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Convergence of paths for pseudo-contractive mappings in Banach spaces. (English) Zbl 0970.47039

An operator T:KX, where K is a subset of a real Banach space X, is said to be pseudo-contractive if

(λ-1)u-v(λI-T)u-(λI-T)v

for all u,vK and λ>1. This generalises the notion of nonexpansive mapping, and is equivalent to the following: for all u,vK, there exists jJ(u-v)={jX * :u-v,j=u-v 2 =j 2 } such that

Tu-Tv,ju-v 2 ·

Furthermore, T is said to satisfy the weakly inward condition if Txcl(I K (x)) for all xK, where I K (x):={x+λ(u-x):uK,λ1} is the inward set of x. If T satisfies these hypotheses and K is closed and convex, it is shown that for each x 0 K, there exists a unique path tx t , t[0,1), satisfying

x t =tTx t +(1-t)x 0 ·

Then the main result of the paper asserts that, if X has a uniformly Gâteaux differentiable norm, and every closed bounded convex subset of K has the fixed point property for nonexpansive self-mappings, and the set E={xK:Tx=λx+(1-λ)x 0 forsomeλ>1} is bounded, then the path defined above converges strongly, as t1 - , to a fixed point of T.

The proof is then adapted to obtain the same conclusion under modified hypotheses.


MSC:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
46B03Isomorphic theory (including renorming) of Banach spaces