Fixed Point Theorems for Nonexpansive Mappings Satisfying Certain Boundary Conditions1
نویسنده
چکیده
Let K be a bounded closed convex subset of a Banach space X with int K 40, and suppose K has the fixed point property with respect to nonexpansive self-mappings (i.e., mappings U: K^K such that \\U(x) U(y)\\ < ||* y||, x,y € K). Let T: K -X be nonexpansive and satisfy inf{||* T(x)\\: x e boundary K, T'x) /Kl > 0. It is shown that if in addition, either (i) T satisfies the Leray-Schauder boundary condition: there exists z £ int K such that T(x) z 4 M* z) for all * e boundary K, A< 1, or (ii) infj||* 7X*)||: * s K\ =0, is satisfied, then T has a fixed point in K.
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