On the Fredholm Alternative for Nonlinear Operators

Let X be a locally convex topological vector space, Y a real Banach space, ƒ a mapping (in general, nonlinear) of X into Y. In several recent papers ([5], [ó], [7]), Pohozaev has studied the concept of normal solvability or the Fredholm alternative for mappings ƒ of class C. If Ax=f'x' is the continuous linear mapping of X into Y which is the derivative of ƒ a t the point x of X, A* the adjoint mapping of F* into X*, his principal results assert that if the nullspace N(A*) is trivial for every x in Xy and if one of the two following hypotheses hold: (1) F is reflexive and f(X) is weakly closed in F ; (2) F is uniformly convex and ƒ(X) is closed in F ; then the image f(X) of ƒ must be all of F. I t is our purpose in the present paper to considerably sharpen and generalize these results by use of a different and more transparent argument. In particular, we establish a corresponding theorem for an arbitrary Banach space F and f{X) closed in F, allow exceptional points x in X a t which the hypothesis on N(A*) may not hold, and derive this theorem from a basic theorem on general rather than differentiable mappings. The techniques which we apply below may be extended to infinite-dimensional manifolds and may be localized to prove the openness of ƒ under stronger hypotheses (as we shall do in another more detailed paper). To state our basic theorem, we use the following definition: DEFINITION 1. Let X be a real vector space, ƒ a mapping of X into the real Banach space Yyxa point of X. Then the element v of the unit sphere 5i( F) of Y is said to lie in the set Rx(f) of asymptotic directions f or f at x if there exists £ ^ 0 in X and a sequence {jj} of positive numbers with 7,—->0 asj—> oo such that for eachj,f(x +7;£) ^ / ( x ) , while