Ulam-Hyers-Rassias Stability of a Hyperbolic Partial Differential Equation
Nicolaie Lungu1and Cecilia Crăciun2
Academic Editor: J. V. Stokman, K. H. Kwon
Received21 Oct 2011
Accepted30 Nov 2011
Published15 Feb 2012
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.
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.