On the asymptotic behavior of solutions to nonlinear ordinary differential equations

Asymptotic Analysis

Volumn 54, Issue 1-2, 2007, Pages 1-50

  Agarwal, Ravi P ^a   Djebali, Smail ^b   Moussaoui, Toufik ^b   Mustafa, Octavian G ^c   Rogovchenko, Yuri V ^d  

^a Florida Institute of Technology ^*  (United States)

^b Ecole Normale Superieure de Kouba   (Algeria)

^c UNIVERSITY OF CRAIOVA   (Romania)

^d EASTERN MEDITERRANEAN UNIVERSITY   (Turkey)

Author keywords

[No Author keywords available]

Indexed keywords

ASYMPTOTIC ANALYSIS; ORDINARY DIFFERENTIAL EQUATIONS; PERTURBATION TECHNIQUES; POLYNOMIALS;

ASYMPTOTIC BEHAVIOR; ASYMPTOTIC REPRESENTATION; NONLINEAR DIFFERENTIAL EQUATIONS;

NONLINEAR EQUATIONS;

EID: 35648959156     PISSN: 09217134     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (78)

1. R.P. Agarwal, S. Djebali, T. Moussaoui and O.G. Mustafa, On the asymptotic integration of nonlinear differential equations, J. Comput. Appl. Math., to appear.
2. N.I. Ahiezer, On the weighted approximation of continuous functions by polynomials on the entire number axis, Amer. Math. Soc. Transl. 22 (1962), 95-137.
3. N.I. Ahiezer and K.I. Babenko, On weighted polynomials of approximation to functions continuous on the whole axis, Dokl. Akad. Nauk SSSR 57 (1947), 315-318.
4. V.M. Alekseev, An estimate for perturbations of the solution of ordinary differential equations, Vestnik Moskov. Univ. Math. Mekh. 2 (1961), 28-36 (Russian).
5. H. Amann, Ordinary Differential Equations. Introduction to Nonlinear Analysis, de Gruyter, Berlin, 1990.
6. K. Atkinson, The numerical solution of integral equations on the half-line, SIAM J. Numer. Anal. 6 (1969), 375-397.
7. C. Avramescu, Sur l'existence des solutions convergentes de systèmes d'équations différentielles non linéaires, Ann. Mat. Pura Appl. 81 (1969), 147-168.
8. D.D. Bainov and P.S. Simeonov, Impulse Differential Equations. Asymptotic Properties of the Solutions, World Scientific, Singapore, 1995.
9. R. Bellman, Stability Theory of Differential Equations, McGraw-Hill, New York, 1953.
10. S.N. Bernstein, Sur les équations du calcul des variations, Ann. Sci. Ecole Norm. Sup. 29 (1912), 431-485.
11. I. Bihari, Notes on a nonlinear integral equation, Studia Sci. Math. Hung. 2 (1967), 1-6.
12. O. Boruvka, Linear Differential Transformations of the Second Order, London Univ. Press, London, 1971.
13. F. Brauer, A nonlinear variation of constants formula for Volterra equations, Math. Syst. Theory 6 (1972), 226-234.
14. H. Brézis, Analyse fonctionelle. Théorie et applications, Masson, Paris, 1992.
15. R.F. Brown, A Topological Introduction to Nonlinear Analysis, Birkhäuser, Boston, 1993.
16. L. Carleson, On Bernstein's approximation problem, Proc. Amer. Math. Soc. 2 (1951), 953-961.
17. A. Constantin, On the existence of positive solutions of second order differential equations, Ann. Mat. Pura Appl. 184 (2005), 131-138.
18. W.A. Coppel, Disconjugacy, Lecture Notes in Math., Vol. 220, Springer, Berlin, 1971.
19. E.T. Copson, Metric Spaces, Cambridge Tracts Math. Phys., Cambridge Univ. Press, 1968.
20. C. Corduneanu, Global existence theorems for differential systems with retarded argument, Stud. Cercet. Ştiinţ. 12 (1961), 249-258 (Romanian).
21. C. Corduneanu, Sur les inégalités différentielles, Mathematica 6 (1964), 31-33.
22. C. Corduneanu, Problèmes globaux dans la théorie des équations intégrales de Volterra, Ann. Mat. Pura Appl. 67 (1965), 349-363.
23. C. Corduneanu, Almost Periodic Functions, Interscience Publ., New York, 1968.
