ON NONLINEAR INTEGRAL INEQUALITIES AND APPLICATIONS

Transcription

1 ON NONLINEAR INTEGRAL INEQUALITIES AND APPLICATIONS A Thesis submitted to the University of Pune for the degree of Doctor of Philosophy in Mathematics By Sitaram Gena Latpate Under the guidance of Dr. S. D. Kendre Department of Mathematics University of Pune Pune (INDIA) June 20, 2014

2 Abstract Inequalities have played a dominant role in the development of all branches of mathematics and hence they have a central place of attention for many mathematicians. Differential and integral inequalities are major tools in the analysis of differential and integral equations. It has been observed that these inequalities developed so far in the literature which provides explicit known bounds on function appearing in differential, integral and other equations. Thus they have found widespread acceptance in a variety of applications. Because of this, it is not surprising that numerous studies of new types of inequalities have been undertaken in order to achieve many new developments in various branches of mathematical science. In 1919, Gronwall [?] established the following integral inequality while investigating the dependence of a system of differential equations with respective parameter. If u(t) is a continuous function defined on the interval J = [α, α + h] and 0 u(t) t α [bu(s) + a]ds, t J, where a and b are nonnegative constants, then 0 u(t) ahe bh, t J. After the discovery of the above integral inequality, a number of mathematicians like Bellman, Bihari, Lakshmikantham, Pachpatte and many more have shown their considerable interest in generalizing the original form of this inequality. 1

3 2 In recent years inequalities have received considerable attention and hence a number of papers and monographs have been appeared in the literature. These inequalities deal properties of solutions of various differential, integral and integro-differential equations. An excellent account of this subject may be found in the monographs by Agarwal R. P. [?,?], Beckenbach E. F. and Bellman R. [?], Hardy G. H., Littlewood J.E. and Polya G. [?], Lakshmikantham V. and Leela S. [?], Pachpatte B. G. [?,?,?] and etc. as well as papers by Abdeldaim A. and Yakout M. [?,?], Byung-IL Kim [?], Thandapani E. [?,?], Oliva Lipovan [?,?] and some others. The Thesis contains number of nonlinear integral inequalities, which find numerous applications in the theory of various classes of differential, integral and integrodifferential equations. These inequalities can be valuable references for researchers into differential and integral equations. The Thesis is divided into five Chapters and followed by references. In the first Chapter, we obtain nonlinear integral and integro-differential inequalities in one variable which are variants and generalizations of Pachpatte s inequalities [?]. These inequalities can be used as powerful tools in the study of various classes of nonlinear differential and integral equations. The second Chapter is concerned with nonlinear integral inequalities in two variables. We obtain nonlinear integral and integro-differential inequalities in two variables to study certain partial differential and integral equations. In particular, we obtain nonlinear generalization of Wendroff s and Pachpatte s inequalities [?] and their variants. The third Chapter contains nonlinear retarded integral inequalities. In this chapter, we generalize inequalities established by Pachpatte [?] and Lipovan [?]. These inequalities can be used as handy tools to study the qualitative as well as the quantitative properties of solutions of some differential and integral equations. The fourth Chapter deals with mixed nonlinear integral inequalities. To study

4 3 Volterra-Fredholm integral equations Pachpatte [?] investigated mixed linear integral inequalities. In this Chapter, we extend the result proved by Pachpatte in [?] using Fangcui Jiang and Fanwei Meng inequality [?]. These inequalities have wide range of applications in the theory nonlinear Volterra-Fredholm integral and integrodifferential equations. The fifth Chapter is devoted to simultaneous integral inequalities. Here, we establish simultaneous integral inequalities in one and two variables using Bellman [?], Bihari [?] and Pachpatte [?] inequalities. These inequalities are useful to study the certain simultaneous differential and integral equations. All the Theorems, Lemmas, Corollary, Definitions and Remarks are numbered consequently in each Chapter without making distinction between them. Figures are sectionally numbered. The references are listed alphabetically and yearwise. The end of the proof is indicated by the symbol. The following are the references cited in the Thesis.

