- 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.
- Difference Equation
- Nonnegative Function
- Integral Inequality
- Discrete Analogue
- Positive Sequence
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.
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]).
If we set in Lemma 2.1, then we can obtain the following corollary.
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.
Since and , this implies that our inequality holds.
Hence, we can obtain a comparison result for linear difference inequalities.
where . Hence, the proof is complete.
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].
If we use Lemma 2.1 in the proof of Theorem 2.5, then we obtain the following bound of which contains double fold summations.
Since , the proof is complete.
The proof of this corollary follows by the similar argument as in the proof of Theorem 2.5. We omit the details.
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.
where and are matrices for each and .
Lemma 3.1 (see [18, Theorem 2.9.1]).
where is a matrix over and . We present the variation of constants formula of difference equations.
where is a fundamental matrix solution of the difference equation such that is the identity matrix.
with initial condition , where .
Suppose that the following conditions hold for , :
(iii) , ,
where and are some positive constants and is a nonnegative sequence defined on with . Then, all solutions of (3.7) are bounded in .
and . Hence, the solutions of (3.7) are bounded in , and the proof is complete.
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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