PERTURBATIONS OF NORMALLY SOLVABLE NONLINEAR OPERATORS,

ABSTRACT. Let X and Y be Banach spaces and let F and be Gateaux differentiable mappings from X to Y In this note we study when the operator F + is surjective for sufficiently small perturbations of a surjective operator F The methods extend previous results in the area of normal solva-bility for nonlinear operators. KEY WORDS AND PHRASES. Solvab2llty, normal solvabit2ty, inverse function theorems, Ekgand ’ s Theorem, contractor detions. AMS/MOS Subject classification. 47H15. Let F be a nonlinear mapping from a Banach space X to a Banach space Y We call F Gateaux differentiable if for each x X there is an operator F ’ (x), not necessarily linear, mapping X to Y for which lim F(x+th) =(x) F’(x)(h) tt-O for all h X This operator F is nory solvable if the injectivity of the set of adjoint operators F’(x) * implies surjectivity of F In this light, the theory of normally solvable operators has been investigate