Subordinate Semimetric Spaces and Fixed Point Theorems

We introduce the concept of subordinate semimetric space. Such notion includes the concept of RS-space introduced by Roldán and Shahzad; therefore the concepts of Branciari’s generalized metric space and Jleli and Samet’s generalized metric space are particular cases. For such spaces we prove a version of Matkowski’s fixed point theorem, and introducing the concept of q-contraction we get a fixed point theorem of Kannan-Ćirić type. Moreover, using such result we characterize complete subordinate semimetric spaces.

Fixed point theorems for contractions in semicomplete semimetric spaces

We introduce the concept of semicompleteness on semimetric space, which is weaker than completeness. We prove fixed point theorems for contractions in semicomplete semimetric spaces. Also, we generalize JachymskiMatkowski-Swia¸ tkowski’s fixed point theorem in semimetric spaces.

Superoptimal Rate of Convergence in Nonparametric Estimation for Functional Valued Processes

We consider the nonparametric estimation of the generalised regression function for continuous time processes with irregular paths when the regressor takes values in a semimetric space. We establish the mean-square convergence of our estimator with the same superoptimal rate as when the regressor is real valued.