Fixed point theorems for middle point linear operators in \(L^{1}\) (Q1008503)

Introducing the notion of middle point linear operators, the authors prove a fixed point theorem for \(\alpha\)-nonexpansive, middle point linear operators in \(L^{1}(\mu)\), where \(\alpha\) is the Kuratowski measure of noncompactness and \(\mu\) is a \(\sigma\)-finite measure. More precisely, any continuous operator \(T: (K,\rho) \to (K,\rho)\) has at least one fixed point in \(K\) whenever \(K\) is a nonempty, bounded, \(\rho\)-closed, and convex subset of \(L^{1}(\mu)\) and \(T\) is a middle point linear, \(\alpha\)-nonexpansive operator. Here, \(\rho\) is the metric of the convergence locally in measure. An example of a fixed point free nonexpansive map due to \textit{D.\,E.\thinspace Alspach} [Proc.\ Am.\ Math.\ Soc.\ 82, 423--424 (1981; Zbl 0468.47036)] illustrates that the assumption of middle point linearity cannot be avoided. Some examples show that this theorem applies, but not \textit{C.\,Lennard}'s theorem [``A new convexity property that implies a fixed point property for \(L^1\)'', Stud.\ Math.\ 100, No.\,2, 95--108 (1991; Zbl 0762.46007)]. As an application of the above theorem, a Markov--Kakutani type fixed point result for commuting family of \(\alpha\)-nonexpansive, middle point linear operators in \(L^{1}(\mu)\) is derived, without assuming compactness for the domain of the family.