Journal of Applied Mathematics

Generalized Nonlinear Volterra-Fredholm Type Integral Inequality with Two Variables

Yusong Lu, Wu-Sheng Wang, Xiaoliang Zhou, and Yong Huang

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Abstract

We establish a class of new nonlinear retarded Volterra-Fredholm type integral inequalities, with two variables, where known function w in integral functions in Q.-H. Ma and J. Pečarić, 2008 is changed into the functions w 1 , w 2 . By adopting novel analysis techniques, such as change of variable, amplification method, differential and integration, inverse function, and the dialectical relationship between constants and variables, the upper bounds of the embedded unknown functions are estimated. The derived results can be applied in the study of solutions of ordinary differential equations and integral equations.

Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 359280, 14 pages.

Dates
First available in Project Euclid: 2 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.jam/1425305702

Digital Object Identifier
doi:10.1155/2014/359280

Mathematical Reviews number (MathSciNet)
MR3200840

Citation

Lu, Yusong; Wang, Wu-Sheng; Zhou, Xiaoliang; Huang, Yong. Generalized Nonlinear Volterra-Fredholm Type Integral Inequality with Two Variables. J. Appl. Math. 2014 (2014), Article ID 359280, 14 pages. doi:10.1155/2014/359280. https://projecteuclid.org/euclid.jam/1425305702


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References

  • T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,” Annals of Mathematics, vol. 20, no. 4, pp. 292–296, 1919.
  • R. Bellman, “The stability of solutions of linear differential equations,” Duke Mathematical Journal, vol. 10, pp. 643–647, 1943.
  • I. Bihari, “A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 7, pp. 81–94, 1956.
  • D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991.
  • D. Baĭnov and P. Simeonov, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
  • B. G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, London, UK, 1998.
  • O. Lipovan, “A retarded Gronwall-like inequality and its applications,” Journal of Mathematical Analysis and Applications, vol. 252, no. 1, pp. 389–401, 2000.
  • B. G. Pachpatte, “Explicit bound on a retarded integral inequality,” Mathematical Inequalities & Applications, vol. 7, no. 1, pp. 7–11, 2004.
  • R. P. Agarwal, S. Deng, and W. Zhang, “Generalization of a retarded Gronwall-like inequality and its applications,” Applied Mathematics and Computation, vol. 165, no. 3, pp. 599–612, 2005.
  • W.-S. Cheung, “Some new nonlinear inequalities and applications to boundary value problems,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 64, no. 9, pp. 2112–2128, 2006.
  • W.-S. Wang, “A generalized retarded Gronwall-like inequality in two variables and applications to BVP,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 144–154, 2007.
  • R. P. Agarwal, C. S. Ryoo, and Y.-H. Kim, “New integral inequalities for iterated integrals with applications,” Journal of Inequalities and Applications, vol. 2007, Article ID 24385, 18 pages, 2007.
  • Q.-H. Ma and J. Pečarić, “Estimates on solutions of some new nonlinear retarded Volterra-Fredholm type integral inequalities,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 69, no. 2, pp. 393–407, 2008.
  • W.-S. Wang and C.-X. Shen, “On a generalized retarded integral inequality with two variables,” Journal of Inequalities and Applications, vol. 2008, Article ID 518646, 9 pages, 2008.
  • Y.-H. Kim, “Gronwall, Bellman and Pachpatte type integral inequalities with applications,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 71, no. 12, pp. e2641–e2656, 2009.
  • R. A. C. Ferreira and D. F. M. Torres, “Generalized retarded integral inequalities,” Applied Mathematics Letters, vol. 22, no. 6, pp. 876–881, 2009.
  • W.-S. Wang, Z. Li, Y. Li, and Y. Huang, “Nonlinear retarded integral inequalities with two variables and applications,” Journal of Inequalities and Applications, vol. 2010, Article ID 240790, 21 pages, 2010.
  • W.-S. Wang, R.-C. Luo, and Z. Li, “A new nonlinear retarded integral inequality and its application,” Journal of Inequalities and Applications, vol. 2010, Article ID 462163, 9 pages, 2010.
  • L. Li, F. Meng, and L. He, “Some generalized integral inequalities and their applications,” Journal of Mathematical Analysis and Applications, vol. 372, no. 1, pp. 339–349, 2010.
  • A. Abdeldaim and M. Yakout, “On some new integral inequalities of Gronwall-Bellman-Pachpatte type,” Applied Mathematics and Computation, vol. 217, no. 20, pp. 7887–7899, 2011.
  • W.-S. Wang, “Some generalized nonlinear retarded integral inequalities with applications,” Journal of Inequalities and Applications, vol. 2013, article 31, 14 pages, 2012.
  • H. Zhou, D. Huang, W.-S. Wang, and J.-X. Xu, “Some new difference inequalities and an application to discrete-time control systems,” Journal of Applied Mathematics, vol. 2012, Article ID 214609, 14 pages, 2012.
  • Sh. S. Behzadi, S. Abbasbandy, T. Allahviranloo, and A. Yildirim, “Application of homotopy analysis method for solving a class of nonlinear Volterra-Fredholm integro-differential equations,” The Journal of Applied Analysis and Computation, vol. 2, no. 2, pp. 127–136, 2012.
  • M. Zarebnia, “A numerical solution of nonlinear Volterra-Fredholm integral equations,” The Journal of Applied Analysis and Computation, vol. 3, no. 1, pp. 95–104, 2013.
  • W.-S. Wang, D. Huang, and X. Li, “Generalized retarded nonlinear integral inequalities involving iterated integrals and an application,” Journal of Inequalities and Applications, vol. 2013, article 376, 17 pages, 2013. \endinput