Existence results for a generalized nonlinear Hammerstein equation on L1L1 spaces

In this paper we are concerned with existence results for a nonlinear Hammerstein integral equation acting on L1L1 vector valued functions. Making use of the De Blasi measure of weak noncompactness we establish a variant of the topological fixed point theorem of Krasnosel’skii for the sum of two operators without requiring weak compactness or weak continuity assumptions. Such a result provides an adequate tool for deriving an existence theory for problem (1.1).