Comparison of eigenvalues for a fourth-order four-point boundary value problem

B. Karna, Marshall University, West Virginia, USA
E. Kaufmann, University of Arkansas at Little Rock, Little Rock, USA
J. Nobles, University of Arkansas at Little Rock, Little Rock, USA

E. J. Qualitative Theory of Diff. Equ., No. 15. (2005), pp. 1-9.

Abstract: We establish the existence of a smallest eigenvalue for the fourth-order four-point boundary value problem

$\left (\phi_p(u''(t)) \right )'' = \lambda h(t) u(t), \, u'(0) = 0, \,
\beta_0 u(\eta_0) = u(1), \, \phi_p'(u''(0)) = 0, \,
\beta_1\phi_p(u''(\eta_1)) = \phi_p(u''(1))$, $p > 2$,
$0 < \eta_1,\eta_0 < 1, 0 < \beta_1, \beta_0 < 1$,

using the theory of u$_0$-positive operators with respect to a cone in a Banach space. We then obtain a comparison theorem for the smallest positive eigenvalues, $\lambda_1$ and $\lambda_2$, for the differential equations

$\left ( \phi_p(u''(t)) \right )'' = \lambda_1 f(t) u(t)$ and
$\left ( \phi_p(u''(t)) \right )'' =\lambda_2 g(t) u(t)$ where $0 \leq f(t) \leq g(t), t \in [0,1]$.

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