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.


Introduction
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 3 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 X and Y are a topological vector space and a normed space, respectively, and that I is an open subset of X. If for any function f : I → Y satisfying the differential inequality a n x y n x a n−1 x y n−1 x · · · a 1 x y x a 0 x y x h x ≤ ε 1.1 for all x ∈ I and for some ε ≥ 0, there exists a solution f 0 : I → Y of the differential equation a n x y n x a n−1 x y n−1 x · · · a 1 x y x a 0 x y x h x 0 1.2 such that f x − f 0 x ≤ K ε for any x ∈ I, where K ε is an expression of ε only, then we say that the above differential equation satisfies the Hyers-Ulam stability or the local Hyers-Ulam stability if the domain I is not the whole space X . 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 14 : if a differentiable function f : I → R is a solution of the differential inequality |y x − y x | ≤ ε, where I is an open subinterval of R, then there exists a solution f 0 : I → R of the differential equation y x y x such that |f x − f 0 x | ≤ 3ε for any x ∈ I. This result of Alsina and Ger has been generalized by Takahasi, Miura, and Miyajima: they proved in 15 that the Hyers-Ulam stability holds for the Banach space-valued differential equation y x λy x see also 16 . 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 λ 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 0 by a Hermite function with an error bound expressed by Cx 2 e x 2 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 α and β are fixed complex numbers and the coefficients a m of the power series are given such that the radius of convergence is ρ > 0. Moreover, using the idea from 17-19 , we will approximate some analytic functions by the solutions of Euler differential equations. In this paper, N 0 denotes the set of all nonnegative integers.

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 c 1 and c 2 are complex constants see 21, Section 2.7 . which implies that the power series given in 2.4 has the same radius of convergence as power series ∞ m 0 a m x m , which is at least ρ. That is, y x given in 2.4 is well defined on its domain 0, ρ .

Approximate Euler Differential Equations
In this section, assume that α and β are complex constants and ρ is a positive constant. For a given K ≥ 0, we denote by C K the set of all functions y : 0, ρ → C with the properties a and b : a y x is expressible by a power series ∞ m 0 b m x m whose radius of convergence is at Let {b m } be a sequence of positive real numbers such that the radius of convergence of the series ∞ m 0 b m x m is at least ρ, and let α and β satisfy either α ≥ 1 and β > 0 or β ≥ α − 1 2 /4. If a function y : 0, ρ → R is defined by y x ∞ m 0 b m x m , then y certainly belongs to C K with K ≥ 1. So, the set C K is not empty if K ≥ 1. In particular, if ρ is small and K is large, then C K is a large class of analytic functions y : 0, ρ → C.
We will now solve the approximate Euler differential equations in a special class of analytic functions, C K . Theorem 3.1. Let α and β be complex constants such that no root of the auxiliary equation 2.2 is a nonnegative integer. If a function y ∈ C K satisfies the differential inequality for all x ∈ 0, ρ and for some ε ≥ 0, then there exists a solution y h : 0, ρ → C of the Euler differential equation 2.1 such that for any x ∈ 0, ρ .
Proof. Since y belongs to C K , it follows from a and b that for each x ∈ 0, ρ . Now, suppose that an arbitrary x ∈ 0, ρ is given. Then we can choose an arbitrary constant ρ 0 ∈ x, ρ . By Abel's formula see 22, Theorem 6.30 , we have