POSITIVE SOLUTIONS FOR A 2n-ORDER BOUNDARY VALUE PROBLEM INVOLVING ALL DERIVATIVES OF ODD ORDERS

We are concerned with the existence, multiplicity and uniqueness of positive solutions for the 2n-order boundary value problem{ (-1)(n)u((2n)) = f(t, u, u, -u, ... ,(-1)(i-1)u((2i-1)), ... , (-1)(n-1)u((2n-1))),u((2i))(0) = u((2i+1))(1) = 0, i = 0, ... , n - 1.where n = 2 and f is an element of C([0,1] x R-+(n+1), R+) (R+ := [0, infinity)) depends on u and all derivatives of odd orders. Our main hypotheses on f are formulated in terms of the linear function g(x) := x(1) + 2 Sigma(n+1)(i=2) x(i). We use fixed point index theory to establish our main results, based on a priori estimates achieved by utilizing some integral identities and an integral inequality. Finally, we apply our main results to establish the existence, multiplicity and uniqueness of positive symmetric solutions for a Lidostone problem involving an open question posed by P. W. Eloe in 2000.