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Bounds of solutions of integrodifferential equations. (English) Zbl 1217.45005

Summary: Some new integral inequalities are given, and bounds of solutions of the following integro-differential equation are determined: \(x'(t) - \mathcal F (t, x(t), \int_0^t k(t, s, x(t), x(s))ds) = h(t)\), \(x(0) = x_0\), where \(h : \mathbb{R}_+ \rightarrow \mathbb{R}\), \(k : \mathbb{R}^2_+ \times \mathbb{R}^2 \rightarrow \mathbb{R}\), \(\mathcal F : \mathbb{R}_+ \times \mathbb{R}^2 \rightarrow \mathbb{R}\) are continuous functions, \(\mathbb{R}_+ = [0, \infty)\).

MSC:

45J05 Integro-ordinary differential equations
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[1] L. Ou Yang, “The boundedness of solutions of linear differential equations y\(^{\prime}\)+A(t)y=0,” Advances in Mathematics, vol. 3, pp. 409-415, 1957.
[2] 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. · Zbl 0912.45012
[3] S. S. Dragomir, “On Volterra integral equations with kernels of L-type,” Analele Universit a tii din Timi soara. Seria Stiin te Matematice, vol. 25, no. 2, pp. 21-41, 1987. · Zbl 0657.45005
[4] S. S. Dragomir and Y.-H. Kim, “On certain new integral inequalities and their applications,” JIPAM: Journal of Inequalities in Pure and Applied Mathematics, vol. 3, no. 4, article 65, p. 8, 2002. · Zbl 1026.26008
[5] H. Engler, “Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity,” Mathematische Zeitschrift, vol. 202, no. 2, pp. 251-259, 1989. · Zbl 0697.73033 · doi:10.1007/BF01215257
[6] A. Haraux, Nonlinear Evolution Equations. Global Behavior of Solutions, vol. 841 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981. · Zbl 0461.35002
[7] O. Lipovan, “A retarded integral inequality and its applications,” Journal of Mathematical Analysis and Applications, vol. 285, no. 2, pp. 436-443, 2003. · Zbl 1040.26007 · doi:10.1016/S0022-247X(03)00409-8
[8] Q. H. Ma and L. Debnath, “A more generalized Gronwall-like integral inequality wit applications,” International Journal of Mathematics and Mathematical Sciences, vol. 15, pp. 927-934, 2003. · Zbl 1014.26015 · doi:10.1155/S0161171203205299
[9] Q.-H. Ma and E.-H. Yang, “On some new nonlinear delay integral inequalities,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 864-878, 2000. · Zbl 0974.26015 · doi:10.1006/jmaa.2000.7134
[10] F. W. Meng and W. N. Li, “On some new integral inequalities and their applications,” Applied Mathematics and Computation, vol. 148, no. 2, pp. 381-392, 2004. · Zbl 1045.26009 · doi:10.1016/S0096-3003(02)00855-X
[11] B. G. Pachpatte, “On some new inequalities related to certain inequalities in the theory of differential equations,” Journal of Mathematical Analysis and Applications, vol. 189, no. 1, pp. 128-144, 1995. · Zbl 0824.26010 · doi:10.1006/jmaa.1995.1008
[12] B. G. Pachpatte, “On some integral inequalities similar to Bellman-Bihari inequalities,” Journal of Mathematical Analysis and Applications, vol. 49, pp. 794-802, 1975. · Zbl 0305.26009 · doi:10.1016/0022-247X(75)90220-6
[13] B. G. Pachpatte, “On certain nonlinear integral inequalities and their discrete analogues,” Facta Universitatis. Series: Mathematics and Informatics, no. 8, pp. 21-34, 1993. · Zbl 0839.26013
[14] B. G. Pachpatte, “On some fundamental integral inequalities arising in the theory of differential equations,” Chinese Journal of Contemporary Mathematics, vol. 22, pp. 261-273, 1994. · Zbl 0820.26009
[15] B. G. Pachpatte, “On a new inequality suggested by the study of certain epidemic models,” Journal of Mathematical Analysis and Applications, vol. 195, no. 3, pp. 638-644, 1995. · Zbl 0842.92022 · doi:10.1006/jmaa.1995.1380
[16] B. G. Pachpatte, Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering 197, Academic Press, San Diego, Calif, USA, 2006. · Zbl 1112.26021
[17] Z. \vSmarda, “Generalization of certain integral inequalities,” in Proceedings of the 8th International Conference on Applied Mathematics (APLIMAT ’09), pp. 223-228, Bratislava, Slovakia, 2009.
[18] E. H. Yang, “On asymptotic behaviour of certain integro-differential equations,” Proceedings of the American Mathematical Society, vol. 90, no. 2, pp. 271-276, 1984. · Zbl 0537.45006 · doi:10.2307/2045354
[19] C.-J. Chen, W.-S. Cheung, and D. Zhao, “Gronwall-Bellman-type integral inequalities and applications to BVPs,” Journal of Inequalities and Applications, vol. 2009, Article ID 258569, 15 pages, 2009. · Zbl 1176.35007 · doi:10.1155/2009/258569
[20] E. H. Yang, “On some new discrete generalizations of Gronwall’s inequality,” Journal of Mathematical Analysis and Applications, vol. 129, no. 2, pp. 505-516, 1988. · Zbl 0643.26013 · doi:10.1016/0022-247X(88)90268-5
[21] J. Ba\vstinec and J. Diblík, “Asymptotic formulae for a particular solution of linear nonhomogeneous discrete equations,” Computers & Mathematics with Applications, vol. 45, no. 6-9, pp. 1163-1169, 2003. · Zbl 1055.39003 · doi:10.1016/S0898-1221(03)00077-4
[22] J. Diblík, E. Schmeidel, and M. Rů\vzi, “Existence of asymptotically periodic solutions of system of Volterra difference equations,” Journal of Difference Equations and Applications, vol. 15, no. 11-12, pp. 1165-1177, 2009. · Zbl 1180.39022 · doi:10.1080/10236190802653653
[23] J. Diblík, E. Schmeidel, and M. Rů\vzi, “Asymptotically periodic solutions of Volterra system of difference equations,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2854-2867, 2010. · Zbl 1202.39013 · doi:10.1016/j.camwa.2010.01.055
[24] S. Salem, “On some systems of two discrete inequalities of gronwall type,” Journal of Mathematical Analysis and Applications, vol. 208, no. 2, pp. 553-566, 1997. · Zbl 0877.26012 · doi:10.1006/jmaa.1997.5257
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