A note on Krasnosel'skii fixed point theorem

In this note, a couple of unclear and unnecessary points made in the two existing papers by Liu and Li (Proc. Am. Math. Soc. 136: 1213-1220, 2008) and Xiang and Yuan (Proc. Am. Math. Soc. 139: 1033-1044, 2011) are first pointed out and clarified. Second, a few additional remarks are observed. Upon these observations, three corresponding refined and unified Krasnosel'skii fixed point theorems in strong topology setup are formulated. As an illustration, several new classes of Krasnosel'skii fixed point theorems are obtained, which expand and complement some known related results by Agarwal, O'Regan and Taoudi (Fixed Point Theory Appl. 2010: 243716, 2010) and Edmunds (Math. Ann. 174: 233-239, 1967).