On a singular Cauchy problem for functional differential equations with non-increasing non-linearities∗

We obtain general conditions sufficient for the solvability of a singular Cauchy problem for functional differential equations with non-increasing non-linearities. 1 Problem setting and introduction The aim of this note is to establish some general conditions sufficient for the existence of a solution of a singular Cauchy problem for a class of non-linear functional differential equations. More presicely, we consider the equation (1) u′(t) = (gu)(t), t ∈ (a,b], where −∞ < a < b < ∞ and g : C((a,b],R)→ L1; loc((a,b],R) is a certain (generally speaking, non-linear) mapping which is assumed to be non-increasing with respect to the natural pointwise ordering (see Definition 4). Solutions of equation (1) are sought for in the class of locally absolutely continuous functions and, in particular, may be unbounded in a neigbourhood of the point a (the precise notion of a solution is given by Definition 2 below). Definition 1. One says that a function u : (a,b]→ R is locally absolutely continuous if its restriction u|[a+ε,b] to the interval [a+ ε,b] is absolutely continuous for any ε ∈ (0,b−a). ∗Research supported in part by AS CR, Institutional Research Plan No. AV0Z10190503. 1 Preprint, Institute of Mathematics, AS CR, Prague. 2009-6-15 I N T IT U TE of M ATH TICS A ca d em y o f Sc ie n ce s C ze ch R ep u b lic