5 Bibliography [1] Abdeldaim A. and Yakout M., On Wendroffï s Inequality and Applications, Int. Journal of Math. Analysis, 4, no. 13, (2010), [2] Abdeldaim A. and Yakout M., On some new inequalities of Gronwall-Bellman- Pachpatte type, Appl. Math. Comput., 217, (2011), [3] Agarwal R. P., Inequalities involving partial derivatives, J. Math. Anal. Appl., 89 no.2, (1982), [4] Agarwal R. P., Difference equations and inequalities: Theory, Methods, and Applications, Marcel Dekker, Inc., New York, [5] Agarwal R. P., Young-Ho Kim and Sk Sen, New retarded integral inequalities with applications, J. Inequal. Appl., (2008), [6] Babolian E. and Shaerlar J. A., On some nonlinear generalizations of Gronwallï s inequality and their applications, Int. J. Contemp. Math. Sciences, 6, 2011, no. 16, [7] Bainov D. and Simeonov P., Integral inequalities and applications, New York, Kluwer Academic Publishers, [8] Baker C., The Numerical treatment of integral equations, Oxford University Press, London, (1977).

6 BIBLIOGRAPHY 5 [9] Beckenbach E. F. and Bellman R., Inequalities, Springer- Verlag, Berlin, New York, [10] Bellman R., The stability of solutions of linear differential equations, Duke Math. J., 10, (1943), 643 -ï 647. [11] Bellman R., Asymptotic series for the solutions of linear differential-difference equations, Rendiconti del circolo Matematica Di Palermo, 7, (1958), 1-9. [12] Bihari I., A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar, 7, (1956), [13] Bondge B. K., Pachpatte B.G. and Walter W., On genrralized Wendroff type inequalities and their applictions, Nonlinear Anal. TMA, 4, (1980), [14] Byung-IL Kim, On some Gronwal type integral inequalities and their applicatios, Bull. korean Math. Soc., 41, (2004), No. 3, [15] Coddington E. A., An introduction to ordinary differential equations, Prentice- Hall, lnc., Englewood Cliffs, N.J., U.S.A., [16] Davis H.T., Introduction to nonlinear differential and integral equations, Dover, Publications, New York, (1962). [17] Dhakne M. B. and Kendre S. D., Invariance for an abstract nonlinear Volterra integro-differential equations, Nonlinear Funct. Anal. Appl., 11, No-2, June [18] Dhakne M. B. and Pachpatte B. G., On perturbed abstract functional integrodifferential equations, Acta Math. Sci., 8, (1988), [19] Dhongade U. D. and Deo S. G., Some genarlizasion of Bellman-Bihari integral inequalities, J. Math. Anal. Appl., 44, (1973), [20] Dhongade U. D. and Deo S. G., A nonlinear genarlizasion of Bihari s inequality, Proc. Am. Math. Soc., 54, (1976),

7 BIBLIOGRAPHY 6 [21] Dragomir S. S. and Young-Ho Kim, Some integral inequalities for functions of two variables, Electron. J. Differential Equations, 10, (2003), [22] El-Owaidy H., Ragab A. and Abdeldaim A., On some new integral inequalities of Growall- Bellman type, Appl. Math. Comput., 106, (1999), [23] Fan Wei Meng and Wei Nian Li, On some new integral inequalitise and their applications, Appl. Math. Comput., 148, (2004), [24] Fangcui Jiang and Fanwei Meng, Explicit bounds on some new nonlinear integral inequality with delay, J. Comput. Appl. Math., 205, (2007), [25] Goldberq R. R., Methods of Real Analysis, Oxford and IBH Publshing Co. Pvt. Ltd, New Delhi, [26] Greene D. E., An inequality for a class of integral system, Proc. Am. Math. Soc., 62, (1977), [27] Gripenberg G., On some epidemic models, Quart. Appl. Math., 39, (1981), [28] Gronwall H. T., Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. Math., 2, ( ), [29] Hardy G. H., Littlewood J.E. and Polya G., Inequalities, Cambridge, Cambridge University Press, [30] Hongxia Zhang and Fanwei Meng, Integral inequalities in two independent variables for retarded Volterra equations, Appl. Math. Comput., 199, (2008), [31] Hongxia Zhang and Fanwei Meng, On certain integral inequalitis in two independent variables for retarded equations, Appl. Math. Comput., 203, (2008), [32] Jerri A., Introduction to integral equations with applications, Wiley, New York, (1999).

