An Operator Method for the Stability of Inhomogeneous Wave Equations

: In this paper, we will apply the operator method to prove the generalized Hyers-Ulam stability of the wave equation, u tt ( x , t ) − c 2 (cid:52) u ( x , t ) = f ( x , t ) , for a class of real-valued functions with continuous second partial derivatives. Finally, we will discuss the stability more explicitly by giving examples.


Introduction
Ulam [1] asked the following question: Under what conditions does there exist an additive function near an approximately additive function? in 1941, Hyers [2] provided an answer for this question, that for all ε ≥ 0 there exists an exact additive function F, such that f (x) − F(x) ≤ ε for all x, if a function f satisfies the inequality f (x + y) − f (x) − f (y) ≤ ε for all x. This theorem of Hyers was the origination for the terminology of the Hyers-Ulam stability.
In this paper, we will study the wave equation in R n : where c > 0 is a constant, = is the Laplace operator, and x = (x 1 , . . . , x n ) ∈ R n . Actually, Choi and Jung [17] investigated the Hyers-Ulam stability of (1) by using the method of dilation invariance. For the fractional calculus, wavelet analysis, and fractal geometry, we refer the reader to [24][25][26][27].
In this paper, we will apply the operator method instead of the method of dilation invariance for investigating the generalized Hyers-Ulam stability of (1). One of the advantages of this present paper over [17] is that there are no limiting conditions in the definitions of U n 1 and U n 2 (see Section 3). We will also consider a more general form of the source term f (x, t) than those in [17] to see the Hyers-Ulam stability of the Equation (1). In addition, concerning the domains of relevant functions, as [17] give a partial answer to the open problem raised in ( [20] Remark 3), this paper will attempt to give a partial answer as well.

Preliminaries
In this section, we will introduce a modified version of ( [18] Theorem 1) which is more suitable for practical applications. This modified version will be applied many times to the proof of our main theorems in the next section (cf. [12] Theorem 2.2). Indeed, the hypotheses of the original theorem ( [18] Theorem 1) were formulated with a instead of a 0 , which imposes a constraint on its usability. The proof of Theorem 1 precisely follows the lines of the proof of ([18] Theorem 1)-hence, we omit the proof. Moreover, assume that ϕ : I → [0, ∞) is a function such that: If a continuously differentiable function v : I → X satisfies the differential inequality: for all t ∈ I, then there exists a unique, continuously differentiable function v 0 :

Main Results
In this section, n is a fixed positive integer, and each point x in R n is expressed as x = (x 1 , . . . , x i , . . . , x n ), where x i denotes the ith coordinate of x. Moreover, |x| denotes the Euclidean distance of x from the origin: Since the propagation speed of each solution u(x, t) to the wave Equation (1) is c, the "shape" of the wave travels at the speed of c. Roughly speaking, u(x, t) seems to have a similar shape at each (x, t), provided |x|/t = c. That is, u(x, t) depends on x and t, mainly through the term |x|/t. For this reason, we will search for approximate solutions to (1), which belong to a special class of scalar functions of the form u(x, t) = tv(|x|/t), where v is a twice continuously differentiable function. Such a method will be called the "method of dilation invariance"-see [17,19,20].
Based on this argument, we define: We may compare these definitions above with the definitions of [17]. It is obvious that (x, t) ∈ D n 1 if, and only if 0 < |x|/t < c. The conditions (2) and (5) may seem to be too strict at first glance. However, we shall see in Corollary 1 that they are not as strict as they look.
exists for a fixed constant r 0 ∈ (0, c) and there exists a positive real number k with: Assume that f : D n 1 → R is a function for which there exists a continuous function g : (0, c) → R such that: exists for all (x, t) ∈ D n 1 . If a u ∈ U n 1 satisfies the inequality: for all (x, t) ∈ D n 1 , then there exists a solution u 0 ∈ U n 1 of the wave Equation (1), such that: for all (x, t) ∈ D n 1 .
Therefore, in view of (4) and (9), we have: and hence, by (6), we get and by considering (3), we further obtain: for all r ∈ (0, c). (Indeed, for each fixed r ∈ (0, c) and for any t > 0, we can select an x t ∈ R n \ {(0, . . . , 0)} such that (x t , t) ∈ D n 1 and r = |x t |/t). We now apply a method for a decomposition of a second-order differential operator into two differential operators of first-order. Let us define the second-order differential operator L 2 n : C 2 (0, c) → C(0, c) by: where C(0, c) and C 2 (0, c) denote the set of all continuous real-valued functions and the set of all twice continuously differentiable real-valued functions defined on (0, c), respectively. We tried to decompose the differential operator L 2 n into the differential operators L − A(r) and L +
We can now apply Theorem 1 to our inequality (17) by considering the substitutions, as seen in the Table 1.
Considering the above table, we easily verify that the following integral exists for any r ∈ (0, c). Thus, the condition (i) of Theorem 1 is fulfilled. Moreover, by considering the Hypotheses (2) and (5) guarantee the validity of conditions (ii) and (iii) of Theorem 1, respectively. According to Theorem 1 and (17), there exists a unique continuously differentiable function w 0 : (0, c) → R, such that: and dy for all r ∈ (0, c). It follows from (16) and the last inequality that: for any r ∈ (0, c). We can again apply Theorem 1 to our inequality (20) by considering the substitutions seen in the following Table 2.
c+r the right side of (20) By substituting ω = c sin θ (0 < θ < π/2), we see that the integral r r 0 exists for every r ∈ (0, c). Indeed, according to the table of integrals (e.g., see [29] §2.518, §2.521), the last integral exists for any given r, r 0 ∈ (0, c). By considering the above table, (2), (5) and (18), we conclude that the conditions (i), (ii), and (iii) of Theorem 1 are fulfilled. Due to Theorem 1 and (20), there exists a unique continuously differentiable function v 0 : (0, c) → R such that: for any r ∈ (0, c). Indeed, on account of (22) and the continuous differentiability of w 0 (r), v 0 (r) is a twice continuously differentiable function. If we define a function u 0 : D n 1 → R by u 0 (x, t) := tv 0 (r), then u 0 ∈ U n 1 , and inequality (7) follows immediately from (8) and (23). Furthermore, by using (19) and (22) and by following the first part of this proof, we can show that u 0 (x, t) is a solution to the wave Equation (1).
In view of (21), if there exist positive real constants k 1 and k 2 such that and |g(r)| ≤ k 2 r 2 c 2 for all r ∈ (0, c), then the conditions (2) and (5) of Theorem 2 are satisfied.
We may compare these definitions with those of D n 2 and U n 2 in [17]. Moreover, we see that (x, t) ∈ D n 2 if, and only if c < |x|/t < ∞. Even if the conditions (26) and (29) below seem somewhat strict at first glance, they are indeed not so strict, as we shall see in Corollary 2.
the hypotheses (26) and (29) ensures the validity of conditions (iii) and (ii) of Theorem 1, respectively. considered in this paper. As can be clearly seen, the main results of this paper, Theorems 2 and 3, are also quite symmetrical to each other.