Sentence examples for by applying inequality from inspiring English sources
Follows directly by applying inequality (1.4).
Now by applying inequality (2.26), and (3.6) and (3.7) thus we establish Theorem 3.2.
On the other hand, by (3.1), we have f ≤ J ( f ) and, by applying inequality (3.5), we see that d ( J ( f ), f ) ≤ 1. Applying Theorem 2.1, it follows that J has a fixed point T ∈ X such that lim n → ∞ d ( J n ( f ), T ) = 0.
By applying inequalities (4), (6), and (7) we get the following corollaries.
So (-X_{1},-X_{2},ldots,-X_{n}) are (mathcal{F} -acceptable random variables that satisF} -acceptable4) forandomq tleq T), and therefore by applying inequalities (5) and (6) for (-S_{n}) we havariables^{mathatl{F}}(satisfyq-epsilon)leq g(n)e^{-fraconditionn^{2}}{2C_{n}}} quad mbox{a.s.
If we apply inequality (4.2) by choosing (411).
If we apply inequality (5.7) by choosing,,, and,, we directly get the desired inequality.
Case 2. Let, then we can apply inequality (2.23) by to obtain (33).
If we apply inequality (2.1) for by choosing,,, and and taking into consideration that for all, we have (215).
Our main object of this article is by applying differential inequality techniques to analyze the exponential stability of model (1.4).
