| New fixed-point theoretic results are obtained for several of the standard classes of mappings of nonlinear functional analysis. The mappings generally are not assumed to be self-mappings of their domains, but rather to satisfy Leray-Schauder type boundary conditions. Specific results obtained include the following.;Let X be a Banach space, D a bounded open subset of X with 0 (ELEM) D, and T:D (--->) X. It is known for various classes of mappings, including the condensing mappings, that if T satisfies the Leray-Schauder condition: T(x) (NOT=) (lamda)x for x (ELEM) (PAR-DIFF) D, (lamda) > 1, then T has a fixed point in D. It is shown in this thesis that if T is a condensing mapping for which I-tT for t (ELEM) (0,1) is one-to-one, then the Leray-Schauder condition in this result may be weakened to: T(x) = (lamda)x for x (ELEM) (PAR-DIFF) D, (lamda) > 1 implies T(x) = (mu)x for some x (ELEM) D and (mu) (ELEM) {1,(lamda)). It is also shown that if T is a continuous pseudo-contractive mapping, then this weaker boundary condition implies inf {(PARLL)x-T(x)(PARLL):x (ELEM) D} = 0. Included among numerous results for mappings satisfying local conditions is the fact that the stronger Leray-Schauder boundary condition is sufficient to guarantee existence of a fixed point for a continous mapping T:D (--->) X which is a local strong pseudo-contraction on D. This result appears to be new even if T is a local contraction. |
