GENERALIZATION OF GRONWALL’S INEQUALITY AND ITS APPLICATIONS IN FUNCTIONAL DIFFERENTIAL EQUATIONS

TINGXIU WANG

Title	GENERALIZATION OF GRONWALL’S INEQUALITY AND ITS APPLICATIONS IN FUNCTIONAL DIFFERENTIAL EQUATIONS
Publication Type	Journal Article
Year of Publication	2015
Authors	WANG, TINGXIU
Volume	19
Issue	4
Start Page	679
Pagination	10
Date Published	2015
ISSN	1083-2564
AMS	26D10, 34A40, 34D20, 34K38
Abstract	In this paper, we briefly review the recent development of research on Gronwall’s inequality. Then obtain a result for the following nonlinear integral inequality: $${ w(u(t)) ≤ K + \sum^{n}_{i=1} \int_{α_i} (t_0)^{α_{i}(t)} f_i(s) \ \prod^{m}_{j=1} H_{ij} (u(s))G_{ij} \left( \underset{s−h≤ξ≤s}{max} \ u(ξ) \right) ds. }$$ As an application, we study the abstract functional differential equation, ${ \frac{du}{dt} = f(t, u_t) }$ with Lyapunov’s second method. Then, we obtain an estimate of solutions of functional differential equations, ${ u ′ = F(t, u_t) }$ with conditions like: $${ i) \ \ \ \ W_1(\|u(t)\|_X) ≤ V (t, u_t) ≤ W_2(\\|{u_t}\\|_{CX}) \\ ii) \ \ \ \ V ′_{(3.1)}(t, u_t) ≤ 0. }$$
URL	http://www.acadsol.eu/en/articles/19/4/15.pdf
Refereed Designation	Refereed
Full Text	REFERENCES [1] R. Bellman, The stability of solutions of linear differential equations, Duke Math. J. 10 (1943) no. 4, 643–647. [2] R. Bellman, Stability Theory of Differential Equations, McGraw-Hill Book Company, Inc., New York, 1953. [3] R. Bellman, Asymptotic series for the solutions of linear differential-difference equations, Rendiconti del Circolo Matematico di Palermo 7 (1958) no. 2, 261–269. [4] M. Bohner, S. Hristova, K. Stefanova, Nonlinear integral inequalities involving maxima of the unknown scalar functions, Math. Inequal. Appl. 15 (2012), no. 4, 811–825. [5] T. A. Burton, Volterra Integral and Differential Equations. Second Edition, in Mathematics in Science and Engineering, 202, Elsevier, New York, 2005. [6] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional-Differential Equations, in Mathematics in Science and Engineering, 178, Academic Press, Orlando, Florida, 1985. [7] T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. of Math. 20 (1919), no. 4, 292–296. [8] J. Hale and S. Lunel, Introduction to Functional-Differential Equations, in Applied Matheamtics Science, 99, Springer, New York, 1993. [9] G. E. Ladas and V. Lakshmikantham, Differential Equations in Abstract Spaces, in Mathematics in Science and engineering, Vol. 85, Academic Press, New York-London, 1972. [10] V. Lakshmikantham, and S. Leela, Differential and Integral Inequalities, Vol. I and II, Academic Press, New York, 1969. [11] Shi-you Lin, Generalized Gronwall inequalities and their applications to fractional differential equations, J. Inequal. Appl. 2013 (2013), 9 pp. [12] B. G. Pachpatte, Inequalities for Differential and Integral Equations, in Mathematics in Science and Engineering, 197, Academic Press, San Diego, 1998. [13] B. G. Pachpatte, On a new integral inequality applicable to certain partial integrodifferential equations. Inequality Theory and Applications. Vol. 5, 131–140, Nova Sci. Publ., New York, 2007. [14] H. Wang, K. Zheng, C. Guo, Chun-xiang, Nonlinear discrete inequality in two variables with delay and its application, Acta Math. Appl. Sin. Engl. Ser. 30 (2014), no. 2, 389–400. [15] T. Wang, Lower and upper bounds of solutions of functional differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Analy. 20 (2013) no. 1, 131-141. [16] T. Wang, Inequalities of Solutions of Volterra Integral and Differential Equations, Electron. J. Qual. Theory Differ. Equ. 2009, Special Edition I, No. 28, 10 pp. [17] T. Wang, Exponential stability and inequalities of abstract functional differential equations, J. Math. Anal. Appl. 342 (2006), no.2, 982–991. [18] T. Wang, Inequalities and stability in a linear scalar functional differential equation, J. Math. Anal. Appl. 298 (2004), no. 1, 33–44. [19] T. Wang, Wazewski’s inequality in a linear Volterra integrodifferential equation, Volterra Equations and Applications, edited by C. Corduneanu and I. W. Sandberg, Vol. 10: Stability and Control: Theory, Methods and Applications, Gordon and Breach, Amsterdam, 2000. [20] T. Wang, Stability in abstract functional-differential equations I: General theorems, J. Math. Anal. Appl. 186 (1994), 534–558. [21] T. Wang, Stability in abstract functional-differential equations II: Applications, J. Math. Anal. ⁀ Appl. 186 (1994), 835–861. [22] J. Wu, Theory and Applications of Partial Functional Differential Equations, in Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996.