- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 973586, 8 pages
On Some Vector-Valued Inequalities of Gronwall Type
Department of Applied Mathematics, I-Shou University, No. 1, Sec. 1, Syuecheng Road, Dashu District,
Kaohsiung City 84001, Taiwan
Received 25 October 2013; Accepted 12 December 2013; Published 22 January 2014
Academic Editor: Weinian Zhang
Copyright © 2014 Dah-Chin Luor. 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.
- 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, no. 4, 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, no. 1, pp. 81–94, 1956.
- 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.
- 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.
- R. A. C. Ferreira and D. F. M. Torres, “Generalized retarded integral inequalities,” Applied Mathematics Letters, vol. 22, no. 6, pp. 876–881, 2009.
- O. Lipovan, “A retarded Gronwall-like inequality and its applications,” Journal of Mathematical Analysis and Applications, vol. 252, no. 1, pp. 389–401, 2000.
- W. N. Li, M. Han, and F. W. Meng, “Some new delay integral inequalities and their applications,” Journal of Computational and Applied Mathematics, vol. 180, no. 1, pp. 191–200, 2005.
- L. Li, F. Meng, and L. He, “Some generalized integral inequalities and their applications,” Journal of Mathematical Analysis and Applications, vol. 372, pp. 339–349, 2010.
- M. Medveď, “A new approach to an analysis of Henry type integral inequalities and their Bihari type versions,” Journal of Mathematical Analysis and Applications, vol. 214, no. 2, pp. 349–366, 1997.
- Q.-H. Ma and J. Pečarić, “Estimates on solutions of some new nonlinear retarded Volterra-Fredholm type integral inequalities,” Nonlinear Analysis, Theory, Methods and Applications, vol. 69, no. 2, pp. 393–407, 2008.
- Q. H. Ma and J. Pečarić, “Explicit bounds on some new nonlinear retarded integral inequalities and their applications,” Taiwanese Journal of Mathematics, vol. 13, pp. 287–306, 2009.
- D. S. Mitrinović, J. Pečarić, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, 1991.
- M. Pinto, “Integral inequalities of Bihari-type and applications,” Funkcialaj Ekvacioj, vol. 33, pp. 387–403, 1990.
- H. Ye and J. Gao, “Henry-Gronwall type retarded integral inequalities and their applications to fractional differential equations with delay,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4152–4160, 2011.
- H. Ye, J. Gao, and Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1075–1081, 2007.
- J. Chandra and B. A. Fleishman, “On a generalization of the Gronwall-Bellman lemma in partially ordered Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 31, no. 3, pp. 668–681, 1970.
- C. Crǎciun and N. Lungu, “Abstract and concrete Gronwall lemmas,” Fixed Point Theory, vol. 10, no. 2, pp. 221–228, 2009.
- I. A. Rus, “Fixed points, upper and lower fixed points: abstract Gronwall lemmas,” Carpathian Journal of Mathematics, vol. 20, no. 1, pp. 125–134, 2004.
- E. Zeidler, Nonlinear Functional Analysis and Its Applications I, Springer, New York, NY, USA, 1986.
- A. L. Brown and A. Page, Elements of Functional Analysis, Van Nostrand-Reinhold, 1970.