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Higher order multi-point boundary value problems

✍ Scribed by John R. Graef; Lingju Kong; Qingkai Kong

Publisher: John Wiley and Sons
Year: 2010
Tongue: English
Weight: 154 KB
Volume: 284
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.200710179

No coin nor oath required. For personal study only.

✦ Synopsis

Solutions, boundary value problems, lower and upper solutions, Nagumo condition, fixed point theorem MSC (2010) 34B15, 34B18

We consider the boundary value problem u, u , . . . , u (n -1) = 0, t ∈ (0, 1),

where n ≥ 2 and m ≥ 1 are integers, tj ∈ [0, 1] for j = 1, . . . , m, and f and gi , i = 0, . . . , n -1, are continuous. We obtain sufficient conditions for the existence of a solution of the above problem based on the existence of lower and upper solutions. Explicit conditions are also found for the existence of a solution of the problem. The differential equation has dependence on all lower order derivatives of the unknown function, and the boundary conditions cover many multi-point boundary conditions studied in the literature. Schauder's fixed point theorem and appropriate Nagumo conditions are employed in the analysis. Examples are given to illustrate the results.

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