An operator , where is a subset of a real Banach space , is said to be pseudo-contractive if
for all and . This generalises the notion of nonexpansive mapping, and is equivalent to the following: for all , there exists such that
Furthermore, is said to satisfy the weakly inward condition if for all , where is the inward set of . If satisfies these hypotheses and is closed and convex, it is shown that for each , there exists a unique path , , satisfying
Then the main result of the paper asserts that, if has a uniformly Gâteaux differentiable norm, and every closed bounded convex subset of has the fixed point property for nonexpansive self-mappings, and the set is bounded, then the path defined above converges strongly, as , to a fixed point of .
The proof is then adapted to obtain the same conclusion under modified hypotheses.