Critical Krasnoselskii-Schaefer type fixed point theorems for weakly sequentially continuous mappings and application to a nonlinear integral equation (Q2802011)

From the authors' abstract: In this paper, we first state some new fixed point theorems for operators of the form \(A + B\) on a bounded closed convex set of a Banach space, where \(A\) is a weakly compact and weakly sequentially continuous mapping and \(B\) is either a weakly sequentially continuous nonlinear contraction or a weakly sequentially continuous separate contraction mapping. Second, we study the fixed point property for a larger class of weakly sequentially continuous mappings under weaker assumptions and they explore this kind of generalization by considering the multivalued mapping \((I-B)^{-1}A\), when \(I-B\) may fail to be injective. To attain this goal, we extend \textit{H. Schaefer}'s theorem [Math. Ann. 129, 415--416 (1955; Zbl 0064.35703)] to multivalued mappings having weakly sequentially closed graph. Finally, we use their abstract results to derive an existence theorem for an integral equation in a reflexive Banach space.