Singular Lidstone boundary value problem with given maximal values for solutions Original Research Article

The singular problem (−1)nx(2n)=μf(t,x,…,x(2n−2)), x(2j)(0)=x(2j)(T)=0(0⩽j⩽n−1), max{x(t):0⩽t⩽T}=A depending on the parameter μ is considered. Here the positive Carathéodory 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 μA>0 such that the above problem with μ=μA has a solution x∈AC2n−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 Vitaliʹs convergence theorem.