Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Difference equations and multipoint boundary value problems

Author: P. W. Eloe
Journal: Proc. Amer. Math. Soc. 86 (1982), 253-259
MSC: Primary 39A10
MathSciNet review: 667284
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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)

$\displaystyle Pu(m) = \sum\limits_{j = 0}^n {{\alpha _j}(m)u(m + j),} \quad {\t... ... \in I,{\alpha _n}(m) \equiv 1,{\text{and }}{\alpha _0}(m) \ne 0{\text{ on }}I.$

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.

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Article copyright: © Copyright 1982 American Mathematical Society