On Different Type Solutions of Boundary Value Problems*. Issue 5 (2nd September 2016) - Dobkevich, Maria; Sadyrbaev, Felix
Abstract: We consider boundary value problems of the type xʺ = f ( t, x, xʹ ), ( ∗ ) x ( a ) = A, x ( b ) = B. A solution ξ ( t ) of the above BVP is said to be of type i if a solution y ( t ) of the respective equation of variations y ʺ = fx ( t, ξ ( t ), ξʹ ( t )) y + fxʹ ( t, ξ ( t ), ξʹ ( t )) yʹ, y ( a ) = 0, yʹ (a) = 1, has exactly i zeros in the interval ( a, b ) and y ( b ) ≠ 0. Suppose there exist two solutions x 1 ( t ) and x 2 ( t ) of the BVP. We study properties of the set S of all solutions x ( t ) of the equation (∗) such that x ( a ) = A, xʹ1 ( a ) ≤ xʹ (a) ≤ xʹ2 ( a ) provided that solutions extend to the interval [ a, b ].