Nonlinear operators in Banach spaces

From the point of view of its applications to nonlinear boundary value problems for partial differential equations (as well as to other problems in nonlinear analysis) the principal result of the Leray-Schauder theory [9] of nonlinear functional equations is embodied in the following theorem: L-S Theorem: Let G be an open subset of the Banach space X, C a mapping of\( \overline G \times [0,1] \)into X whose image is precompact and such that if C t (x) = C(x, t), each mapping C t of\( \overline G \)into X has no fixed points on the boundary of G. Then if I — C0is a homeomorphism of G on an open set of X containing 0, the equation (I — C1) u = 0 has a solution u in G (i.e, C1has a fixed point in G).