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

PHILOS, CH. G.

Glasgow mathematical journal, Tome 49 (2007) no. 2, pp. 197-211

Résumé

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.

Détail

DOI : 10.1017/S0017089507003667

Mots-clés : 34K10, 34B18, 34B40

Bibliographie

[1] 1. Agarwal, R. P. and O'Regan, D., Infinite interval problems for differential, difference and integral equations (Kluwer Academic Publishers, Dordrecht, 2001). Google Scholar | DOI

[2] 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. Google Scholar | DOI

[3] 3. Constantin, A., On the existence of positive solutions of second order differential equations, Ann. Mat. Pura Appl. 184 (2005), 131–138. Google Scholar | DOI

[4] 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). Google Scholar | DOI

[5] 5. Hale, J. K. and Verduyn Lunel, S. M., Introduction to functional differential equations (Springer-Verlag, 1993). Google Scholar | DOI

[6] 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. Google Scholar | DOI

[7] 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. Google Scholar | DOI

[8] 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. Google Scholar | DOI

[9] 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] 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. Google Scholar | DOI

[11] 11. Lovelady, D., Positive bounded solutions for a class of linear delay differential equations, Hiroshima Math. J. 6 (1976), 451–456. Google Scholar | DOI

[12] 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. Google Scholar | DOI

[13] 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. Google Scholar | DOI

[14] 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. Google Scholar | DOI

[15] 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. Google Scholar | DOI

[16] 16. Mustafa, O. and Rogovchenko, Y. V., Global existence and asymptotic behavior of solutions of nonlinear differential equations, Funkcial. Ekvac. 47 (2004), 167–186. Google Scholar | DOI

[17] 17. Mustafa, O. and Rogovchenko, Y. V., Asymptotic integration of nonlinear differential equations, Nonlinear Anal. 63 (2005), e2135–e2143. Google Scholar | DOI

[18] 18. Mustafa, O. and Rogovchenko, Y. V., Asymptotic integration of a class of nonlinear differential equations, Appl. Math. Lett. 19 (2006), 849–853. Google Scholar | DOI

[19] 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. Google Scholar | DOI

[20] 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] 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. Google Scholar | DOI

[22] 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] 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. Google Scholar | DOI

[24] 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. Google Scholar | DOI

[25] 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] 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. Google Scholar | DOI

[27] 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. Google Scholar | DOI

[28] 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. Google Scholar | DOI

[29] 29. Yin, Z., Monotone positive solutions of second-order nonlinear differential equations, Nonlinear Anal. 54 (2003), 391–403. Google Scholar | DOI

[30] 30. Zhao, Z., Positive solutions of nonlinear second order ordinary differential equations, Proc. Amer. Math. Soc. 121 (1994), 465–469. Google Scholar | DOI