Two-point boundary value problem for nonlinear differential equation of nth order

The conditions of solvability and the structure of the set of solutions of the equation x(n) = f(t,x,x′, ..., x(n-1)), on a given interval [a,b] (-∞<a<b<∞) under the boundary conditions x(i)(a) = Ai (i = 0, ..., k), x(j)(b) = Bj (j = 0, ..., l) and additional restriction α(t)≤x(t)≤β(t) (a≤t≤b) were investigated. It is assumed that n≥4; the function f:[a,b]×Rn→R satisfies the Caratheodory conditions; k, l≥0; (k+1)+(l+1)≤n; α and β (α<βi.e.α(t)<β(t), for every t∈[a,b]) are solutions of x(n) = f(t,x,x′, ..., x(n-1)). The main statements are obtained as the consequence of special suppositions of the uniqueness for solutions of auxiliary boundary value problems.