A-Properness and Fixed Point Theorems for Sums of Dissipative and Ball-Condensing Maps

Abstract

We obtain new results on A-properness of sums of dissipative and ball-condensing maps in π1-Banach spaces. The maps involved need not be self-maps, and the dissipative maps are not required to satisfy any extra range conditions. Moreover, the domains of the maps may be closed subsets. It is known that, in general, it is difficult to prove A-properness at all points of the whole space for a map defined on a closed subset. An open question on A-properness of a k-contraction defined on a ball in a general π1-Banach space, raised by Petryshyn in 1975 and 1993, has not been completely solved so far. Our result will give a partial answer to the open question. New fixed point theorems for sums of the above maps and range results for dissipative maps are derived. An application to eigenvalue problems for homogeneous integral equations is provided.