Sentence examples for maps and hence from inspiring English sources

The next example motivates the study of multivalued mappings by showing that control problems may be translated in terms of multivalued maps and, hence, a study of their fixed points could provide new solution schemes to such problems.

Since f and g are surjective maps and hence there exist two points y and (y^{prime}) in X such that (x^=fy) and (x^=gy ^{prime}).

(2) In Example 2.2, the pair is a Banach operator but and are not -commuting maps and hence not a subcompatible pair.  .

In Example 2.2, the pair is a Banach operator but and are not -commuting maps and hence not a subcompatible pair.

Experimental results show that the proposed method constructs plausible ghost maps and hence yields pleasing HDR images without noticeable ghost artifacts.

For our exposition, we introduce a new notion of density which has serious advantages over the standard form because it is independent of index maps — and hence has much broader application.

We observe from (1.1), (1.2) and (1.3) that every nonexpansive mapping is an α-strictly pseudocontractive mapping and every α-strictly pseudocontractive mapping is a pseudocontractive mapping, and hence a class of pseudocontractive mappings is a more general class of mappings.

Now by [21, Lemma  2.4], is a linear mapping and hence is a linear mapping.

and the mapping, and hence, generates a Nemytskii operator by (3.12).

Lemma 2.2 shows that every is a firmly nonexpansive mapping and hence nonexpansive and.

This implies that is a contraction mapping and hence the conclusion follows.