On Approximate Euler Differential Equations
We solve the inhomogeneous Euler differential equations of the form and apply this result to the approximation of analytic functions of a special type by the solutions of Euler differential equations.
The stability problem for functional equations starts from the famous talk of Ulam and the partial solution of Hyers to the Ulam's problem (see [1, 2]). Thereafter, Rassias  attempted to solve the stability problem of the Cauchy additive functional equation in a more general setting.
The stability concept introduced by Rassias’ theorem significantly influenced a number of mathematicians to investigate the stability problems for various functional equations (see [4–10] and the references therein).
Assume that and are a topological vector space and a normed space, respectively, and that is an open subset of . If for any function satisfying the differential inequality
for all and for some , there exists a solution of the differential equation
such that for any , where is an expression of only, then we say that the above differential equation satisfies the Hyers-Ulam stability (or the local Hyers-Ulam stability if the domain is not the whole space ). We may apply these terminologies for other differential equations. For more detailed definition of the Hyers-Ulam stability, refer to [1, 3, 5, 6, 8–11].
Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [12, 13]). Here, we will introduce a result of Alsina and Ger (see ): if a differentiable function is a solution of the differential inequality , where is an open subinterval of , then there exists a solution of the differential equation such that for any .
This result of Alsina and Ger has been generalized by Takahasi, Miura, and Miyajima: they proved in  that the Hyers-Ulam stability holds for the Banach space-valued differential equation (see also ).
Using the conventional power series method, the first author investigated the general solution of the inhomogeneous Hermite differential equation of the form
under some specific condition, where is a real number and the convergence radius of the power series is positive. This result was applied to prove that every analytic function can be approximated in a neighborhood of by a Hermite function with an error bound expressed by (see [17–20]).
In Section 2 of this paper, using power series method, we will investigate the general solution of the inhomogeneous Euler (or Cauchy) differential equation
where and are fixed complex numbers and the coefficients of the power series are given such that the radius of convergence is . Moreover, using the idea from [17–19], we will approximate some analytic functions by the solutions of Euler differential equations.
In this paper, denotes the set of all nonnegative integers.
2. General Solution of Inhomogeneous Euler Equations
The second-order Euler differential equation
which is sometimes called the second-order Cauchy differential equation, is one of the most famous differential equations and frequently appears in applications.
The quadratic equation
where and are complex constants (see [21, Section 2.7]).
Theorem 2.1. Let and be complex constants such that no root of the auxiliary equation (2.2) is a nonnegative integer. If the radius of convergence of power series is at least , then every solution of the inhomogeneous Euler differential equation (1.4) can be expressed by for all , where is a solution of the Euler differential equation (2.1).
Proof. Assume that is a function given by (2.4), where is a solution of the homogeneous Euler differential equation (2.1). We first prove that the function , defined by , satisfies the inhomogeneous equation (1.4).
Indeed, we have which proves that is a particular solution of the inhomogeneous equation (1.4). Moreover, each power series appearing in the above equalities has the same radius of convergence as (which can be verified by using the ratio test). Since every solution to (1.4) can be expressed as a sum of a solution of the homogeneous equation and a particular solution of the inhomogeneous equation, every solution of (1.4) is certainly of the form (2.4).
We will now apply the ratio test to the power series in (2.4). Indeed, by setting , we get and hence we have which implies that the power series given in (2.4) has the same radius of convergence as power series , which is at least . That is, given in (2.4) is well defined on its domain .
3. Approximate Euler Differential Equations
In this section, assume that and are complex constants and is a positive constant. For a given , we denote by the set of all functions with the properties (a) and (b):(a) is expressible by a power series whose radius of convergence is at least ; (b) for any where we set for .
Let be a sequence of positive real numbers such that the radius of convergence of the series is at least and let and satisfy either and or . If a function is defined by , then certainly belongs to with . So, the set is not empty if . In particular, if is small and is large, then is a large class of analytic functions .
We will now solve the approximate Euler differential equations in a special class of analytic functions, .
Theorem 3.1. Let and be complex constants such that no root of the auxiliary equation (2.2) is a nonnegative integer. If a function satisfies the differential inequality for all and for some , then there exists a solution of the Euler differential equation (2.1) such that for any .
Proof. Since belongs to , it follows from (a) and (b) that
for all . By considering (3.1) and (3.3), we get
for any . This inequality, together with (b), yields that
for each .
Now, suppose that an arbitrary is given. Then we can choose an arbitrary constant . By Abel's formula (see [22, Theorem 6.30] ), we have where we set Since for any and , it follows from (3.5) and (3.6) that If we let in the above inequality, then we obtain for all and for any . Since as , we get for every .
Finally, it follows from (3.3), Theorem 2.1, and (3.10) that there exists a solution of the Euler differential equation (2.1) such that for any .
4. An Example
We fix , and suppose that a small is given. We can choose a constant such that
We will consider the function which can be expressed by the power series , whose radius of convergence is .
If we set and for any , then it follows from (b) that
Thus, we have
for any , which enables us to choose . Thus, it holds that .
Moreover, we have
for all . It now follows from Theorem 3.1 that there exist complex numbers and with
for any .
The authors would like to express their cordial thanks to the referees for their useful remarks and constructive comments which have improved the first version of this paper. This work was supported by National Research Foundation of Korea Grant funded by the Korean Government (no. 2009-0071206).
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