24. C. Corduneanu, Principles of Differential and Integral Equations, Chelsea, New York, 1977.
25. C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York, 1973.
26. J. Dugundji and A. Granas, Fixed Point Theory, I, Monogr. Mat., Vol. 61, PWN, Warszawa, 1982.
27. V.K. Dzyadyk, Approximation Methods for Solutions of Differential and Integral Equations, VSP, Utrecht, 1995.
28. M.S.P. Eastham, Theory of Ordinary Differential Equations, Van Nostrand, London, 1970.
29. M. Frigon and D. O'Regan, Some general existence principles for ordinary differential equations, Topology Methods Nonlin. Anal. 2 (1993), 35-54.
30. M.K. Grammatikopoulos, Oscillatory and asymptotic behavior of differential equations with deviating arguments, Hiroshima Math. J. 6 (1976), 31-53.
31. A. Granas and Z. Guennoun, Quelques résultats dans la théorie de Bernstein-Carathéodory de l'équation y″ = f(t, y, y′), CR. Acad. Sci. Paris 306 (1988), 703-706.
32. A. Granas, R.B. Guenther and J.W. Lee, Some general existence principles in the Carathéodory theory of nonlinear differential systems, J. Math. Pures Appl. 70 (1991), 153-196.
33. J.K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, 1969.
34. P. Hartman, A remark on a theorem of Arzela, Amer. J. Math. 59 (1937), 436.
35. P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.
36. P.F. Hsieh and Y. Sibuya, Basic Theory of Ordinary Differential Equations, Springer, New York, 1999.
37. E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, Reading, 1969.
38. M. Hukuhara, Sur la fonction S(x) de M.E. Kamke, Japanese J. Math. 17 (1940), 289-298.
39. A.G. Kartsatos, Advanced Ordinary Differential Equations, Mariner Publ. Comp., Tampa, 1980.
40. J. Kato, The asymptotic relations of two systems of ordinary differential equations, Contributions Differential Equations 3 (1964), 141-161.
41. J.L. Kelley, General Topology, Van Nostrand, New York, 1970.
42. W. Kryzewski and B. Przeradzki, The topological degree and fixed points of DC-mappings, Fund. Math. 126 (1985), 15-26.
43. T. Kusano and W.F. Trench, Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary differential equations, Ann. Mat. Pura Appl. 142 (1985), 381-392.
44. T. Kusano and W.F. Trench, Global existence theorems for solutions of nonlinear differential equations with prescribed. asymptotic behavior, J. London Math. Soc. 31 (1985), 478-486.
45. G.E. Ladas and V. Lakshmikantham, Differential Equations in Abstract Spaces, Academic Press, New York, 1972.
46. V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, I, II, Academic Press, New York, 1969.
47. V. Lakshmikantham, S. Leela and A.A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, New York, 1989.
48. O. Lipovan, On the asymptotic behavior of the solutions to a class of second order nonlinear differential equations, Glasgow Math. J. 45 (2003), 179-187.
49. H. Mâagli and S. Masmoudi, Sur les solutions d'un opérateur différentiel singulier semi-linéaire, Potential Anal. 10 (1999), 289-304.
50. L. Maatoug and M. Zribi, On the existence of positive solutions of nonlinear differential equations of high order, Nonlinear Anal. 43 (2001), 721-731.
51. R.K. Miller and G.R. Sell, Existence, uniqueness and continuity of solutions of integral equations, Ann. Mat. Pura Appl. 80 (1968), 135-152.
52. P. Montel, Sur l'intégrale supérieure et l'intégrale inférieure d'une équation différentielle, Bull. Sci. Math. 50 (1926), 204-217.
53. M. Müller, Über das Fundamentaltheorem in der Theorie der gewöhnlichen Differentialgleichungen, Math. Z. 26 (1927), 619-645.
