Comparison of smallest eigenvalues and extremal points for third and fourth order three point boundary value problems.
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Neugebauer, Jeffrey T.
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The theory of u₀-positive operators with respect to a cone in a Banach space is applied to the linear differential equations u⁽⁴⁾ + λ₁p(x)u = 0 and u⁽⁴⁾ + λ₂q(x)u = 0, 0 ≤ x ≤ 1, with each satisfying the boundary conditions u(0) = u′(r) = u″(r) = u‴(1) = 0, 0 < r < 1. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenvalues is obtained. These results are then extended to the nth order problem using two different methods. One method involves finding the Green's function for –u⁽ⁿ⁾ = 0 satisfying the higher order boundary conditions, and the other involves making a substitution that allows us to work with a variation of the fourth order problem. Extremal points via Krein-Rutman theory are then found. Analogous results are then obtained for the eigenvalue problems u‴ + λ₁p(x)u = 0 and u‴ + λ₂q(x)u = 0, with each satisfying u(0) = u′(r) = u″(1) = 0, 0 < 1/2 < r < 1.