On the solvability of a Dirichlet‐type problem for one equation with variable coefficients

Abstract

The paper investigates a boundary value problem in a rectangular domain for a high even order equation with discontinuous coefficients. A criterion for the uniqueness of a solution is obtained using the spectral method. The solution is built in the form of a Fourier series. When justifying the convergence of the series, the problem of small denominators arises. We obtain sufficient conditions for the convergence of the series.

A Study of the Growth Results for the Hadamard Product of Several Dirichlet Series with Different Growth Indices

“…Thus, the sum function f (s) of ( 1) is an entire function. For convenience, allow D to denote the set of all functions f (s) with the form (1), which is analytic in the region s < +∞ and the sequence {λ n } satisfy (2). Usually, we utilize the order and type to estimate the growth in f (s), which are defined as follows.…”

“…Remark 2. Generally, 2-order is always called an order, that is, ρ [2] = ρ. Similarly, for the lower 2-order, 2-type and lower 2-type, we have χ [2] = χ, T [2] = T and τ [2] = τ.…”

“…To describe the growth in f (s) when ρ [2] = 0, the definitions, including the logarithmic order, logarithmic type, lower logarithmic order and lower logarithmic type, can be introduced as follows Definition 2 (see [30,39]). If f (s) ∈ D, and is of zero-order ρ [2] = 0, then we define the logarithmic order and lower logarithmic order of f (s) as follows…”

“…In view of the arbitrariness of ε, it follows that T [2] (G) ≤ 0, by combining this with the fact that T [2] (G) ≥ 0, we can obtain T [2] (G) = 0. In view of Case (ii 1 ) and Case (ii 2 ), we have T [q 1 ] (G) = 0 for q 1 = 2, 3, 4, .…”

“…If we take σ = 0 and λ n = n, then this series (1) could be a complex Fourier series. It is widely known that the Dirichlet series can be used in many fields of mathematics, such as analytic number theory, functional equations, and certain areas of theoretical and applied probability (see [1][2][3][4][5]).…”