Sentence examples for extensions are proper from inspiring English sources

We show by examples that the obtained extensions are proper.

Some examples support our results and show that the obtained extensions are proper.

An example showing that this extension is proper is given.

An example which shows that this extension is proper is given.

Some concrete examples in a Hilbert space showing that our extension is proper are also given.

Examples are given to show that our results are proper extensions of the known ones.

Examples show that the obtained results are proper extensions of the existing ones.

Thus, Theorem 3.1 and its corollaries are proper extensions of these results.

Our results are proper extensions and generalizations of results of Kutbi and Sintunavarat [1] on quasi b-metric spaces.

This example also shows (as in [23], Remark 2.4]) the importance of the second generalized altering distance function ψ 2 in Theorems 1 and 2. The next example shows that Theorems 1 and 2 are proper extensions of the respective results in standard metric spaces.

Most results obtained in this paper seem new and are proper extensions of the known results in [2, 16]; in particular, our Theorem 3.3 extends and improves the result in [16], Theorem 5.1, and, even in the special case when (U_{1}) and (U_{2}) are singletons, our Corollary 5.1 improves the corresponding result in [2], Theorem 6.9.