The Euler buckling problem is
$$ - u'' = \lambda \sin u, $$
Even though Lu = -u″ isn’t invertible with respect to the given boundary condition, we can apply Theorem 19.9 to the modified problem
$$ u'(0) = u'(\pi ) = 0. $$
$$ - u'' - \in u = \lambda \sin u, $$
to prove Theorem 20.1. For each k = 1, 2, …, the integer k2 is a bifurcation point for the Euler buckling problem (E) and the branch C k of nontrivial solutions containing (k2, 0) is an unbounded subset of R × X such that if(λ, u) ∈ C k with u ≠ 0, then u ∈ S k .
$$ u'(0) = u'(\pi ) = 0 $$
KeywordsShape Function Constant Function Bifurcation Point Nontrivial Solution Tangent Line
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© Springer Science+Business Media New York 2004