Approximation of Analytic Functions by Solutions of Cauchy-Euler Equation

for any x ∈ I, where K(ε) depends on ε only and satisfies limε→0K(ε) = 0, then we say that the differential equation (1) satisfies (or has) the Hyers-Ulam stability (or the local HyersUlam stability if the domain I is not the whole space R). If the above statement also holds when we replace ε and K(ε) with some appropriate φ(x) and Φ(x), respectively, then we say that the differential equation (1) satisfies the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability). We may apply these terminologies for other differential equations. For more detailed definition of the Hyers-Ulam stability and recent papers on this subject, refer to [1–4]. Obłoza seems to be the first author who investigated the Hyers-Ulam stability of linear differential equations (see [5, 6]). Let g, r : (a, b) → R be continuous functions with


Introduction
Throughout this paper, let  be a positive integer, let  be a nondegenerate interval of R, and let K denote either R or C. We will consider the (linear) differential equation of th order F ( () ,  (−1) , . . .,   , , ) = 0 (1) defined on , where  :  → K is an  times continuously differentiable function.
We may apply these terminologies for other differential equations.For more detailed definition of the Hyers-Ulam stability and recent papers on this subject, refer to [1][2][3][4].
In this paper, we consider the (inhomogeneous) Cauchy-Euler equation where  and  are real-valued constants and  : R → R is a continuous function, and we investigate the approximation properties of twice continuously differentiable functions by solutions of the Cauchy-Euler equation which is associated with (4).
Theorem 1. Assume that the real-valued constants ,  are given with (−1) 2 −4 > 0 and  is an arbitrarily given positive constant.Let  be a positive real-valued constant and let  1 ,  2 be given as If  : (0, ∞) → R is a differentiable function and  : (0, ∞) → R is a twice continuously differentiable function such that the inequality holds for any  ∈ (0, ∞), then there exists a solution   : (0, ∞) → R of the inhomogeneous Cauchy-Euler equation ( 4) such that for all  ∈ (0, ∞).
Remark 4. Cîmpean and Popa [14] proved the Hyers-Ulam stability of the linear differential equations of th order with constant coefficients.Indeed, they proved a general theorem for the Hyers-Ulam stability which includes Theorems 1, 2, and 3 as its corollaries with the inequality However, Theorems 1, 2, and 3 have the advantage of more exact local approximation over the result of Cîmpean and Popa as we see in Theorems 5, 6, and 7.
If we define for all  1 ,  2 ∈ B(; ) and  ∈ R, then B(; ) is a vector space over R.This fact implies that the set B(; ) is large enough to be a vector space.
In the following theorems, we investigate approximation properties of functions of B(; ) by solutions of the Cauchy-Euler equation (5).
We will only estimate the following limit for the case of  1 = 0 by applying L'Hospital's rule: which implies the validity of this theorem.
We now consider the case of ( − 1) 2 − 4 = 0 and use Theorem 2 to prove the following theorem.as  → .