Solvability of multi-point boundary value problem at resonance (III)

Authors:
Bing Liu;Jianshe Yu
Affiliations:
College of Mathematics and Statistics, Wuhan University, Wuhan 430072, People's Republic of China;College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, People's Republic of China
Venue:
Applied Mathematics and Computation
Year:
2002

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Abstract

In this paper, we consider the following second-order ordinary differential equation x" = f(t,x(t),x'(t)) + e(t), t ∈ (0, 1), (E) subject to one of the following boundary value conditions: x(0) = ∑ αix(ξi), x(1) = βx(η), (B1) x(0) = ∑ αix(ξi), x'(1) = βx'(η), (B2) x'(0) = ∑ αix'(ξi), x(1) = βx(η), (B3) x'(0) = ∑ αix'(ξi), x'(1) = βx'(η), (B4) where αi (1≤i≤m-2), β ∈ R, 0 1 2 m-2 i's have no the same sign, some existence results are given for (E) with boundary conditions (B1), (B2), (B3), (B4) at resonance case. We also give some examples to demonstrate our results.