Sentence examples for contained properly from inspiring English sources

The monotone operator B is said to be maximal monotone if the graph of B, (G(B)), is not contained properly in any other monotone subset of (E times E^).

They were warned that radioactive waste, if not contained properly, could contaminate soil, water and air for hundreds of miles.

The following result contains properly the recent main result (Theorem 2.1) of Shi and Xu [39].

Our results contain properly the main theorem due to Suzuki [24] and many of its extensions [23] (see also [28]).

(2) The class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-nonexpansive mappings as a subclass, but the converse is not true.

The class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of relatively nonexpansive mappings (see [17]) as a subclass.

The class of quasi-ϕ-nonexpansive multi-valued mappings contains properly the class of relatively nonexpansive multi-valued mappings as a subclass, but the converse is not true.

It is obvious that the class of relatively nonexpansive multivalued mappings contains properly the class of relatively nonexpansive multivalued mappings (for example, see [20]).

In [6], Rhoades gave an example to show that the class of asymptotically pseudocontractive mappings contains properly the class of asymptotically nonexpansive mappings, see [6] for more details.

It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains properly the class of asymptotically nonexpansive mappings.

The class of asymptotically pseudocontraction contains properly the class of asymptotically nonexpansive mappings and every asymptotically nonexpansive mapping is a uniformly -Lipschitzian [2, 24].