A Note on the Integral Inequalities with Two Dependent Limits
© Allaberen Ashyralyev et al. 2010
Received: 4 October 2009
Accepted: 5 July 2010
Published: 4 August 2010
The theorem on the Gronwall's type integral inequalities with two dependent limits is established. In application, the boundedness of the solutions of nonlinear differential equations is presented.
Integral inequalities play a significant role in the study of qualitative properties of solutions of integral, differential and integro-differential equations (see, e.g., [1–4] and the references given therein). One of the most useful inequalities in the development of the theory of differential equations is given in the following lemma (see ).
The present study involves some Gronwall's type integral inequalities with two dependent limits. Section 2 includes some new integral inequalities with two dependent limits and relevant proofs. Subsequently, Section 3 includes an application on the boundedness of the solutions of nonlinear differential equations.
2. A Main Statement
Our main statement is given by the following theorem.
The inequality (2.8) follows from (2.29), (2.55), and (2.64). Theorem 2.1 is proved.
3. An Application
Let us give a corollary of Theorem 2.1.
Therefore, the inequality (3.4) follows from the last inequality. Theorem 3.1 is proved.
The authors thank professor O. Celebi (Turkey), professor R. P. Agarwal (USA), and anonymous reviewers for their valuable comments.
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