Abstract
AbstractIn 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).
Highlights
1 Introduction Sixty years ago, Krasnosel’skii [, ] observed that many problems in analysis can be formulated abstractly as a mapping which is the sum of a contraction and a compact map
More and more scholars are interested in the study of the existence of solutions of nonlinear abstract operator equation or the fixed point for the sum of two operators of the form
There appears a huge number of papers studying the fixed point problems for the sum of two operators, the generalizations and its variants of Krasnosel’skii fixed point theorem and their applications in real-world problems, see, for instance, [ – ] and the references therein
Summary
Krasnosel’skii [ , ] observed that many problems in analysis can be formulated abstractly as a mapping which is the sum of a contraction and a compact map.
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