Sentence examples for modular spaces as from inspiring English sources

Now, we recall some basic notions and facts about modular spaces as formulated by Kozlowski [18].

As commuting mappings are symmetric Banach operator pairs, we obtain an analog to DeMarr's common fixed point theorem [8] in modular spaces as follows.

The notion of modular spaces, as a generalize of metric spaces, was introduced by Nakano [11] and was intensively developed by Koshi, Shimogaki, Yamamuro [11 13] and others.

Indeed we look at these spaces as the nonlinear version of the classical modular spaces as introduced by Nakano [15] on vector spaces and modular function spaces introduced by Musielack [16] and Orlicz [17].

The notion of modular spaces, as a generalization of metric spaces, was introduced by Nakano [1] in 1950 in connection with the theory of order spaces and redefined and generalized by Musielak and Orlicz [2] in 1959.

Indeed we look at these spaces as the nonlinear version of the classical modular spaces as introduced by Nakano [3] on vector spaces and modular function spaces introduced by Musielak [4] and Orlicz [5].

Mongkolkeha and Kumam [30] proved the existence of a fixed point generalized weak contractive mapping in modular space as follows.

The above definition will allow us to introduce the growth function in the modular metric spaces as was done in the linear case.

Now, we recall some basic facts and notations as regards modular spaces.

Indeed, we look at these spaces as a nonlinear version of the classical modular spaces, introduced by Nakano [3], on vector spaces and modular function spaces, introduced by Musielak [4] and Orlicz [5].

Visitors will be allowed to enter the interiors of the modular spaces in small groups.