ON A SINGULAR TWO-POINT BOUNDARY VALUE PROBLEM FOR THE NONLINEAR mTH-ORDER DIFFERENTIAL EQUATION WITH DEVIATING ARGUMENTS
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Sufficient conditions of solvability and unique solvability of the boundary value problem u(m)(t) = f ( t, u(τ11(t)), . . . , u(τ1k(t)), . . . , u(τm1(t)), . . . . . . , u(τmk(t)) ) , u(t) = 0, for t 6∈ [a, b], u(i−1)(a) = 0 (i = 1, . . . , m− 1), u(m−1)(b) = 0, are established, where τij : [a, b] → R (i = 1, . . . , m; j = 1, . . . , k) are measurable functions and the vector function f : ]a, b[×Rkmn → Rn is measurable in the first and continuous in the last kmn arguments; moreover, this function may have nonintegrable singularities with respect to the first argument. In the present paper we consider on a segment I = [a, b] the n-dimensional mth-order vector differential equation u(m)(t) = f ( t, u(τ11(t)), . . . , u(τ1k(t)), . . . . . . , u(τm1(t)), . . . , u(τmk(t)) ) (1) with the conditions u(t) = 0, for t 6∈ I, (2) u(i−1)(a) = 0 (i = 1, . . . , m− 1), u(m−1)(b) = 0, (3) where k ≥ 1, m ≥ 2, n ≥ 1, the functions τij : I → R (i = 1, . . . , m; j = 1, . . . , k) are measurable, the vector function f : ]a, b[×Rkmn → Rn is such that f(·, x11, . . . , x1k, . . . , xm1, . . . , xmk) : [a, b[→ Rn is measurable for all 1991 Mathematics Subject Classification. 34K10.
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تاریخ انتشار 2001