8 BIBLIOGRAPHY 7 [33] Kanwal R. P., Linear integral equations Theory and Techique, Academic Press, New York, 1997 [34] Lakshmikantham V. and Leela S., Differential and integral inequalities, Vol. I, New York, Academic Press, [35] Lianzhong Li, Fanwei Mengb and Leliang Hea; Some generalized integral inequalities and their applications, J. Math. Anal. Appl., 372, (2010), [36] Linz D., Analytical and numerical methods for Volterra equations, SIAM, philadelphia, (1985). [37] Miller R. K., Nonlinear Volterra integral equations, W. A. Benjamin, New York, (1971). [38] Mitrinovic D. S., Analytic inequalities, Berlin, New York, Springer-Verlag, [39] Lipovan Olivia, A retarded integral inequality and its applications, J. Math. Anal. Appl., 285, (2003), [40] Lipovan Olivia, Integral inequalities for retarded Volterra equations, J. Math. Anal. Appl., 322, (2006), [41] Olmstead W.E. and Handelsman R.A., Asymptotic solution to a class of nonlinear Volterra integral equations, II, SIAM J. Appl. Math., 30, (1976), [42] Pachpatte B. G., A note on Gronwall-Bellman inequality, J. Math. Anal. Appl., 44, (1973), [43] Pachpatte B. G., On some new integral inequalities similar to Bellman-Bihari inequalities, J. Math. Anal. Appl., 49, (1975), [44] Pachpatte B. G., A note on Gronwall type integral and integro differential inequalities, Tamkang J. Math., 8, (1977), [45] Pachpatte B. G., On some fundamental integro differential inequalities for differential equations, Chinese J. Math., 6 (1), (1978),

9 BIBLIOGRAPHY 8 [46] Pachpatte B. G., On some new integrodifferential inequalities of the Wendroff type, J. Math. Anal. Appl., 73, (1980), [47] Pachpatte B. G., On some new inequality Suggested by the Study of Certain Epidemic Models, J. Math. Anal. Appl., 195, (1995), [48] Pachpatte B. G., On some new inequalities related to certain inequalities in the theory of differential equations, J. Math. Anal. Appl., 189 (13), (1995), [49] Pachpatte B. G., On Gronwall type inequalities occurring in the theory of differential equations, Fasciculi Mathematici, No. 26, (1996), [50] Pachpatte B. G., Inequalities for differential and integral equations, Academic Press, New York and London, (1998). [51] Pachpatte B. G., On some fundamental integral inequalities and their discrete analogues, J. Ineq. Pure Appl. Math., 2 (2001), Article 15. [ONLINE: [52] Pachpatte B. G., On generalizations of Bihari s inequality, Soochow J. Math., 31, (2005), [53] Pachpatte B. G., Integral and finite difference inequalities and applications, North-Holland Mathematics studies, 205, [54] Pachpatte B. G., Multidimensional integral equations and inequalities, Series: Atlantis Studies in Mathematics for Engineering and Science, 9, (2011), [55] Peano G., Sull integrabilita delle equazioni differenziali del primo ordine, Atti. R. Accad. Sc. Toriao, 21, ( ), [56] Qing-Hua Ma and Josip Pecaric, Explicit bounds on some new nonlinear retarded integral inequalities and their applications, Taiwanese Jouranal of Mathematics, 13, No. 1, (2009), [57] Thandapani E. and Agarwal R. P., On some new integro-differential integral inequalities and applications, Tamkang J. Math., 11, (1980),

10 BIBLIOGRAPHY 9 [58] Thandapani E. and Agarwal R. P., On some new inequalities in n independent variables and applications, J. Math. Anal. Appl., 86, (1982), [59] Walter Rudin, Principles of mathematical analysis, Mcgraw-Hill Book Company, Londan, Tokyo, [60] Waltman P., Determinstic threshold models in the theory of epidemics, Lecture Notes in Biomathematics, Springer Verlag, New York, Vol. 1, (1974). [61] Wei Nian Li, Some delay integral inequalities on time scales, Comuters and Mathemtics with Applications, 59, (2010), [62] Wu-Sheng Wang, A class of retarded nonlinear integral inequalities and its application in nonlinear differential-integral equation, J. Inequal. Appl., (2012), 154. [63] Wu-Sheng Wang, Ri-Cai Luo and Zizun Li, A new nonlinear retarded integral inequality and its application, J. Inequal. Appl., (2010), [64] Young-Ho Kim, Gronwall-Bellman and Pachpatte type integral inequalities with applications, Nonlinear Anal., 71, (2009), e2641-e2656. [65] Zhao C. J., Some integral inequalities for differential equations, J. Binzhou Teachers College, 17(4), (2001), Research Student Research Guide Mr. Sitaram Gena Latpate Place: Pune Date: June 20, Dr. S. D. Kendre