Nonlinear alternatives of Schauder and Krasnoselʹskij types with applications to Hammerstein integral equations in L1 spaces

This paper is devoted to establishing new variants of some nonlinear alternatives of Leray–Schauder and Krasnoselʹskij type involving the weak topology of Banach spaces. The De Blasi measure of weak noncompactness is used. An application to solving a nonlinear Hammerstein integral equation in L1 spaces is given. Our results complement recent ones in [K. Latrach, M.A. Taoudi, A. Zeghal, Some fixed point theorems of the Schauder and the Krasnoselʹskij type and application to nonlinear transport equations, J. Differential Equations 221 (2006) 256–2710] and [K. Latrach, M.A. Taoudi, Existence results for a generalized nonlinear Hammerstein equation on L1 spaces, Nonlinear Anal. 66 (2007) 2325–2333].