Sentence examples for modular spaces and from inspiring English sources

In 1950 [1], Nakano initiated the theory of modular spaces and further it was generalized and redefined by Musielak and Orlicz [2, 3] in 1959.

In this paper, we introduce the notions of nonlinear contractions in modular spaces and establish their fixed points theorems in modular spaces.

We will start with a brief recollection of basic concepts and facts in modular spaces and modular metric spaces (see [14, 15, 27 29] for more details).

In 2015, Jleli and Samet [1] introduced a very interesting concept of a generalized metric space, which covers different well-known metric structures including classical metric spaces, b-metric spaces, dislocated metric spaces, modular spaces, and so on.

Recently several years, the researchers in [3 14] and others continued the study of Rhoades and Branciari, proved some fixed point and common fixed point theorems for various generalized weakly contraction mappings and contractive mappings of integral type in complete metric spaces, Banach spaces, modular spaces and symmetric spaces.

Let ({X_{rho}}) be a ρ-complete modular space and (T:{X_{rho}}rightarrow{X_{rho}}).

Let ((X_{rho},rho)) be a complete modular space and (f: Xto X) be a mapping.

Let ({X_{rho}}) be a modular space and (T:{X_{rho}}rightarrow{X_{rho}}) be a self-map.

Let (X_{rho}) be a modular space and let ({ x_{n} } ) be a sequence in (X_{rho}).

Let ({X_{rho}}) be a ρ-modular space and (Delta ={(x,x):xin X}) denote the diagonal of ({X_{rho}}times{X_{rho}}). Let G be a directed graph such that the set (V(G)) of its vertices coincides with X and (E(G)) be the set of edges of the graph such that (Deltasubseteq E(G)).

For a current review of the theory of Musielak-Orlicz spaces and modular spaces, the reader is referred to Musielak [12] and Kozlowski [10].