Abstract

We prove the generalized Hyers-Ulam stability of the first-order linear homogeneous matrix differential equations . Moreover, we apply this result to prove the generalized Hyers-Ulam stability of the th order linear differential equations with variable coefficients.

1. Introduction

For a given positive integer , let be a nondegenerate interval of . We will consider the (linear) differential equation of th orderdefined on , where is an times continuously differentiable function.

For an arbitrary , assume that an times continuously differentiable function satisfies the differential inequalityfor all . If for each function , satisfying inequality (2), there exists a solution of the differential equation (1) such that for any , where depends on only and satisfies , then we say that the differential equation (1) satisfies (or has) the Hyers-Ulam stability (or the local Hyers-Ulam stability if the domain is not the whole space ). If the above statement also holds when we replace and with some appropriate and , respectively, then we say that the differential equation (1) satisfies the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability). For more detailed definition of the Hyers-Ulam stability, refer to [1].

Obłoza seems to be the first author who investigated the Hyers-Ulam stability of linear differential equations (see [2, 3]). Given real-valued constants and , let be continuous functions with . Assume that is an arbitrary real number. Obłoza proved that if a differentiable function satisfies the inequality for all and if a function satisfies for all and for some , then there exists a constant such that for all .

Thereafter, Alsina and Ger [4] proved that if a differentiable function satisfies the differential inequality , then there exists a function such that and for all . This result of Alsina and Ger was generalized by Takahasi et al. [5]. Indeed, they proved the Hyers-Ulam stability of the Banach space valued differential equation (see also [6–12]).

In this paper, we prove the generalized Hyers-Ulam stability of the first-order linear homogeneous matrix differential equations:Moreover, we apply this result to prove the generalized Hyers-Ulam stability of the th order linear differential equations with variable coefficients.

2. Preliminaries

In what follows, let be an arbitrary open interval with and let be a fixed integer larger than . The first-order linear inhomogeneous differential equation with matrix coefficient is of the formwhere is an unknown vector function to be determined, is an matrix whose entries are complex-valued continuous functions on , and is a continuous column vector function.

According to [13, Section ], if the coefficient matrix is an matrix whose entries are complex-valued continuous functions, then there exist linearly independent solutions to the homogeneous matrix differential equation (4) and each of its solutions can be written in the formwhere is a constant column vector and is a fundamental matrix whose columns consist of linearly independent solutions to the homogeneous differential equation (4); that is, is an matrix defined by

In view of [13, Section ], the general solution to the inhomogeneous differential equation (5) is given bywhere is a fundamental matrix for the corresponding homogeneous differential equation (4) and is an arbitrarily given real number included in .

Let be a complex normed space, where each of its elements is a column vector, and let be a vector space consisting of all complex matrices. We choose a norm on which is compatible with ; that is, both norms obeyfor all and .

Some of the most important matrix norms are induced by -norms. For , the matrix norm induced by the -norm, is called the matrix -norm. For example, we get It is known that the matrix -norm, together with the -norm, satisfies the conditions in (9), wherefor any .

3. Generalized Hyers-Ulam Stability of (4)

We prove the main theorem of this paper concerning the generalized Hyers-Ulam stability of the first-order linear homogeneous matrix differential equation (4).

Theorem 1. Assume that is an arbitrary open interval with . Given a fixed integer , let and be complex normed spaces with the property (9), whose elements are column vectors, respectively, complex matrices. Assume that the coefficient matrix is an matrix whose entries are complex-valued continuous functions on . Let be an arbitrary real number included in and a continuous function. If a continuously differentiable vector function satisfies the inequalityfor all , then there exists a solution to the homogeneous matrix differential equation (4) such that for all , where is a fundamental matrix of the homogeneous matrix differential equation (4).

Proof. If we define a continuous vector function by , then it follows from (13) thatfor all .
In view of (5) and (8), we have for all , where is a constant column vector and is a fundamental matrix whose columns consist of linearly independent solutions to the homogeneous equation (4) (see [13, Section ]).
If we define a continuously differentiable vector function by for each , then is a solution to the homogeneous matrix differential equation (4) in view of (6). (We know that every function is a solution to (4).) Moreover, by (9) and (15), we have for any .

If we set , (a constant matrix), and with , then Theorem 1 yields the following corollary.

Corollary 2. Let and be complex normed spaces with property (9), whose elements are column vectors, respectively, complex matrices. Assume that the coefficient matrix is given by where ’s are complex numbers withAssume that is an arbitrary open interval with and is an arbitrary real number included in . Let be a continuous function. If a continuously differentiable vector function satisfies the inequalityfor all , then there exists a solution to the homogeneous matrix differential equation (4) with such that for all , where and are given in (33) and (34).

Proof. The complex numbers are the eigenvalues of and the vectors are the eigenvectors of corresponding to and , respectively.
In view of [14, Theorems 1.4.5 and 7.1.4], the conditions in (20) imply that is invertible and . Hence, the eigenvectors and are linearly independent.
If we set then is invertible and Due to a well known theorem of linear algebra, it holds thatIf we define a vector function byfor all , then is continuously differentiable, and it follows from (21) and (27) that for all .
According to Theorem 1, there exists a vector function satisfyingfor every , where is a fundamental matrix of the homogeneous matrix differential equation (30).
If we set then and are the solutions to (30) which are linearly independent. Therefore, the fundamental matrix of the matrix differential equation (30) may be expressed asThen, we havefor any .
If we set for all , then it follows from (27) and (30) that for all ; that is, is a solution to the matrix differential equation (4) with .
Finally, it follows from (28) and (31) that for all .

4. Applications

We now consider the linear th order differential equations with variable coefficientswhere are continuous functions.

In this section, we setfor all .

Theorem 3. Assume that is an arbitrary open interval with . For a fixed integer , assume that continuous functions are given. Let be an arbitrary real number included in and a continuous function. If an times continuously differentiable function satisfies the inequalityfor all , then there exists a solution to the homogeneous matrix differential equation (37) such that for all , where is a fundamental matrix of the first-order linear homogeneous matrix differential equation (46).

Proof. If we define continuous functions bythen it follows from (39) and (41) that and moreover for all . That is, we have for all , where the coefficient matrix is defined in (38), the vector norm is defined in (12), and the continuously differentiable vector function is defined byAccording to Theorem 1, there exists a continuously differentiable vector function satisfyingfor all , where is a fundamental matrix of the first-order linear homogeneous matrix differential equation (46).
We now setand then we consider the first elements of (46) to show thatMoreover, we use the last equality of (49) and then the last element of (46) to seefor all . The continuous differentiability of implies that is times continuously differentiable. Using (49) and (50), we further havefor every .
If we define a function by then it follows from (51) that the times continuously differentiable function is a solution to the homogeneous matrix differential equation (37). Moreover, by (45), (47), and (48), we get for all .