ON AN EFFECTIVE CRITERION OF SOLVABILITY OF BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATION OF n-TH ORDER

New sufficient conditions for the existence of a solution of the boundary value problem for an ordinary differential equation of n-th order with certain functional boundary conditions are constructed by a method of a priori estimates. Introduction In this paper we give new sufficient conditions for the existence of a solution of the ordinary differential equation (1) u(t) = f ( t, u(t), . . . , u(t) ) with the boundary conditions (2) Φ0i ( u ) = φi(u) i = 1, . . . , n, resp. (31) li ( u, u, . . . , u0 ) = 0 i = 1, . . . , k0, (32) Φ0i ( u ) = φi ( u0 ) i = k0 + 1, . . . , n, where f : [a, b]× R → R satisfies the local Carathéodory conditions, n ≥ 2, and 1 ≤ k0 ≤ n− 2. For each index i, the functional Φ0i in the conditions (2), resp. (32), is supposed to be linear, nondecreasing, nontrivial, continuous on C([a, b]), and concentrated on [ai, bi] ⊆ [a, b] (i.e., the value of functional Φ0i depends only on a function restricted to [ai, bi] and this segment can be degenerated to a point). In general Φ0i(1) ∈ R, without loss of generality we can suppose that Φ0i(1) = 1, which simplifies the notation. 2000 Mathematics Subject Classification: 34A15, 34B10.