Sentence examples for by using the contraction from inspiring English sources

Also by using the contraction mapping theorem we get the uniqueness result.

First, by using the contraction principle, the existence and uniqueness of mild solutions are given.

Proof To prove this lemma, one proceeds as in Theorem 3.1 by using the contraction argument.

Then the proof now can be finished by using the contraction mapping principle.

In section 4, we prove Theorem 1.2 by using the contraction mapping principle.

Uniqueness criteria of the steady state in porous catalysts are derived by using the contraction mapping principle with different norms.

Since Δ < 1, F is a contraction mapping, therefore, by using the Banach contraction principle, F has a unique fixed point, which is the unique solution of problem (1) by using Lemma 2.2.

The uniqueness of solution was established by using the Banach contraction mapping principle.

The second result is the existence and uniqueness of a solution to problem (1.1), by using the Banach contraction principle.

The uniqueness of solutions of the fractional differential system is established by using the Banach contraction principle.

By using the Banach contraction mapping principle, we know that (T_{mu}) has a unique fixed point (x^).