Abstract
Abstract Criteria for the existence and uniqueness of a solution of the boundary value problem are established, where ƒ :]a, b[×R 2 → R satisfies the local Carathéodory conditions, and μ : [a, b] → R is the function of bounded variation. These criteria apply to the case where the function ƒ has nonintegrable singularities in the first argument at the points a and b.
Highlights
We shall use the following notations: R is a set of real numbers
Α t u(s+) and u(s−) are the limits of the function u at the point s from the right and from the left
In the present paper we concern ourselves with the problem of the existence and uniqueness of a solution of equation (1.1) satisfying the boundary conditions b u(a+) = 0, u(b−) = u(s) dμ(s), a where μ : [a, b] → R is the function of bounded variation
Summary
We shall use the following notations: R is a set of real numbers. L([a, b]) is the set of functions p :]a, b[→ R which are Lebesgue integrable on [a, b]. Let there exist t1 ∈]a, b[ and a positive function p ∈ Lloc(]a, b[) such that σab(pi2)p ∈ L([a, b]), i = 1, 2, and (1.12) is fulfilled, where p2 ∈ K0 ]a, b[×R2 and p12(t) ≤ p2(t, x, y) ≤ p22(t) for (t, x, y) ∈]t1, b[×R2. Let there exist a point t1 ∈]a, b[ and a positive function p ∈ Lloc(]a, b[) such that conditions (1.14) and (1.15) are fulfilled. Let there exist numbers λi ∈ [0, 1[, li ∈ [0, +∞[, γi ∈ [0, +∞[, i = 1, 2, c ∈]a, b[, t1 ∈]a, b[ and a function p :]a, b[→]0, +∞[ such that conditions (1.15)–(1.17) and (1.22) are fulfilled, where p1 and p2 are the functions defined by equalities (1.18) and (1.19).
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