Semigroups generated by pseudo-contractive mappings under the Nagumo condition
Department of Mathematical Sciences, UAH
September 28, 2007
302 Madison Hall
Let C be a closed subset of a Banach space X whose topological dual space X* is uniformly convex. Using strong measurability, the Pettis integral, the weak derivative, and other concepts from the calculus of vector-valued functions, one may show that, for any demicontinuous weakly Nagumo k-pseudo-contractive mapping T: C → X, A = T − I weakly generates a semigroup of type k − 1 on C. if K < 1 (id est, if T is strongly pseudo-contractive), then the semigroup consists of contractive operators. A family of commuting contraction operators on C necessarily has a unique common fixed point, consequently, T has a unique fixed point. This implies that, if T is pseudo-contractive (k = 1) and C is also bounded and convex, then T has at least one fixed point. But T is weakly inward when C is convex and self-mappings are always weakly inward, hence, any demicontinuous pseudo-contractive mapping T: C → C has a fixed point when C is closed, bounded, and convex. This answers an important question in fixed point theory which as been open for quite some time.
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