Solvability of multi-point boundary value problem at resonance-part IV

Authors:
Bing Liu
Affiliations:
Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, People's Republic of China
Venue:
Applied Mathematics and Computation
Year:
2003

Citing 4
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A generalized multi-point boundary value problem for second order ordinary differential equations

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

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First-order three-point boundary value problems at resonance

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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), (1.1) subject to one of the following boundary value conditions: x(0) = Σi=1m-2 αix(ξi), x(1) = Σj=1n-2 βjx(ηj), (1.2) x(0) = Σi=1m-2 αix(ξi, x'(1) = Σj=1n-2 βjx'(ηj), (1.3) x'(0) = Σi=1m-2 αix'(ξi, x(1) = Σj=1n-2 βjx(ηj), (1.4) where αi (1 ≤ i ≤ m - 2), βj (1 ≤ j ≤ n - 2) ∈ R, 0 1 2 m-2 1 2 n-2 . When all the αis have no the same sign and all the βjs have no the same sign, some existence results are given for (1.1) with boundary conditions (1.2)-(1.4) at resonance case. We also give some examples to demonstrate our results.