In this paper, we present a Pata-type fixed point theorem in modular spaces which generalizes and improves some old results.
In this paper, we will consider modular spaces which lie somewhere in between the abstract modular theory and Musielak-Orlicz theory, i.e. the class of modular spaces given by modulars not of any particular form but, nevertheless, having much more convenient properties than the abstract modulars can possess.
On the other hand, in 2010, Chistyakov [14] introduced the notion of a modular metric space which is raised in an attempt to avoid some restrictions of the concept of a modular space (for the literature of a modular space, see e.g. [15 21] and references therein).
In 2010, Şimşek et al. [12] introduced a new modular sequence space which is more general than the Cesàro sequence space defined by Shiue [13] and the generalized Cesàro sequence space defined by Suantai.
The main purpose of this paper is to investigate the property and the uniform Opial property equipped with the Luxemburg norm of the new modular sequence space, which is defined by using the generalized de la Vallée-Poussin mean with generalized Riesz transformation.
Let us finish this section with such terminology for the modular space mapping which will be studied throughout.
The purpose of this paper is to give an outline of fixed point theory for nonexpansive mappings (i.e., mappings with the modular Lipschitz constant 1) on subsets of modular metric spaces which are natural generalization of classical modulars over linear spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and many others.
The importance for applications of mappings defined within modular function spaces consists in the richness of the structure of modular function spaces, which, besides being Banach spaces, are equipped with modular equivalents of norm or metric notions and also are equipped with almost everywhere convergence and convergence in sub-measure.
We introduce the notion of α-type F-contraction in the setting of modular metric spaces which is independent from one given in (Hussain et al. in Fixed Point Theory Appl. 2015:158, 2015).
