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 G₁ be a group and let G₂ be a metric group with the metric d(·, ·). Given ε > 0, does there exist a δ > 0 such that if a function h : G₁ → G₂ satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G₁, then there exists a homomorphism H : G₁ → G₂ with d(h(x), H(x)) < ε for all x ∈ G₁?

The case of approximately additive functions was solved by Hyers [2] under the assumption that G₁ and G₂ are the Banach spaces. Indeed, he proved that each solution of the inequality ∥f(x + y) − f(x) − f(y)∥ ≤ ε, for all x and y, can be approximated by an exact solution, say an additive function. In this case, the Cauchy additive functional equation, f(x + y) = f(x) + f(y), 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 c and t₀ be fixed real numbers with c > 0. For any differentiable function h : ℝ × ℝ → ℂ, the function defined as

()

is a solution of the wave equation

()

as we see

()

from which we know that u(x, t) satisfies the wave equation (3).

Conversely, we know that every solution u : ℝ × ℝ → ℂ of the wave equation (3) can be expressed by

()

where f, g : ℝ × ℝ → ℂ are arbitrary twice differentiable functions. If these f(x, t) and g(x, t) satisfy

()

for all x, t ∈ ℝ, then u(x, t) 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 φ(x, t) is a given function and u(x, t) is an arbitrary and continuous function which satisfies the integral inequality:

()

If there exist a function u₀(x, t) and a constant C > 0 such that

()

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

2. Preliminaries

For a nonempty set X, we introduce the definition of the generalized metric on X. A function d : X × X → [0, ∞] is called a generalized metric on X if and only if d satisfies

M₁ d(x, y) = 0 if and only if x = y;
M₂ d(x, y) = d(y, x) for all x, y ∈ X;
M₃ d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.

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 (X, d) be a generalized complete metric space. Assume that Λ : X → X is a strictly contractive operator with the Lipschitz constant L < 1. If there exists a nonnegative integer k such that d(Λ^k+1x, Λ^kx) < ∞ for some x ∈ X, then the following are true:

(a)
the sequence {Λⁿx} converges to a fixed point x^* of Λ;
(b)
x^* is the unique fixed point of Λ in
()
(c)
if y ∈ X^*, then
()

3. The Generalized Hyers-Ulam Stability

In the following theorem, for given real numbers a, b, c, and t₀ satisfying c > 0, t₀ > 0, and a + ct₀ < b − ct₀, let I : = [a, b], T : = (0, t₀], and I₀ : = [a + ct₀, b − ct₀] be finite intervals. Assume that L and M are positive constants with 0 < L < 1. Moreover, let φ : I × T → (0,1] be a continuous function satisfying

()

for all x ∈ I₀ and t ∈ T.

We denote by X the set of all functions f : I × T → ℂ with the following properties:

(a)
f(x, t) is continuous for all x ∈ I₀ and t ∈ T;
(b)
f(x, t) = 0 for all x ∈ I∖I₀ and t ∈ T;
(c)
|f(x, t)| ≤ Mφ(x, t) for all x ∈ I₀ and t ∈ T.

Moreover, we introduce a generalized metric on X as follows:

()

Theorem 2. If a function u ∈ X satisfies the integral inequality:

()

for all x ∈ I₀ and t ∈ T, then there exists a unique function u₀ ∈ X which satisfies

()

for all x ∈ I₀ and t ∈ T.

Proof. First, we show that (X, d) is complete. Let {h_n} be a Cauchy sequence in (X, d). Then, for any ε > 0 there exists an integer N_ε > 0 such that d(h_m, h_n) ≤ ε for all m, n ≥ N_ε. In view of (13), we have

()

If x and t are fixed, (17) implies that {h_n(x, t)} is a Cauchy sequence in ℂ. Since ℂ is complete, {h_n(x, t)} converges for any x ∈ I₀ and t ∈ T. Thus, considering (b), we can define a function h : I × T → ℂ by

()

Since φ is bounded on I₀ × T, (17) implies that

converges uniformly to

in the usual topology of ℂ. Hence, h is continuous and |h| is bounded on I₀ × T with an upper bound Mφ(x, t); that is, h ∈ X. (It has not been proved yet that {h_n} converges to h in (X, d).)

