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On the quasilinear Cauchy problem for a hyperbolic functional differential equation

Puźniakowska-Gałuch E.

Opuscula Mathematica

2015

Vol. 35, no. 6

915--933

The Cauchy problem for hyperbolic functional differential equations is considered. Volterra and Fredholm dependence are considered. A theorem on the local existence of generalized solutions defined on the Haar pyramid is proved. A result on differentiability of a solution with respect to initial data is proved.

Existence and regularity of solutions for hyperbolic functional differential problems

Kamont Z.

Opuscula Mathematica

2014

Vol. 34, no. 2

217--242

A generalized Cauchy problem for quasilinear hyperbolic functional differential systems is considered. A theorem on the local existence of weak solutions is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions for this system is proved by using a method of successive approximations. We show a theorem on the differentiability of solutions with respect to initial functions which is the main result of the paper.