Bounds of Solutions of Integrodifferential Equations
ZdenΔk Ε marda1
Academic Editor: Miroslava RΕ―ΕΎiΔkovΓ‘
Received20 Jan 2011
Accepted24 Feb 2011
Published14 May 2011
Abstract
Some new integral inequalities are given, and bounds of solutions of the following integro-differential equation are determined: , , where , , are continuous functions, .
1. Introduction
Ou Yang [1] 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, [2β19]. 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 [21β23].
Pachpatte [11, 12, 14β16] and Salem [24] have given some new integral inequalities of the Gronwall-Ou Yang type involving functions and their derivatives. Lipovan [7] used the modified Gronwall-Ou Yang inequality with logarithmic factor in the integrand to the study of wave equation with logarithmic nonlinearity. Engler [5] 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 [3] 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 .
Proof. Since the function is positive and nondecreasing, we obtain from (2.10)
Applying Lemma 2.1 to inequality (2.12), we obtain desired inequality (2.11).
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 .
Proof. Multiplying both sides of (3.1) by and integrating from 0 to we obtain
From (3.2) and (3.4), we get
Using inequality (2.14) in Lemma 2.3, we have
where
which is the desired inequality (3.3).
Remark 3.2. It is obvious that inequality (3.3) gives the bound of the solution of (3.1) in terms of the known functions.
Acknowledgment
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.
References
L. Ou Yang, βThe boundedness of solutions of linear differential equations ,β Advances in Mathematics, vol. 3, pp. 409β415, 1957.
D. Baınov and P. Simeonov, Integral Inequalities and Applications, vol. 57 of Mathematics and Its Applications (East European Series), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
S. S. Dragomir, βOn Volterra integral equations with kernels of L-type,β Analele Universit a tii din Timi soara. Seria Stiin te Matematice, vol. 25, no. 2, pp. 21β41, 1987.
S. S. Dragomir and Y.-H. Kim, βOn certain new integral inequalities and their applications,β JIPAM: Journal of Inequalities in Pure and Applied Mathematics, vol. 3, no. 4, article 65, p. 8, 2002.
H. Engler, βGlobal regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity,β Mathematische Zeitschrift, vol. 202, no. 2, pp. 251β259, 1989.
A. Haraux, Nonlinear Evolution Equations. Global Behavior of Solutions, vol. 841 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981.
O. Lipovan, βA retarded integral inequality and its applications,β Journal of Mathematical Analysis and Applications, vol. 285, no. 2, pp. 436β443, 2003.
Q. H. Ma and L. Debnath, βA more generalized Gronwall-like integral inequality wit applications,β International Journal of Mathematics and Mathematical Sciences, vol. 15, pp. 927β934, 2003.
Q.-H. Ma and E.-H. Yang, βOn some new nonlinear delay integral inequalities,β Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 864β878, 2000.
F. W. Meng and W. N. Li, βOn some new integral inequalities and their applications,β Applied Mathematics and Computation, vol. 148, no. 2, pp. 381β392, 2004.
B. G. Pachpatte, βOn some new inequalities related to certain inequalities in the theory of differential equations,β Journal of Mathematical Analysis and Applications, vol. 189, no. 1, pp. 128β144, 1995.
B. G. Pachpatte, βOn some integral inequalities similar to Bellman-Bihari inequalities,β Journal of Mathematical Analysis and Applications, vol. 49, pp. 794β802, 1975.
B. G. Pachpatte, βOn certain nonlinear integral inequalities and their discrete analogues,β Facta Universitatis. Series: Mathematics and Informatics, no. 8, pp. 21β34, 1993.
B. G. Pachpatte, βOn some fundamental integral inequalities arising in the theory of differential
equations,β Chinese Journal of Contemporary Mathematics, vol. 22, pp. 261β273, 1994.
B. G. Pachpatte, βOn a new inequality suggested by the study of certain epidemic models,β Journal of Mathematical Analysis and Applications, vol. 195, no. 3, pp. 638β644, 1995.
B. G. Pachpatte, Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering 197, Academic Press, San Diego, Calif, USA, 2006.
Z. Šmarda, βGeneralization of certain integral inequalities,β in Proceedings of the 8th International Conference on Applied
Mathematics (APLIMAT '09), pp. 223β228, Bratislava, Slovakia, 2009.
E. H. Yang, βOn asymptotic behaviour of certain integro-differential equations,β Proceedings of the American Mathematical Society, vol. 90, no. 2, pp. 271β276, 1984.
C.-J. Chen, W.-S. Cheung, and D. Zhao, βGronwall-Bellman-type integral inequalities and applications to BVPs,β Journal of Inequalities and Applications, vol. 2009, Article ID 258569, 15 pages, 2009.
E. H. Yang, βOn some new discrete generalizations of Gronwall's inequality,β Journal of Mathematical Analysis and Applications, vol. 129, no. 2, pp. 505β516, 1988.
J. Baštinec and J. Diblík, βAsymptotic formulae for a particular solution of linear nonhomogeneous discrete equations,β Computers & Mathematics with Applications, vol. 45, no. 6–9, pp. 1163β1169, 2003.
J. Diblík, E. Schmeidel, and M. Růžičková, βExistence of asymptotically periodic solutions of system of Volterra difference equations,β Journal of Difference Equations and Applications, vol. 15, no. 11-12, pp. 1165β1177, 2009.
J. Diblík, E. Schmeidel, and M. Růžičková, βAsymptotically periodic solutions of Volterra system of difference equations,β Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2854β2867, 2010.
S. Salem, βOn some systems of two discrete inequalities of gronwall type,β Journal of Mathematical Analysis and Applications, vol. 208, no. 2, pp. 553β566, 1997.