The fixed point property in Banach spaces via the strict convexity and the Kadec–Klee property

Abstract

Let X be a strictly convex Banach space, whose predual space is $$ Y (X=Y^{\prime })$$, having the weak star sequentially compact unit ball for the topology $$ \sigma (X, Y) $$ and the weak star Kadec–Klee property. Furthermore, we suppose that the unit ball of the dual space $$X^{\prime }$$ is weak star sequentially compact for the topology $$\sigma (X^{\prime }, X)$$. Let C be a nonempty convex bounded closed subset of X; then every nonexpansive mapping $$ T:C \rightarrow C $$ has a fixed point. As consequences of this result, we generalize the Browder (Proc Natl Acad Sci USA 54:1041–1044, 1965) and Gohde (Math Nachr 301:251–258, 1965) theorems, where X is a uniformly convex Banach space and the Lin’s theorem by Lin (Nonlinear Anal 68:2303–2308, 2008) and Lin (J Math Anal Appl 362:534–541, 2010), where $$ X=l^{1} $$.