This paper is concerned with the existence of exact analytic solutions to a class of linear functional differential equations with proportional delays (see (1.2) in introduction) and a class of linear functional differential equations with constant delays (see (1.3) in introduction) under the different hypotheses, by means of the majorant series method. For the linear functional differential Eq. (1.2) with proportional delays, the following results are obtained: (a) In the case of \(k>l\ge 1,\) we prove that Eq. (1.2) has an analytic solution in a neighborhood of the origin, and the analytical region of such solution is the same as that of known functions (Theorem 2.1). This result improves and generalizes the classical existence theorem and previous results (Ifantis in J Differ Equ 29:86–104, 1978; Chambers in Appl Math 32:445–456, 1975; Si in Acta Math Appl Sin 14:262–276, 1991 (in Chinese)); (b) In the case of \(k=l\ge 1,\) Eq. (1.2) is a neutral differential equation with proportional delays. Firstly, we consider the analytic solutions near a ordinary point. For \(0<|p|<1,\) we obtain the same result as that found in Theorem 2.1. For \(|p|=1,\) we consider two cases: \(|c|\ne 1\) and \(c=\pm 1.\) In the first case, \(|c|\ne 1,\) we obtain the same result as that found in Theorem 2.2. However, for the second case, \(c=\pm 1,\) when \(|p|=1\) but not a root of the unity, i.e. \(p=e^{2\pi i\theta }\) , \(\theta \in {\mathbb {R}}\backslash {\mathbb {Q}},\) we encounter difficulties in proving the convergence of formal solution due to the appearance of a small denominators \(|1+cp^m|\) . In this case, we prove that Eq. (1.2) has an analytic solution in a neighborhood of the origin provided that \(\theta \) is a Brjuno number (see (2.25)). The main contribution of this result is to apply the ideas of treating a small divisor in dynamical systems to the functional differential equations. Secondly, we consider the analytic solutions near a regular singular point. When \(g(z)=0\) Eq. (1.2) is a linear neutral homogeneous equation with proportional delays. If the origin \(\mathcal {O}\) is a regular singular point of all known functions, we prove that this equation has a special form of power series solution in a neighborhood of the origin. This result can be regard as a generalization of the classical Fuchs’ theorem (Fuchs in J Reine Angew Math (Crelles J) 66:121–160, 1866; J Reine Angew Math (Crelles J) 68:354–385, 1868) (see also Frobenius in J Reine Angew Math (Crelles J) 76:214–235, 1873) from linear ordinary differential equations to linear functional differential equations with proportional delays. For the linear functional differential Eq. (1.3) with constant delays, to our knowledge, the existence of its analytic solutions is known relatively little. In this paper analytic solutions for Eq. (1.3) are investigated. We prove that Eq. (1.3) has an analytic solution in the negative half plane under appropriate hypotheses.
