On F-differentiable Fredholm operators of nonstationary initial-boundary value problems

Vladimir Durikovic and Monika Durikovicova

Address. Vladimir Durikovic, Department of Matematical Analysis of Komensky University, Mlynska dolina,
842 48 Bratislava, SLOVAK REPUBLIC

Monika Durikovicova, Department of Mathematics of Slovak Technical University, nam. Slobody 17,
812 31 Bratislava, SLOVAK REPUBLIC

E-mail: vdurikovic@fmph.uniba.sk, durikovi@sjf.stuba.sk

Abstract. We are dealing with Dirichlet, Neumann and Newton type initial-boundary value problems for a general second order nonlinear evolution equation. Using the Fredholm operator theory we establish some sufficient conditions for Fr\'echet differentiability of associated operators to the given problems. With help of these results the generic properties, existence and continuous dependency of solutions for initial-boundary value problems are studied.

AMSclassification. 58D25, 35R15, 35K22, 47H15.

Keywords. H\"older spaces, Fr\'echet differentiable Fredholm operator of the zero index, critical and singular points of the mixed problem.