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Asymptotic behaviour of solutions of some general nonlinear differential equations and integral inequalities

Published online by Cambridge University Press:  14 November 2011

Paul R. Beesack
Paul R. Beesack
Affiliation:
Carleton University, Ottawa, Ontario, CanadaK1S 5B6
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Synopsis

We deal with the asymptotic behaviour, as t→∞, of complex-valued solutions of nonlinear differential equations

Upper bounds for ∣x(l)(t)∣, 0≦j≦n, are obtained by obtaining upper bounds for solutions u(t) of Bihari-type integral inequalities of the form

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 98 , Issue 1-2 , 1984 , pp. 49 - 67
Copyright
Copyright © Royal Society of Edinburgh 1984

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References

1Aliev, R. M.. On new integral inequalities and their applications. Izv. Akad. Nauk Azerbaidzan. SSR Ser. Fiz.-Tehn. Mat. Nauk 2 (1976), 133–139 (Russian, English Summary).Google Scholar
2Agarwal, R. P. and Thandapani, E.. Remarks on generalization of Gronwall's inequality. Chinese J. Math. 9 (1981), 1–22.Google Scholar
3Beesack, P. R.. Gronwall Inequalities. Carleton Mathematical Lecture Notes, No. 11, May, 1975.Google Scholar
4Beesack, P. R.. On integral inequalities of Bihari type. Acta Math. Acad. Sci. Hungar. 28 (1976), 81–88.CrossRefGoogle Scholar
5Beesack, P. R.. On Lakshmikantham's comparison method for Gronwall inequalities. Ann.Polon. Math. 35 (1977), 187–222.Google Scholar
6Beesack, P. R.. Zbl. Math. 323 (1976), 159. (A review of reference No. 12).Google Scholar
7Bihari, I.. A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Ada Math. Acad. Sci. Hungar. 7 (1956), 71–94.Google Scholar
8Cohen, D. S.. The asymptotic behaviour of a class of nonlinear differential equations. Proc. Amer. Math. Soc. 18 (1967), 607–609.CrossRefGoogle Scholar
9Dannan, F. M.. Integral inequalities of Gronwall-Bellman-Bihari type and asymptotic behaviour of certain second order nonlinear differential equations. J. Math. Anal. Appl. to appear.Google Scholar
10Dhongade, U. D. and Deo, S. G.. Some generalizations of Bellman-Bihari integral inequalities. J. Math. Anal. Appl. 44 (1973), 218–226.Google Scholar
11Dhongade, U. D. and Deo, S. G.. Pointwise estimates of solutions of some Volterra integral equations. J. Math. Anal. Appl. 45 (1974), 615–628.CrossRefGoogle Scholar
12Dhongade, U. D. and Deo, S. G.. A nonlinear generalization of Bihari's inequality. Proc. Amer. Math. Soc. 54 (1976), 211–216.Google Scholar
13LaSalle, J. P.. Uniqueness theorems and successive approximations. Ann. of Math. 50 (1949), 722–730.CrossRefGoogle Scholar
14Moore, R. A. and Nehari, Z.. Nonoscillation theorems for a class of nonlinear differential equations. Trans. Amer. Math. Soc. 93 (1959), 30–52.CrossRefGoogle Scholar
15Pachpatte, B. G.. A note on some integral inequalities. Math. Student 42 (1974), 409–417.Google Scholar
16Pachpatte, B. G.. On some generalizations of Bellman's lemma. J. Math. Anal. Appl. 51 (1975), 141–150.Google Scholar
17Pachpatte, B. G.. On an integral inequality of Gronwall-Bellman. J. Math. Phys. Sci. 9 (1975), 405–416.Google Scholar
18Pachpatte, B. G.. On some new integral inequalities for differential and integral equations.J. Math. Phys. Sci. 10 (1976), 101–116.Google Scholar
19Pachpatte, B. G.. On some fundamental integral inequalities of the Bellman-Bihari type. Chinese J. Math. 5 (1977), 71–80.Google Scholar
20Reddy, K. Narsimha. Integral inequalities and applications. Bull. Austral. Math. Soc. 21 (1980), 13–20.CrossRefGoogle Scholar
21Thandapani, E. and Agarwal, R. P.. On some new integrodifferential inequalities: theory and applications. Tamkang J. Math. 11 (1981), 169–184.Google Scholar
22Tong, Jingcheng. The asymptotic behaviour of a class of nonlinear differential equations of secondorder. Proc. Amer. Math. Soc. 84 (1982), 235–236.Google Scholar
3Waltman, P.. On the asymptotic behaviour of a nonlinear equation. Proc. Amer. Math. Soc. 15 (1964), 918–923.CrossRefGoogle Scholar
24Waltman, P.. On the asymptotic behaviour of solutions of an nth order equation. Monatsh. Math. 69 (1965), 427–430.Google Scholar
25Yang, En Hao. Boundedness conditions for solutions of differential equation (a(t)x′)′ +f(t, x) =0. J. Nonlinear Anal. T.M.A., to appear.Google Scholar
26Yeh, C.-C.. Bellman-Bihari integral inequalities in several independent variables. J. Math. Anal. Appl. 87 (1982), 311–321.CrossRefGoogle Scholar
27Young, E. C.. On Bellman-Bihari integral inequalities. Intemat. J. Math. Math. Sci. 5 (1982), 97–103.Google Scholar