Coincidence degree and periodic solutions of neutral equations

This paper is devoted to the problem of existence of periodic solutions for some nonautonomous neutral functional differential equations. It is essentially an application of a basic theorem on the Fredholm alternative for periodic solutions of some linear neutral equations recently obtained by one of the authors [2] and of a generalized Leray-Schauder theory developed by the second one [3, 41. Although their proofs are surprisingly simple, the obtained results are nontrivial extensions to the neutral case of a number of recent existence theorems for periodic solutions of functional differential equations. In particular, Section 3 generalizes some existence criteria due to one of the authors [5] and a corresponding recent extension by J. Cronin [6], the example following Theorem 4.1 improves a condition for existence given by Lopes [ 141 for the equation of a transmission line problem, and Theorem 5.1 generalizes a result due to R. E. Fennel1 [7]. Lastly, criteria analogous to Theorem 5.2 for the retarded case can be found in [8]. For partly related results concerning periodic solutions of neutral functional differential equations, see [9].