Ważewski's Principle for retarded functional differential equations

AbstractWażewski Principle is an important tool in the study of the asymptotic behavior of solutions of ordinary differential equations. A direct extension of this principle to retarded functional differential equations (RFDEs) can be obtained by noticing that solutions of RFDEs generate processes on C = C([−r, 0], Rn) and by using the general version of Ważewski Principle for flows on topological spaces. The resulting method is of little use in applications, due to the infinite-dimensionality of the space C. This paper presents a “Razumikhin-type” extension of Ważewski's Principle, which is widely applicable to concrete examples. The main results are Corollaries 3.1 and 3.2. Also, an extension of the method to RFDEs with a merely continuous right-hand side is given, and a few examples illustrate the use of the method. Throughout the paper, a standard notation is used