An extension of Guo's theorem via k -ψψ-contractive retractions

Let X be a infinite-dimensional Banach space. We generalize Guo's Theorem [D.J. Guo, Eigenvalues and eigenvectors of nonlinear operators, Chinese Ann. Math. 2 (1981) 65–80 [English]] to k  -ψψ-contractions and condensing mappings, under a condition which depends on the infimum kψkψ of all k⩾1k⩾1 for which there exists a k  -ψψ-contractive retraction of the closed unit ball of the space X onto its boundary.