On Different Type Solutions of Boundary Value Problems

This research is supported by the European Social Fund within the project Nr. 2013/0024/1DP/1.1.1.2.0/13/APIA/VIAA/045.

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 yʺ = f_x(t, ξ(t), ξʹ(t))y + f_xʹ (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 x₁(t) and x₂(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].

Keywords:

• boundary value problem
• multiple solutions
• existence

AMS Subject Classification:

• 34B15

Notes

This research is supported by the European Social Fund within the project Nr. 2013/0024/1DP/1.1.1.2.0/13/APIA/VIAA/045.