by (5) satisfies the integral equation (7). These facts imply that the integral equation (7) is strongly connected with the wave equation (3).

Cădariu and Radu [13] applied the fixed point method to the investigation of the Cauchy additive functional equation. Using such a clever idea, they could present another proof for the Hyers-Ulam stability of that equation [14–19].

In this paper, we introduce the integral equation:

which may be considered as a special form of (2), and prove the generalized Hyers-Ulam stability of the integral equation (7) by using ideas from [13, 15, 19, 20]. More precisely, assume that is a given function and is an arbitrary and continuous function which satisfies the integral inequality:

If there exist a function and a constant such that

then we say that the integral equation (7) has the generalized Hyers-Ulam stability.

2. Preliminaries

For a nonempty set , we introduce the definition of the generalized metric on . A function is called a generalized metric on if and only if satisfies

() if and only if ; () for all ; () for all .

We remark that the only one difference of the generalized metric from the usual metric is that the range of the former is permitted to include the infinity.

We now introduce one of fundamental results of fixed point theory. For the proof, we refer to [21]. This theorem will play an important role in proving our main theorems.

Theorem 1. Let be a generalized complete metric space. Assume that is a strictly contractive operator with the Lipschitz constant . If there exists a nonnegative integer such that for some , then the following are true:

(a)the sequence converges to a fixed point of ; (b) is the unique fixed point of in

(c)if , then

3. The Generalized Hyers-Ulam Stability

In the following theorem, for given real numbers , , , and satisfying , , and , let , , and be finite intervals. Assume that and are positive constants with . Moreover, let be a continuous function satisfying

for all and .

We denote by the set of all functions with the following properties:

(a) is continuous for all and ; (b) for all and ; (c) for all and .

Moreover, we introduce a generalized metric on as follows:

Theorem 2. If a function satisfies the integral inequality:

for all and , then there exists a unique function which satisfies

for all and .

Proof. First, we show that is complete. Let be a Cauchy sequence in . Then, for any there exists an integer such that for all . In view of (13), we have

If and are fixed, (17) implies that is a Cauchy sequence in . Since is complete, converges for any and . Thus, considering , we can define a function by

Since is bounded on , (17) implies that converges uniformly to in the usual topology of . Hence, is continuous and is bounded on with an upper bound ; that is, . (It has not been proved yet that converges to in .)

If we let increase to infinity, it follows from (17) that

By considering (13), we get

This implies that the Cauchy sequence converges to in . Hence, is complete.

We now define an operator by

for all . Then, according to the fundamental theorem of calculus, is continuous on . Furthermore, it follows from (12), , and (21) that

for any and . Hence, we conclude that .

We assert that is strictly contractive on . Given any , let be an arbitrary constant with . That is,

for all and . Then, it follows from (12), (21), and (23) that

for all and . That is, . Hence, we may conclude that for any and we note that .

We prove that the distance between the first two successive approximations of is finite. Let be given. By , , and (13) and from the fact that , we have

for any and . Thus, (13) implies that

Therefore, it follows from Theorem 1 that there exists a such that in and .

In view of and (13), it is obvious that , where was chosen with the property (26). Now, Theorem 1 implies that is the unique element of which satisfies for any and .

Finally, Theorem 1, together with (13) and (14), implies that

since (14) means that . In view of (13), we can conclude that (16) holds for all and .

Remark 3. Even though condition (12) seems to be strict, the condition can be satisfied provided that and are chosen so that is small enough and is a large number.

Conflict of Interests

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

Acknowledgments

The author would like to express his cordial thanks to the referees for useful remarks. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2005557).

References

S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, NY, USA, 1960.

D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, pp. 222–224, 1941.

T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, pp. 297–300, 1978.

T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, no. 1-2, pp. 64–66, 1950.

L. Cădariu, “The generalized Hyers-Ulam stability for a class of the Volterra nonlinear integral equations,” Buletinul Stiintific al Universitatii “Politehnica” din Timisoara, vol. 56, no. 70, pp. 30–38, 2011.

L. Cǎdariu, L. Gǎvruţa, and P. Gǎvruţa, “Weighted space method for the stability of some nonlinear equations,” Applicable Analysis and Discrete Mathematics, vol. 6, no. 1, pp. 126–139, 2012.

G. L. Forti, “Hyers-Ulam stability of functional equations in several variables,” Aequationes Mathematicae, vol. 50, no. 1-2, pp. 143–190, 1995.

P. Găvrută, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994.

D. H. Hyers, G. Isac, and M. Th. Rassias, Stability of Functional Equations of Several Variables, Birkhäuser, Basel, Switzerland, 1998.

D. H. Hyers and T. M. Rassias, “Approximate homomorphisms,” Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125–153, 1992.

S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2011.

T. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23–130, 2000.

L. Cădariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,” Grazer Mathematische Berichte, vol. 346, pp. 43–52, 2004.

M. Akkouchi, “Hyers-Ulam-Rassias stability of nonlinear Volterra integral equations via a fixed point approach,” Acta Universitatis Apulensis, vol. 26, pp. 257–266, 2011.

L. Cădariu and V. Radu, “Fixed points and the stability of Jensen's functional equation,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, Article ID 4, 2003.

L. P. Castro and A. Ramos, “Hyers-Ulam-Rassias stability for a class of nonlinear Volterra integral equations,” Banach Journal of Mathematical Analysis, vol. 3, no. 1, pp. 36–43, 2009.

K. Ciepliński, “Applications of fixed point theorems to the Hyers-Ulam stability of functional equations-a survey,” Annals of Functional Analysis, vol. 3, no. 1, pp. 151–164, 2012.

S.-M. Jung, “A fixed point approach to the stability of isometries,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 879–890, 2007.

V. Radu, “The fixed point alternative and the stability of functional equations,” Fixed Point Theory, vol. 4, no. 1, pp. 91–96, 2003.

S.-M. Jung, “A fixed point approach to the stability of a volterra integral equation,” Fixed Point Theory and Applications, vol. 2007, Article ID 57064, 9 pages, 2007.

B. Margolis and J. Diaz, “A fixed point theorem of the alternative for contractions on a generalized complete metric space,” Bulletin of the American Mathematical Society, vol. 74, pp. 305–309, 1968.