Difference equations and multipoint boundary value problems

Abstract:

Let $I = \left \{ {a,a + 1, \ldots ,b} \right \}$ be finite, let $n \geqslant 1$, and let ${I^j} = \left \{ {a,a + 1, \ldots ,b + j} \right \}$, $j = 1, \ldots ,n$. Let $B$ be the set of mappings from ${I^n}$ into the reals and define the linear difference operator $P$ by (1) \[ Pu(m) = \sum \limits _{j = 0}^n {{\alpha _j}(m)u(m + j),} \quad {\text {where }}m \in I,{\alpha _n}(m) \equiv 1,{\text {and }}{\alpha _0}(m) \ne 0{\text { on }}I.\] Existence of solutions theorems and iteration schemes that approximate solutions are given for boundary value problems of the form $Pu(m) = f(m,u,Eu, \ldots ,{E^{n - 1}}u)$, with boundary conditions $Tu(m) = r$, where $P$ is defined by (1), ${E^j}u(m) = u(m + j)$, $j = 0,1, \ldots ,n - 1$, $f:I \times {{\mathbf {R}}^n} \to {\mathbf {R}}$ is continuous, and $T:B \to {{\mathbf {R}}^n}$ is a continuous linear operator. The results are based on solutions of difference inequalities and sign properties of associated Green’s functions.