Sentence examples for using the monotonicity of from inspiring English sources

By using the monotonicity of, we obtain (2.14).

By adding (1.19) with (1.20) and using the monotonicity of, we have (1.21).

Hence, summing up this two inequalities and using the monotonicity of and, (3.44).

by multiplying and using the monotonicity of, then we obtain (3.38).

Integrating (3.7) over and using the monotonicity of we have (3.9).

Using the monotonicity of and the choosing rule of, we have (3.27).

Now, we use the monotonicity of ( 1 − | x | a n + δ n ) 1 4 ϕ ( x ).

In the following text, we will often use the monotonicity of (F_{xy}).

here we use the monotonicity of φ ( t ) and φ ( t ) / t.

Secondly, we use the monotonicity of the successor function, the geometric theory of differential equations, and the definition of the successor function to prove the uniqueness of an order-one periodic solution.

It holds