We find the eigenvalues and eigenfunctions of two nonlocal problems for a mixed-type equation with the singular coefficients of lower-order terms.Consider the finite simply connected domain Ω bounded for y > 0 by the arc σ 0 = {(x, y) : x 2 + y 2 = 1, x > 0} and the straight line segment OA = {(x, y) : x = 0, 0 < y < 1}, and for y < 0, by the characteristics OC, x + y = 0, and BC, x − y = 1, of the equationwhere λ is some numerical parameter and α, β ∈ R are such that 0 < 2α < 1, 0 < 2β < 1, α < β. Introduce the following notation:where a, b, c ∈ R, k ∈ N; also, Γ(z) is the Euler gamma function, J m (z) is the Bessel function of the first kind of order m, and F (a, b, c; z) is the Gauss hypergeometric function.Note that A 1, √ λ 0x and F 0x are the operators introduced in [1] and [2] respectively. Finding the eigenvalues and eigenfunctions of boundary value problems for various equations of mixed type is the topic of many studies among which we should mention the articles by Moiseev, Ponomarev, Kal menov, and others; for instance, see [3][4][5][6][7][8][9].In this article we find the eigenvalues and the corresponding eigenfunctions of two nonlocal problems for (1) in Ω. Similar problems for the Lavrent ev-Bitsadze equation and an equation of mixed type with one singular coefficient were previously studied in [10,11].Henceforth by a regular solution to (1) in Ω 1 we understand some function u(x, y) with the properties u(x, y) ∈ C(Ω 1 ) ∩ C 2 (Ω 1 ), (−y) 2β u y ∈ C(Ω 1 ∪ OB), u(x, 0) ∈ C[0, 1] ∩ C 2 (0, 1), lim y→0 (−y) 2β u y ∈ C 2 (0, 1) that meets (1) in Ω 1 ; here lim y→0 (−y) 2β u y could have singularity of order less than 1 − 2β as x → 0 or x → 1.Tashkent.
