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POSITIVE INCREASING SOLUTIONS ON THE HALF-LINE TO SECOND ORDER NONLINEAR DELAY DIFFERENTIAL EQUATIONS

Published online by Cambridge University Press:  09 August 2007

CH. G. PHILOS
CH. G. PHILOS*
Affiliation:
Department of Mathematics, University of Ioannina, P. O. Box 1186, 451 10 Ioannina, Greece e-mail: [email protected]
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Abstract

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Second order nonlinear delay differential equations with positive delays are considered, and sufficient conditions are given that guarantee the existence of positive increasing solutions on the half-line with first order derivatives tending to zero at infinity. The approach is elementary and is essentially based on an old idea which appeared in the author's paper Arch. Math. (Basel)36 (1981), 168–178. The application of the result obtained to second order Emden-Fowler type differential equations with constant delays and, especially, to second order linear differential equations with constant delays, is also presented. Moreover, some (general or specific) examples demonstrating the applicability of the main result are given.

Keywords

34K1034B1834B40
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 49 , Issue 2 , May 2007 , pp. 197 - 211
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

REFERENCES

1. Agarwal, R. P. and O'Regan, D., Infinite interval problems for differential, difference and integral equations (Kluwer Academic Publishers, Dordrecht, 2001).CrossRefGoogle Scholar
2. Agarwal, R. P., Philos, Ch. G. and Tsamatos, P. Ch., Global solutions of a singular initial value problem to second order nonlinear delay differential equations, Math. Comput. Modelling 43 (2006), 854–869.CrossRefGoogle Scholar
3. Constantin, A., On the existence of positive solutions of second order differential equations, Ann. Mat. Pura Appl. 184 (2005), 131–138.CrossRefGoogle Scholar
4. Diekmann, O., Gils, S. A. van, Lunel, S. M. Verduyn and Walther, H.-O., Delay equations: Functional-, Complex-, and Nonlinear Analysis (Springer-Verlag, 1995).CrossRefGoogle Scholar
5. Hale, J. K. and Verduyn Lunel, S. M., Introduction to functional differential equations (Springer-Verlag, 1993).CrossRefGoogle Scholar
6. Kusano, T. and Trench, W. F., Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary differential equations, Ann. Mat. Pura Appl. 142 (1985), 381–392.CrossRefGoogle Scholar
7. Kusano, T. and Trench, W. F., Global existence theorems for solutions of nonlinear differential equations with prescribed asymptotic behaviour, J. London Math. Soc. (2) 31 (1985), 478–486.CrossRefGoogle Scholar
8. Lipovan, O., On the asymptotic behaviour of the solutions to a class of second order nonlinear differential equations, Glasgow Math. J. 45 (2003), 179–187.CrossRefGoogle Scholar
9. Liu, Y., Existence and unboundedness of positive solutions for singular boundary value problems on the half-line, Appl. Math. Comput. 144 (2003), 543–556.Google Scholar
10. Liu, Y., Boundary value problems on the half-line for functional differential equations with infinite delay in a Banach space, Nonlinear Anal. 52 (2003), 1695–1708.CrossRefGoogle Scholar
11. Lovelady, D., Positive bounded solutions for a class of linear delay differential equations, Hiroshima Math. J. 6 (1976), 451–456.CrossRefGoogle Scholar
12. Mavridis, K. G., Philos, Ch. G. and Tsamatos, P. Ch., Existence of solutions of a boundary value problem on the half-line to second order nonlinear delay differential equations, Arch. Math. (Basel) 86 (2006), 163–175.CrossRefGoogle Scholar
13. Mavridis, K. G., Philos, Ch. G. and Tsamatos, P. Ch., Multiple positive solutions for a second order delay boundary value problem on the half-line, Ann. Polon. Math. 88 (2006), 173–191.CrossRefGoogle Scholar
14. Mustafa, O., On the existence of solutions with prescribed asymptotic behaviour for perturbed nonlinear differential equations of second order, Glasgow Math. J. 47 (2005), 177–185.CrossRefGoogle Scholar
15. Mustafa, O. and Rogovchenko, Y. V., Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations, Nonlinear Anal. 51 (2002), 339–368.CrossRefGoogle Scholar
16. Mustafa, O. and Rogovchenko, Y. V., Global existence and asymptotic behavior of solutions of nonlinear differential equations, Funkcial. Ekvac. 47 (2004), 167–186.CrossRefGoogle Scholar
17. Mustafa, O. and Rogovchenko, Y. V., Asymptotic integration of nonlinear differential equations, Nonlinear Anal. 63 (2005), e2135–e2143.CrossRefGoogle Scholar
18. Mustafa, O. and Rogovchenko, Y. V., Asymptotic integration of a class of nonlinear differential equations, Appl. Math. Lett. 19 (2006), 849–853.CrossRefGoogle Scholar
19. Philos, Ch. G., On the existence of nonoscillatory solutions tending to zero at infty; for differential equations with positive delays, Arch. Math. (Basel) 36 (1981), 168–178.CrossRefGoogle Scholar
20. Philos, Ch. G., Asymptotic behaviour of a class of nonoscillatory solutions of differential equations with deviating arguments, Math. Slovaca 33 (1983), 409–428.Google Scholar
21. Philos, Ch. G., Purnaras, I. K. and Tsamatos, P. Ch., Asymptotic to polynomials solutions for nonlinear differential equations, Nonlinear Anal. 59 (2004), 1157–1179.CrossRefGoogle Scholar
22. Philos, Ch. G., Purnaras, I. K. and Tsamatos, P. Ch., Global solutions approaching lines at infinity to second order nonlinear delay differential equations, Funkcial. Ekvac., to appear.Google Scholar
23. Philos, Ch. G., Sficas, Y. G. and Staikos, V. A., Some results on the asymptotic behavior of nonoscillatory solutions of differential equations with deviating arguments, J. Austral. Math. Soc. Series A 32 (1982), 295–317.CrossRefGoogle Scholar
24. Philos, Ch. G. and Staikos, V. A., A basic asymptotic criterion for differential equations with deviating arguments and its applications to the nonoscillation of linear ordinary equations, Nonlinear Anal. 6 (1982), 1095–1113.CrossRefGoogle Scholar
25. Philos, Ch. G. and Tsamatos, P. Ch., Solutions approaching polynomials at infinity to nonlinear ordinary differential equations, Electron. J. Differential Equations 2005 (2005), No. 79, pp. 1–25.Google Scholar
26. Wei, Z. and Chen, S., Positive solution of singular boundary value problems on a half-line, Acta Math. Appl. Sinica (English Ser.) 21 (2005), 553–564.CrossRefGoogle Scholar
27. Yan, B., Multiple unbounded solutions of boundary value problems for second-order differential equations on the half-line, Nonlinear Anal. 51 (2002), 1031–1044.CrossRefGoogle Scholar
28. Yan, B. and Liu, Y., Unbounded solutions of the singular boundary value problems for second order differential equations on the half-line, % Appl. Math. Comput. 147 (2004), 629–644.CrossRefGoogle Scholar
29. Yin, Z., Monotone positive solutions of second-order nonlinear differential equations, Nonlinear Anal. 54 (2003), 391–403.CrossRefGoogle Scholar
30. Zhao, Z., Positive solutions of nonlinear second order ordinary differential equations, Proc. Amer. Math. Soc. 121 (1994), 465–469.CrossRefGoogle Scholar