Solutions of m-Point Boundary Value Problems

Here, first we shall find connections between the solutions of the initial value problem (6.1), and its perturbed system (6.2) satisfying, instead of the same initial condition y(k₀) = x₀, the m-point boundary conditions 33.1 $${A_1}y({k_1}) + {A_2}y({k_2}) + \cdots + {A_m}y({k_m}) = \gamma ,$$ where k₀ ≤ k₁ ≤ k₂ ⋯ ≤ k _m ≤ k₀ + J, k _i ∈ I_0,J = {k₀, k₀ + 1, ⋯ , k₀ + J}, A _i ∈ ℝn×n, i = 1, 2, ⋯ , m are constant matrices, and γ ∈ ℝn is a constant vector. Then, these connections will be used to study the existence and uniqueness of the solutions of the boundary value problem (6.2), (33.1) in ‘generalized (vector) normed spaces’. An iterative scheme which can be used to compute approximate solutions of (6.2), (33.1) will also be provided. In what follows, it is sufficient to assume that the systems (6.1) and (6.2) are defined only on I_0,J . We begin with the following: