Singular Lidstone boundary value problem with given maximal values for solutions

The singular problem (-1)(n)x((2n))=muf(t,x,...,x((2n-2))), x((2j))(0)=x((2j))(T)=0 (0less than or equal tojless than or equal ton - 1), max{x(t) : 0 less than or equal to t less than or equal to T} = A depending on the parameter mu is considered. Here the positive Caratheodory function f may be singular at the zero value of all its phase variables. The paper presents conditions which guarantee that for any A 0 there exists mu(A) 0 such that the above problem with mu = mu(A) has a solution x is an element of AC(2n - 1)([0, T]) which is positive on (0, T). The proofs are based on the regularization and sequential techniques and use the Leray-Schauder degree and Vitalis convergence theorem. (C) 2003 Elsevier Ltd. All rights reserved.