Sentence examples for modifying the proof from inspiring English sources

Modifying the proof of Chen [4, Proposition 2.5].

By slightly modifying the proof of Theorem 1 and Theorem 2 in [15], we get the following two lemmas.

By modifying the proof of Theorem 3.1 of [7] (or see [25]), we can prove the following lemma.

We will extend Theorem 1 to maps f : ( B 2, S 1 ) → ( R 2, S 1 ) by modifying the proof of Theorem 1 in [2].

The following lemma is needed to prove Theorems 1.4 and 1.5 and it is established by modifying the proof of Lemma 3 in Fakhre-Zakeri and Lee [9].

The following result can be proved by slightly modifying the proof of Lemma 3.2 of Chen [52] and it will play an important role in finding the bounds for positive solutions to (1.3).

We modify the proof of Theorem 3.1 of [29].

Restrictions on the function h allow us to modify the proof of the directed case.

For the case (u(x ge0), we modify the proof of Theorem 2.1 in the following way.

Our next intention is to show that considering a simplified form of (4.12) we can modify the proof of Theorem 3.1 to obtain a stronger estimate of solutions.

In this paper, lower semi-continuous functions are used to modify the proof of Caristi's fixed point theorems on partial metric spaces.