Maximum principles and boundary value problems for first-order neutral functional differential equations

We obtain the maximum principles for the first-order neutral functional differential equation (M x) (t) x ′ (t) - (S x ′) (t) - (A x) (t) + (B x) (t) = f (t), t [ 0, ], where A: C [ 0, ] → L [ 0, ] ∞, B: C [ 0, ] → L [ 0, ] ∞, and S: L [ 0, ] ∞ → L [ 0, ] ∞ are linear continuous operators, A and B are positive operators, C [ 0, ] is the space of continuous functions, and L [ 0, ] ∞ is the space of essentially bounded functions defined on [ 0, ]. New tests on positivity of the Cauchy function and its derivative are proposed. Results on existence and uniqueness of solutions for various boundary value problems are obtained on the basis of the maximum principles.