Sentence examples for using the above theorem from inspiring English sources

Using the above theorem, we can obtain the following multiplicity result of positive solutions to (1.1).

Using the above Theorem 5, we can get the following result.

Using the above theorem, we obtain the proximity of LMS attractors and the WFs in the following.

Using the above theorems and mathematical induction, we obtain the following theorem.

This will be done by using the above theorems and the corrected three-point quadrature formulae recently introduced in [10].

In the sequel we use the above theorems to obtain generalizations of the results proved in the previous section.

Some concrete results have been derived by using the above two theorems.

Some concrete results are derived by using the above two theorems.

Using the above Lemmas and Theorem 1, we can get the following Corollaries describing the uniqueness results.

By using the above lemmas and Theorem 1.2, as in [2], we can prove the following results.

By using the above lemma, we obtain Theorem 4.3.