54. O.G. Mustafa, On the existence of solutions with prescribed asymptotic behavior for perturbed nonlinear differential equations of second order, Glasgow Math. J. 47 (2005), 177-185.
55. O.G. Mustafa, Positive solutions of nonlinear differential equations with prescribed decay of the first derivative, Nonlinear Anal. 60 (2005), 179-185.
56. O.G. Mustafa, Banach function spaces in the study of asymptotic behavior of solutions of a class of differential equations of arbitrary order (Romanian), Ph.D. thesis, Univ. Craiova, Romania, 2000.
57. O.G. Mustafa and Y.V. Rogovchenko, Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations, Nonlinear Anal. 51 (2002), 339-368.
58. O.G. Mustafa and Y.V. Rogovchenko, Existence of square integrable solutions of perturbed nonlinear differential equations, in: Proc. Fourth Internat. Conf. Dynam. Systems Differential Equations, Wilmington, USA, Discrete Contin. Dynam. Systems supplement volume, 2003, pp. 647-655.
59. O.G. Mustafa and Y.V. Rogovchenko, Asymptotic integration of nonlinear differential equations, Nonlinear Anal. 63 (2005), e2135-e2145.
60. O.G. Mustafa and Y.V. Rogovchenko, Oscillation of second-order perturbed differential equations, Math. Nachr. 278 (2005), 460-469.
61. F. Neuman, Global Properties of Linear Ordinary Differential Equations, Kluwer, London, 1991.
62. N. Onuchic, Applications of the topological method of Ważewski to certain problems of asymptotic behavior in ordinary differential equations, Pacific J. Math. 1961, 1511-1527.
63. C.G. Philos, Asymptotic behavior of a class of nonoscillatory solutions of differential equations with deviating arguments, Math. Slovaca 33 (1983), 409-428.
64. C.G. Philos and V.A. Staikos, A basic asymptotic criterion for differential equations with deviating arguments and its applications to the nonoscillation of linear ordinary differential equations, Nonlinear Anal. 6 (1982), 1095-1113.
65. C.G. Philos and P.C. Tsamatos, Solutions approaching polynomials at infinity to nonlinear ordinary differential equations, Electron. J. Differential Equations 79 (2005), 1-25.
66. C.G. Philos, I.K. Purnaras and P.C. Tsamatos, Asymptotic to polynomials solutions for nonlinear differential equations, Nonlinear Anal. 59 (2004), 1157-1179.
67. H. Pollard, Solution of Bernstein's approximation problem, Proc. Amer. Math. Soc. 4 (1953), 869-875.
68. B. Przeradzki, On a two-point boundary value problem for differential equations on the half-line, Ann. Pol. Math. 50 (1989), 53-61.
69. C.R. Putnam, An oscillation criterion involving a minimum principle, Duke Math. J. 16 (1949), 633-636.
70. W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966.
71. T. Satō, Sur l'équation intégrale nonlinéaire de Volterra, Comp. Math. 11 (1953), 271-290.
72. A. Stokes, The Applications of a Fixed Point Theorem to a Variety of Non-linear Stability Problems, Contributions Differential Equations V, Princeton Univ. Press, Princeton, 1960.
73. W.F. Trench, Global solutions of nonlinear perturbations of linear differential equations, WSSIAA 1 (1992), 543-557.
74. F.G. Tricomi, Differential Equations, Blackie & Son, Glasgow, 1966.
75. S. Vasilache, Sur la détermination d'un système fondamental de solutions d'une équation différentielle linéaire d'ordre n, Ann. Pol. Math. 3 (1957), 172-182.
76. W. Walter, Ordinary Differential Equations, Springer, New York, 1998.
77. K. Yosida, Functional Analysis, Springer, Berlin, 1966.
78. Z. Yin, Monotone positive solutions of second-order nonlinear differential equations, Nonlinear Anal. 54 (2003), 391-403.