Sentence examples for using the differential inequality from inspiring English sources
We find that it is difficult to study the existence of blow-up and global solutions of problem (1.1) by using the differential inequality technique in [1].
By using the differential inequality technique, they gave the upper and lower bounds for blow-up time when the blow-up of the solution occurred.
By constructing some auxiliary functions and using the differential inequality technique, they established the conditions on functions f, g, h, and (u_{0}) to ensure that the solution u blows up at some time.
To overcome this difficulty, we used the differential inequality theory, and finally proved the following results: if the positive equilibrium exists, then it is globally attractive.
However, it is difficult to use the differential inequality technique employed in [20, 24, 25] to study the blow-up problem of (1.1).
Our research relies mainly on constructing some auxiliary functions and using the parabolic maximum principles and the differential inequality technique.
Some results on the existence of positive almost periodic solutions for the equations are obtained by using the contracting mapping principle and the differential inequality technique.
New criteria for asymptotic stability of the zero solution are established using the fixed point method and the differential inequality techniques.
So, by employing an auxiliary function p t) on the contraction condition, we get new criteria for asymptotic stability of the zero solution by using the fixed point method and the differential inequality techniques which not only includes the results on sufficient part in [17 19], but also includes several equations that previously known related results can not be applied to.
Using these auxiliary functions, the parabolic maximum principle, and the differential inequality technique, we complete the study of (1.1).
Using Lemma 4.6, we find that the function y n satisfies the differential inequality y ˙ n ( t ) ≤ C ( 1 + y n 8 ( t ) ). (109).
