A LOCAL FIXED POINT THEOREM AND ITS APPLICATION TO LINEAR OPERATORS

In the article, first we prove a local version of a fixed point theorem concerning the mapping T satisfying the following condition of F-contractive type: for all u, v in an open ball of the Banach space parallel to T(u) - T(v) parallel to <= parallel to u-v parallel to / 1 + tau parallel to u - v parallel to, where tau > 0. Next, for this type of mapping T defined on a non-empty open subset of the Banach space, we prove that the mapping I-T is a homeomorphism. Finally, the results obtained are applied in order to show that some linear operators defined on a Banach space are invertible.