Higher order multi-point boundary value problems

作者：Graef John R^*; Kong Lingju; Kong Qingkai

来源：Mathematische Nachrichten, 2011, 284(1): 39-52.

DOI：10.1002/mana.200710179

• Solutions
• boundary value problems
• lower and upper solutions
• Nagumo condition
• fixed point theorem

We consider the boundary value problem
u((n)) + f(t, u, u', ... , u((n-1))) = 0, t is an element of (0, 1),
u((i))(0) = g(i)(u((i))(t(1)), ... , u((i))(t(m))), i = 0, ... , n - 2,
u((n-2))(1) = g(n-1)(u((n-2))(t(1)), ... , u((n-2))(t(m))),
where n >= 2 and m >= 1 are integers, t(j) is an element of [0,1] for j = 1, ... , m, and f and g(i), 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.

• 出版日期2011-1