MR2449047 (2009j:47108) Chermisi, Milena; Martellotti, Anna Fixed point theorems for middle point linear operators in $L^1$. Fixed Point Theory Appl. 2008, Art. ID 648591, 13 pp. (Reviewer: Diana Caponetti) 47H10 (47H09)

Abstract

In the paper under review the notion of middle point operator is introduced. The authors prove that for a given nonempty, bounded, $\rho$-closed, convex subset K of L1(μ), where $\rho$ is the metric of the convergence locally in measure, if T from (K, $\rho$) to(K, $\rho$) is a continuous, $\rho$-nonexpansive, middle point linear operator, then T has at least one fixed point in K. To prove the theorem they use results of A. V. Bukhvalov [in Operator theory in function spaces and Banach lattices, 95–112, Birkh¨auser, Basel, 1995; MR1322501 (95m:46123)] and M. Furi and A. Vignoli [Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 48 (1970), 195–198; MR0279792 (43 #5513)]. Then they derive a Markov-Kakutani type fixed point theorem for commuting family of $\rho$-nonexpansive and middle point linear operators in L1(μ).