- Research Article
- Open Access
Volterra Discrete Inequalities of Bernoulli Type
© K. Choi and N. Koo 2010
Received: 27 April 2010
Accepted: 12 December 2010
Published: 27 January 2011
We obtain the discrete versions of integral inequalities of Bernoulli type obtained in Choi (2007) and give an application to study the boundedness of solutions of nonlinear Volterra difference equations.
Integral inequalities of Gronwall type have been very useful in the study of ordinary differential equations. Sugiyama  proved the discrete analogue of the well-known Gronwall-Bellman inequality [2–5] which find numerous applications in the theory of finite difference equations. See [6–11] for differential inequalities and difference inequalities.
Willett and Wong  established some discrete generalizations of the results of Gronwall . The discrete analogue of the result of Bihari  was partially given by Hull and Luxemburg  and was used by them for the numerical treatment of ordinary differential equations. Pachpatte  obtained some general versions of Gronwall-Bellman inequality. Oguntuase  established some generalizations of the inequalities obtained in . However, there were some defects in the proofs of Theorems 2.1 and 2.7 in . Choi et al.  improved the results of  and gave an application to boundedness of the solutions of nonlinear integrodifferential equations.
In this paper, we establish the discrete analogues of integral inequalities of Bernoulli type in  and give an application to study the boundedness of solutions of nonlinear Volterra difference equations.
2. Main Results
Pachpatte  proved the following useful discrete inequality which can be used in the proof of various discrete inequalities. Let and for fixed nonnegative integers and .
Lemma 2.1 (see [11, Theorem 2.3.4]).
Willet and Wong [12, Theorem 4] proved the nonlinear difference inequality by using the mean value theorem. We obtain the following result which is slightly different from Willet and Wong's Theorem 4.
Hence, we can obtain a comparison result for linear difference inequalities.
If we set in Theorem 2.5, then we obtain the following corollary from Theorem 2.5. This is an analogue of the nonlinear difference inequality in [17, Corollary 2.8].
The proof of this corollary follows by the similar argument as in the proof of Theorem 2.5. We omit the details.
3. An Application
In this section, we present an application of nonlinear difference inequalities established in Theorem 2.5 to study the boundedness of the solutions of nonlinear Volterra difference equations.
Lemma 3.1 (see [18, Theorem 2.9.1]).
This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (NRF-2010-0008835). The authors are thankful to the referee for giving valuable comments for the improvement of this paper.
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