ISRN Mathematical AnalysisVolume 2012 (2012), Article ID 609754, 10 pagesdoi:10.5402/2012/609754

Research Article

Ulam-Hyers-Rassias Stability of a Hyperbolic Partial Differential Equation

Nicolaie Lungu¹ and Cecilia Crăciun²

¹Department of Mathematics, Technical University of Cluj-Napoca, Street C. Daicoviciu 15, 400020, Cluj-Napoca, Romania²Department of Mathematics, Colfe's School, Horn Park Lane, London SE12 8AW, UK

Received 21 October 2011; Accepted 30 November 2011

Academic Editors: K. H. Kwon and J. V. Stokman

Copyright © 2012 Nicolaie Lungu and Cecilia Crăciun. 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 consider a nonlinear hyperbolic partial differential equation in a general form. Using a Gronwall-type lemma we prove results on the Ulam-Hyers stability and the generalised Ulam-Hyers-Rassias stability of this equation.

1. Introduction

Results on Ulam stability for the functional equations are well known (see, e.g., [1–4]). In this paper we consider different types of Ulam stability of a nonlinear hyperbolic partial differential equation. Following results from Rus [5] and Rus and Lungu [6], we present some Ulam-Hyers and Ulam-Hyers-Rassias stabilities for the following equation: where and is a real or complex Banach space.

2. Preliminaries

In this section we present some definitions and results on different types of Ulam stability for a hyperbolic partial differential equation. We also present a Gronwall type lemma, which will be useful in the proof of the main results.

Let , , and be a real or complex Banach space.

We consider the following hyperbolic partial differential equation: where .

We also consider the following inequalities:

Definition 2.1. A function is a solution to (2.1) if and satisfies (2.1).

From [5, 6] we have the following definitions and results.

Definition 2.2. Equation (2.1) is Ulam-Hyers stable if there exist the real numbers , , and such that for any and for any solution to the inequality (2.2) there exists a solution to (2.1) with

Definition 2.3. Equation (2.1) is generalised Ulam-Hyers-Rassias stable if there exist the real numbers , , and such that for any and for any solution to the inequality (2.3) there exists a solution to (2.1) with

Remark 2.4. A function is a solution to the inequality (2.2) if and only if, there exists a function , which depends on , such that(i)For all , for all , for all ;(ii)For all , for all :

Remark 2.5. A function is a solution to the inequality (2.3) if and only if, there exists a function , which depends on , such that(i), for all , for all ;(ii)For all , for all :

Remark 2.6. A function is a solution to the inequality (2.4) if and only if, there exists a function , which depends on , such that(i)For all , for all , for all ;(ii)For all , for all :

Throughout this paper we denote

Theorem 2.7. If is a solution to the inequality (2.2), then satisfies the following system of integral inequalities: for all and .

Proof. By Remark 2.4 we have that and we have the following inequalities:

The following two theorems are obtained in a similar fashion.

Theorem 2.8. If is a solution to the inequality (2.3), then satisfies the following system of integral inequalities: for all and .

Theorem 2.9. If is a solution to the inequality (2.4), then satisfies the following system of integral inequalities: for all and .

The following Gronwall type lemma is an important tool in proving the main results of this paper.

Lemma 2.10 (see [7], also [8]). One assumes that (i); (ii)for any one has (iii) is positive and nondecreasing.Then,

3. Ulam-Hyers Stability

In this section we present a result on the existence and uniqueness of the solution to (2.1) and derive a result on Ulam-Hyers stability for the same equation in the case of and .

Theorem 3.1. One assumes that (i), ; (ii);(iii)there exists such that for all , and . Then, (a)for and (2.1) has a unique solution, which satisfies (b)(2.1) is Ulam-Hyers stable.

Proof. (a) This is a known result (see, e.g, [9, 10]).(b) Let be a solution to the inequality (2.2) and let be the unique solution to (2.1), which satisfies the following conditions: From Theorem 2.7, the hypothesis (iii), and Gronwall Lemma 2.10, it follows that Similarly we have

Remark 3.2. In general, if or , then (2.1) is not Ulam-Hyers stable.

4. Generalised Ulam-Hyers-Rassias Stability

In this section we prove the generalised Ulam-Hyers-Rassias stability of the hyperbolic partial differential equation (2.1). We consider (2.1) and the inequality (2.3) in the case and .

Theorem 4.1. One assumes that (i);(ii)there exists such that for all ;(iii)there exist such that (iv) is increasing.Then (2.1) ( and ) is generalised Ulam-Hyers-Rassias stable.

Proof. Let be a solution to the inequality (2.3). Denote by the unique solution to the Darboux problem: If is a solution to the Darboux problem (4.3), then is a solution to the following system: From Theorem 2.8 and the hypothesis (iii), it follows that Using (4.5) gives us From Lemma 2.10 it follows that where .Similarly, we have and from Lemma 2.10 we get where .Also, By using Lemma 2.10 we obtain where .So, (2.1) is generalised Ulam-Hyers-Rassias stable.

References

L. Cădariu and V. Radu, “The fixed points method for the stability of some functional equations,” Carpathian Journal of Mathematics, vol. 23, no. 1-2, pp. 63–72, 2007.

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

D. Popa, Functional Equations. Set-Valued Solutions Stability, Technical University Press, Cluj-Napoca, Romania, 2006.

D. Popa, “On the stability of the general linear equation,” Results in Mathematics, vol. 53, no. 3-4, pp. 383–389, 2009.

I. A. Rus, “Ulam stability of ordinary differential equations,” Studia. Universitatis Babeş-Bolyai. Mathematica, vol. 54, no. 4, pp. 125–133, 2009.

I. A. Rus and N. Lungu, “Ulam stability of a nonlinear hyperbolic partial differential equation,” Carpathian Journal of Mathematics, vol. 24, no. 3, pp. 403–408, 2008.

V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, vol. 125, Marcel Dekker Inc., New York, NY, USA, 1989.

B. G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, San Diego, Calif, USA, 1998.

N. Lungu, Qualitative Problems in the Theory of Hyperbolic Differential Equations, Digital Data, Cluj-Napoca, Romania, 2006.

N. Lungu and I. A. Rus, “Hyperbolic differential inequalities,” Libertas Mathematica, vol. 21, pp. 35–40, 2001.