Atlas: Solvability of some nonlinear equations in Banach spaces by S. A. Brykalov

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Functional Analysis Valencia 2000
July 3-7, 2000
Technical University of Valencia (UPV) and University of Valencia (UV)
Valencia, Spain

Organizers
R.M. Aron (Kent State U., USA), K.D. Bierstedt (U. Paderborn, Germany), J. Bonet (UPV), J. Cerdà (U. Barcelona, Spain), H. Jarchow (U. Zürich, Switzerland), M. Maestre (UV), J. Schmets (U. Liège, Belgium)

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Solvability of some nonlinear equations in Banach spaces
by
S. A. Brykalov
Inst. Mathematics & Mechanics, Ur. Br. of Russian Acad. Sci.

Sufficient conditions for the solvability are given for a system of equations Dx=F(x), l(x)x=g(x) in a Banach space. Here D is a linear bounded operator with finite-dimensional kernel, F is a completely continuous nonlinear mapping, l and g take values in a finite- dimensional space, l(z)x is linear with respect to x, the map g is nonlinear. In particular, it is assumed that there exists a family B of linear operators which contains l(z) for any fixed z (possibly, together with some other linear operators so that B is closed in an appropriate sense) and such that for any L from B the problem Dx=0, Lx=0 possesses only the trivial solution. Some particular cases and possible generalisations of the results are discussed. Applications to differential equations can be given.

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Date received: November 22, 1999