On Some New Impulsive Integral Inequalities
© Jianli Li. 2008
Received: 4 June 2008
Accepted: 21 July 2008
Published: 29 July 2008
We establish some new impulsive integral inequalities related to certain integral inequalities arising in the theory of differential equalities. The inequalities obtained here can be used as handy tools in the theory of some classes of impulsive differential and integral equations.
Differential and integral inequalities play a fundamental role in global existence, uniqueness, stability, and other properties of the solutions of various nonlinear differential equations; see [1–4]. A great deal of attention has been given to differential and integral inequalities; see [1, 2, 5–8] and the references given therein. Motivated by the results in [1, 5, 7], the main purpose of this paper is to establish some new impulsive integral inequalities similar to Bihari's inequalities.
To prove our main results, we need the following result (see [1, Theorem 1.4.1]).
2. Main Results
In this section, we will state and prove our results.
This proof is similar to that of Theorem 2.1; thus we omit the details here.
If , then (2.1), (2.8), (2.10), and (2.21) have no impulses. In this case, it is clear that Theorems 2.2-2.3 improve the corresponding results of [5, Theorem 1].
An interesting and useful special version of Theorem 2.6 is given in what follows.
This proof is complete.
This work is supported by the National Natural Science Foundation of China (Grants nos. 10571050 and 60671066). The project is supported by Scientific Research Fund of Hunan Provincial Education Department (07B041) and Program for Young Excellent Talents at Hunan Normal University.
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