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

()

By considering (13), we get

()

This implies that the Cauchy sequence {h_n} converges to h in (X, d). Hence, (X, d) is complete.

We now define an operator Λ : X → X by

()

for all h ∈ X. Then, according to the fundamental theorem of calculus, Λh is continuous on I₀ × T. Furthermore, it follows from (12), (c), and (21) that

()

for any x ∈ I₀ and t ∈ T. Hence, we conclude that Λh ∈ X.

We assert that Λ is strictly contractive on X. Given any f, g ∈ X, let C_fg ∈ [0, ∞] be an arbitrary constant with d(f, g) ≤ C_fg. That is,

()

for all x ∈ I₀ and t ∈ T. Then, it follows from (12), (21), and (23) that

()

for all x ∈ I₀ and t ∈ T. That is, d(Λf, Λg) ≤ LC_fg. Hence, we may conclude that d(Λf, Λg) ≤ Ld(f, g) for any f, g ∈ X and we note that 0 < L < 1.

We prove that the distance between the first two successive approximations of Λ is finite. Let h₀ ∈ X be given. By (b), (c), and (13) and from the fact that Λh₀ ∈ X, we have

()

for any x ∈ I₀ and t ∈ T. Thus, (13) implies that

()

Therefore, it follows from Theorem 1(a) that there exists a u₀ ∈ X such that Λⁿh₀ → u₀ in (X, d) and Λu₀ = u₀.

In view of (c) and (13), it is obvious that {f ∈ X∣d(h₀, f) < ∞} = X, where h₀ was chosen with the property (26). Now, Theorem 1(b) implies that u₀ is the unique element of X which satisfies (Λu₀)(x, t) = u₀(x, t) for any x ∈ I₀ and t ∈ T.

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

()

since (14) means that d(Λu, u) ≤ 1. In view of (13), we can conclude that (16) holds for all x ∈ I₀ and t ∈ T.

