311 QUARTERLY OF APPLIED MATHEMATICS Vol. XXXV OCTOBER 1977 No. 3 ON A FUNCTIONAL EQUATION ARISING IN THE STABILITY THEORY OF DIFFERENCE-DIFFERENTIAL EQUATIONS* By W. B. CASTELAN Brown University and Universidade Federal de Santa Catarina Florianopolis, Brasil AND E. F. INFANTE Brown University Abstract. The functional differential equation Q'(t) = AQ(t) + BQT(->- t), -co<t<co, where A, B are n X n constant matrices, x > 0, Q(t) is a differentiable n X n matrix and QT{t) is its transpose, is studied. Existence, uniqueness and an algebraic representation of its solutions are given. This equation, of considerable interest in its own right, arises naturally in the construc- tion of Liapunov functionals of difference-differential equations of the type x(t) = Cx(t) + Dx{t - t), where C, D are constant n X n matrices. The role played by the matrix Q(t) is analogous to the one played by a positive definite matrix in the construction of Liapunov functions for ordinary differential equations. In this paper, we show that, in spite of the functional nature of this equation, the linear vector space of its solutions is n2\ moreover, we give a complete algebraic characterization of its solutions and indicate computationally simple methods for obtaining these solu- tions, which we illustrate through an example. Finally, we briefly indicate how to obtain solutions for the nonhomogeneous problem, through the usual variation of constants method. 1. Introduction. The study of difference-differential equations has received consid- erable attention in recent times [2, 6, 7], the overwhelming interest being devoted to equations with positive delays. * Received April 15, 1977. This research was supported in part by CAPES (Coordenac;ao do Aperfei^oa- mento de Pessoal de Nivel Superior—Brasil) under Processo no. 510/76, in part by the Office of Naval Research (NONR N000 14-75-C-0278A04), in part by the United States Army (AROD AAG 29-76-6-005), and in part by the National Science Foundation (MPS 71-02923).