Fixed points of nearly quasi-closed and nearly almost convex mappings

Abstract In this paper, we first obtain a characterization of transfer weakly lower continuous functions. Then, by introducing the class of nearly quasi-closed set-valued mappings, we obtain some characterizations of set-valued mappings whose displacement functions are transfer weakly lower continuous. We also present some fixed point theorems for nearly quasi-closed set-valued mappings which are either nearly almost convex or almost affine. Finally, we construct an almost affine mapping T : [ 0 , 1 ) → R , which is not α-almost convex for any continuous and strictly increasing function α : [ 0 , + ∞ ) → [ 0 , + ∞ ) with α ( 0 ) = 0 . This example gives an affirmative response to the Question 3 of Jachymski [J. Jachymski, J. Nonlinear Convex Anal., 6 (2015), 1055-1068].