THE INVARIANCE OF DOMAIN FOR $k$-SET-PSEUDO-CONTRACTIVE OPERATORS IN BANACH SPACES

Abstract

We introduce a new family of nonlinear operators called $k$-set-pseudo-contractions where several well-known mappings, such as, the condensing mappings (for $k=1$) and the compact perturbations of $k$-pseudo-contractive mappings are embraced in the class of $k$-set-pseudo-contractions. We prove an invariance of domain theorem and (as a consequence) a fixed point theorem for a $k$-set-pseudo-contraction ($0 \lt k \lt 1$) which is also an $L$-set-contraction ($L \geq 0$). Several well known results can be deduced from our theorems.