The theory of u(0)-positive operators with respect to a cone in a Banach sp
ace is applied to the class of even-order differential equation on measure
chains
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satisfying the lidstone boundary conditions x(Delta 2i)(0) = 0, 0 less than
or equal to i less than or equal to m - 1; x(Delta 2j) (sigma(2)(1)) = 0,
0 less than or equal to j less than or equal to m - i. The existence of a s
mallest positive eigenvalue is obtained and then a theorem is established c
omparing the smallest positive eigenvalues for two problems of this type. (
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