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

1 Ulam S. M., A Collection of Mathematical Problems, 1960, Interscience Publishers, New York, NY, USA.
Web of Science® Google Scholar
2 Hyers D. H., On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America. (1941) 27, no. 4, 222–224, https://doi.org/10.1073/pnas.27.4.222.
10.1073/pnas.27.4.222
CAS PubMed Web of Science® Google Scholar
3 Rassias T. M., On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society. (1978) 72, 297–300, https://doi.org/10.1090/S0002-9939-1978-0507327-1.
10.1090/S0002-9939-1978-0507327-1
Web of Science® Google Scholar
4 Aoki T., On the stability of the linear transformation in Banach spaces, Journal of the Mathematical Society of Japan. (1950) 2, no. 1-2, 64–66, https://doi.org/10.2969/jmsj/00210064.
10.2969/jmsj/00210064
Google Scholar
5 Cădariu L., The generalized Hyers-Ulam stability for a class of the Volterra nonlinear integral equations, Buletinul Stiintific al Universitatii “Politehnica” din Timisoara. (2011) 56, no. 70, 30–38.
Google Scholar
6 Cǎdariu L., Gǎvruţa L., and Gǎvruţa P., Weighted space method for the stability of some nonlinear equations, Applicable Analysis and Discrete Mathematics. (2012) 6, no. 1, 126–139, https://doi.org/10.2298/AADM120309007C.
10.2298/AADM120309007C
Web of Science® Google Scholar
7 Forti G. L., Hyers-Ulam stability of functional equations in several variables, Aequationes Mathematicae. (1995) 50, no. 1-2, 143–190, 2-s2.0-0000469257, https://doi.org/10.1007/BF01831117, ZBL0836.39007.
10.1007/BF01831117
Google Scholar
8 Găvrută P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, Journal of Mathematical Analysis and Applications. (1994) 184, no. 3, 431–436, 2-s2.0-0001036579, https://doi.org/10.1006/jmaa.1994.1211, ZBL0818.46043.
10.1006/jmaa.1994.1211
Web of Science® Google Scholar
9 Hyers D. H., Isac G., and Rassias M. Th., Stability of Functional Equations of Several Variables, 1998, Birkhäuser, Basel, Switzerland.
10.1007/978-1-4612-1790-9
Google Scholar
10 Hyers D. H. and Rassias T. M., Approximate homomorphisms, Aequationes Mathematicae. (1992) 44, no. 2-3, 125–153, 2-s2.0-0002713514, https://doi.org/10.1007/BF01830975, ZBL0806.47056.
10.1007/BF01830975
Google Scholar
11 Jung S.-M., Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, 2011, 48, Springer, New York, NY, USA, Springer Optimization and Its Applications.
10.1007/978-1-4419-9637-4
Google Scholar
12 Rassias T. M., On the stability of functional equations and a problem of Ulam, Acta Applicandae Mathematicae. (2000) 62, no. 1, 23–130, 2-s2.0-0002028296, https://doi.org/10.1023/A:1006499223572, ZBL0981.39014.
10.1023/A:1006499223572
Web of Science® Google Scholar
13 Cădariu L. and Radu V., On the stability of the Cauchy functional equation: a fixed point approach, Grazer Mathematische Berichte. (2004) 346, 43–52.
Google Scholar
14 Akkouchi M., Hyers-Ulam-Rassias stability of nonlinear Volterra integral equations via a fixed point approach, Acta Universitatis Apulensis. (2011) 26, 257–266.
Google Scholar
15 Cădariu L. and Radu V., Fixed points and the stability of Jensen′s functional equation, Journal of Inequalities in Pure and Applied Mathematics. (2003) 4, no. 1, 4.
Google Scholar
16 Castro L. P. and Ramos A., Hyers-Ulam-Rassias stability for a class of nonlinear Volterra integral equations, Banach Journal of Mathematical Analysis. (2009) 3, no. 1, 36–43, 2-s2.0-65549156647.
10.15352/bjma/1240336421
Web of Science® Google Scholar
17 Ciepliński K., Applications of fixed point theorems to the Hyers-Ulam stability of functional equations-a survey, Annals of Functional Analysis. (2012) 3, no. 1, 151–164.
10.15352/afa/1399900032
Web of Science® Google Scholar
18 Jung S.-M., A fixed point approach to the stability of isometries, Journal of Mathematical Analysis and Applications. (2007) 329, no. 2, 879–890, 2-s2.0-33846599774, https://doi.org/10.1016/j.jmaa.2006.06.098, ZBL1153.39309.
10.1016/j.jmaa.2006.06.098
Web of Science® Google Scholar
19 Radu V., The fixed point alternative and the stability of functional equations, Fixed Point Theory. (2003) 4, no. 1, 91–96.
Google Scholar
20 Jung S.-M., A fixed point approach to the stability of a volterra integral equation, Fixed Point Theory and Applications. (2007) 2007, 9, 2-s2.0-34848815053, https://doi.org/10.1155/2007/57064, 57064, ZBL1155.45005.
10.1155/2007/57064
Google Scholar
21 Margolis B. and Diaz J., A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bulletin of the American Mathematical Society. (1968) 74, 305–309, https://doi.org/10.1090/S0002-9904-1968-11933-0.
10.1090/S0002-9904-1968-11920-2
Web of Science® Google Scholar

Citing Literature

All articles

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

Abstract

1. Introduction

2. Preliminaries

3. The Generalized Hyers-Ulam Stability

Conflict of Interests

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley