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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 571795, 7 pages
Bounds of Solutions of Integrodifferential Equations
Department of Mathematics, Faculty of Electrical Engineering and Communication, Technická 8, Brno University of Technology, 61600 Brno, Czech Republic
Received 20 January 2011; Accepted 24 February 2011
Academic Editor: Miroslava Růžičková
Copyright © 2011 Zdeněk Šmarda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Some new integral inequalities are given, and bounds of solutions of the following integro-differential equation are determined: , , where , , are continuous functions, .
Ou Yang  established and applied the following useful nonlinear integral inequality.
Theorem 1.1. Let and be nonnegative and continuous functions defined on and let be a constant. Then, the nonlinear integral inequality implies
This result has been frequently used by authors to obtain global existence, uniqueness, boundedness, and stability of solutions of various nonlinear integral, differential, and integrodifferential equations. On the other hand, Theorem 1.1 has also been extended and generalized by many authors; see, for example, [2–19]. Like Gronwall-type inequalities, Theorem 1.1 is also used to obtain a priori bounds to unknown functions. Therefore, integral inequalities of this type are usually known as Gronwall-Ou Yang type inequalities.
In the last few years there have been a number of papers written on the discrete inequalities of Gronwall inequality and its nonlinear version to the Bihari type, see [13, 16, 20]. Some applications discrete versions of integral inequalities are given in papers [21–23].
Pachpatte [11, 12, 14–16] and Salem  have given some new integral inequalities of the Gronwall-Ou Yang type involving functions and their derivatives. Lipovan  used the modified Gronwall-Ou Yang inequality with logarithmic factor in the integrand to the study of wave equation with logarithmic nonlinearity. Engler  used a slight variant of the Haraux's inequality for determination of global regular solutions of the dynamic antiplane shear problem in nonlinear viscoelasticity. Dragomir  applied his inequality to the stability, boundedness, and asymptotic behaviour of solutions of nonlinear Volterra integral equations.
In this paper, we present new integral inequalities which come out from above-mentioned inequalities and extend Pachpatte's results (see [11, 16]) especially. Obtained results are applied to certain classes of integrodifferential equations.
2. Integral Inequalities
Lemma 2.1. Let , , and be nonnegative continuous functions defined on . If the inequality holds where is a nonnegative constant, , then for .
Proof. Define a function by the right-hand side of (2.1) Then, , and Define a function by then , , Integrating (2.7) from 0 to , we have Using (2.8) in (2.6), we obtain Integrating from 0 to and using , we get inequality (2.2). The proof is complete.
Lemma 2.2. Let , , and be nonnegative continuous functions defined on , be a positive nondecreasing continuous function defined on . If the inequality holds, where is a nonnegative constant, , then where .
Lemma 2.3. Let , , , and be nonnegative continuous functions defined on , and let be a nonnegative constant.
If the inequality holds for , then where
Proof. Define a function by the right-hand side of (2.13) Then , and Differentiating and using (2.17), we get Integrating inequality (2.18) from 0 to , we have where is defined by (2.15), is positive and nondecreasing for . Now, applying Lemma 2.2 to inequality (2.19), we get Using (2.20) and the fact that , we obtain desired inequality (2.14).
3. Application of Integral Inequalities
Consider the following initial value problem where , , are continuous functions. We assume that a solution of (3.1) exists on .
Theorem 3.1. Suppose that where , are nonnegative continuous functions defined on . Then, for the solution of (3.1) the inequality holds on .
This author was supported by the Council of Czech Government grant MSM 00216 30503 and MSM 00216 30529 and by the Grant FEKTS-11-2-921 of Faculty of Electrical Engineering and Communication.
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