Uniqueness Implies Existence and Uniqueness Conditions for a Class of (k + j)-Point Boundary Value Problems for n-th Order Differential Equations

Abstract

AbstractFor the n-th order nonlinear differential equation, y(n) = f (x, y, y′, … , y(n–1)), we consider uniqueness implies uniqueness and existence results for solutions satisfying certain (k + j)-point boundary conditions for 1 ≤ j ≤ n – 1 and 1 ≤ k ≤ n – j. We define (k; j)-point unique solvability in analogy to k-point disconjugacy and we show that (n – j0; j0)-point unique solvability implies (k; j)-point unique solvability for 1 ≤ j ≤ j0, and 1 ≤ k ≤ n – j. This result is analogous to n-point disconjugacy implies k-point disconjugacy for 2 ≤ k ≤ n – 1.