Cauchy problem for quasilinear systems of functional differential equations

TL;DR

This paper proves the existence and uniqueness of solutions for quasilinear systems of functional differential equations with retarded and advanced arguments using the contracting mapping principle, under mild conditions.

Contribution

It establishes new existence and uniqueness results for both linear and quasilinear functional differential equations with retarded and advanced arguments.

Findings

Unique solutions exist under mild restrictions.

Results apply to linear systems with locally integrable coefficients.

Similar results hold for advanced argument systems.

Abstract

We use the contracting mapping principle for proving that under some mild restrictions the Cauchy problem for quasilinear systems of functional differential equations with retarded arguments has the unique solution. As a consequence from this result we obtain that the Cauchy problem for linear systems of functional differential equations with locally integrable coeffcients and with locally measurable retarded arguments has the unique solution. We show that similar results can be obtained for the Cauchy co problem of quasilinear systems of functional differential equations with advanced arguments.