Abstract
We will solve the inhomogeneous Bessel′s differential equation , where ν is a positive nonintegral number and apply this result for approximating analytic functions of a special type by the Bessel functions of fractional order.
Highlights
The stability problem for functional equations starts from the famous talk of Ulam and the partial solution of Hyers to the Ulam problem see 1, 2
A function is called a Bessel function of fractional order if it is a solution of the Bessel differential equation 1.3, where ν is a positive nonintegral number
The Bessel differential equation plays a great role in physics and engineering
Summary
The stability problem for functional equations starts from the famous talk of Ulam and the partial solution of Hyers to the Ulam problem see 1, 2. We will introduce a result of Alsina and Ger see 17 : 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, there exists a constant c such that |f x − cex| ≤ 3ε for any x ∈ I This result of Alsina and Ger has been generalized by Takahasi et al They proved in 18 that the Hyers-Ulam stability holds for the Banach space-valued differential equation y x λy x see 19. Throughout this paper, we denote by x the largest integer not exceeding x for any x ∈ R, and we define Iρ −ρ, 0 ∪ 0, ρ for any ρ > 0
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