# zbMATH — the first resource for mathematics

Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations. (English) Zbl 1017.34005
The differential equation under consideration is $$u''+ f(t,u,u')= 0$$. A typical result is the following. If there exist continuous positive nondecreasing functions $$h$$, $$p_1,p_2: \mathbb{R}_+\to \mathbb{R}_+$$ such that $|f(t,u,v)|\leq h(t)\Biggl[p_1\Biggl({|u|\over t}\Biggr)+ p_2(|v|)\Biggr],$ where $$h(s)$$ satisfies $$\int^\infty_{h_0} h(s) ds= H< \infty$$. Assume also that $$G(+\infty)= +\infty$$, where, for $$x\geq t_0$$, $G(x)= \int^x_{t_0} {dx\over p_1(s)+ p_2(s)}.$ Then, for every $$u_0,u_1\in \mathbb{R}$$, the initial value problem $u''+ f(t,u,u')= 0,\quad t\geq t_0,\quad u(t_0)= u_0,\quad u'(t_0)= u_1,$ has at least one solution $$u(t)$$ defined on $$[t_0,+\infty)$$ with the asymptotic representation $$u(t)= at+ o(t)$$ as $$t\to+\infty$$, where $$a$$ is a real constant. Further results with variations of these conditions are investigated in some detail.
Reviewer: P.Smith (Keele)

##### MSC:
 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 34E05 Asymptotic expansions of solutions to ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations
Full Text:
##### References:
 [1] Avramescu, C., Sur l’existence des solutions convergentès de systèmes d’équations différentielles non linéaires, Ann. mat. pura appl., 81, 4, 147-168, (1969) · Zbl 0196.10701 [2] Bellman, R., Stability theory of differential equations, (1953), McGraw-Hill London · Zbl 0052.31505 [3] Bernstein, S.N., Sur certaines equations differentielles ordinaires du second ordre, C. R. acad. sci. Paris, 138, 950-951, (1904) · JFM 35.0341.02 [4] Bihari, I., A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta math. acad. sci. hung., 7, 83-94, (1956) · Zbl 0070.08201 [5] Bihari, I., Researches of the boundedness and stability of the solutions of non-linear differential equations, Acta math. acad. sci. hung., 8, 261-278, (1957) · Zbl 0097.29301 [6] Bôcher, M., On regular singular points of linear differential equations of second order whose coefficients are not necessarily analytic, Trans. amer. math. soc., 1, 40-52, (1900) · JFM 31.0345.01 [7] Brauer, F.; Nohel, J.A., The qualitative theory of ordinary differential equations: an introduction, (1989), Dover Publications Inc New York [8] Cohen, D.S., The asymptotic behavior of a class of nonlinear differential equations, Proc. amer. math. soc., 18, 607-609, (1967) · Zbl 0152.28501 [9] Constantin, A., Some observations on a Conti’s result, Atti accad. naz. lincei, cl. sci. fis. mat. natur. IX ser. rend. lincei, mat. appl., 2, 137-145, (1991) · Zbl 0732.34011 [10] Constantin, A., On the asymptotic behavior of second order nonlinear differential equations, Rend. mat. appl., 7, 13, 627-634, (1993) · Zbl 0808.34050 [11] Conti, R., Sulla prolungabilità delle soluzioni di un sistema di equazioni differenziali ordinarie, Boll. un. mat. ital., 11, 3, 510-514, (1956) · Zbl 0072.30403 [12] Corduneanu, C., Principles of differential and integral equations, (1977), Chelsea Publishing Company The Bronx, New York · Zbl 0208.10701 [13] Cronin, J., Fixed points and topological degree in nonlinear analysis, Mathematical surveys and monographs, Vol. 11, (1964), American Mathematical Society Providence, RI [14] Dannan, F.M., Integral inequalities of gronwall – bellman – bihari type and asymptotic behavior of certain second order nonlinear differential equations, J. math. anal. appl., 108, 151-164, (1985) · Zbl 0586.26008 [15] Deo, S.G.; Raghavendra, V., Ordinary differential equations and stability theory, (1980), Tata McGraw-Hill Publishing Company Ltd New Delhi · Zbl 0482.34006 [16] J. Dugundji, A. Granas, Fixed Point Theory, Vol. 1, Monografie Matematyczne, Tom 61, PWN, Warszawa, 1982. · Zbl 0483.47038 [17] Fubini, G., Studi asintotici per alcune equazioni differenziali, Rend. reale accad. lincei, 25, 253-259, (1937) · JFM 63.0434.01 [18] Gallavotti, G., The elements of mechanics, (1983), Springer New York [19] Hallam, T.G., Asymptotic integration of second order differential equation with integrable coefficients, SIAM J. appl. math., 19, 430-439, (1970) · Zbl 0208.10804 [20] Hartman, P., Ordinary differential equations, (1964), Wiley New York, London, Sydney · Zbl 0125.32102 [21] Hartman, P.; Wintner, A., On the non-increasing solutions of y″=f(x, y, y′), Amer. J. math., 73, 390-404, (1951) · Zbl 0042.32601 [22] Hartman, P.; Wintner, A., On non-oscillatory linear differential equations, Amer. J. math., 75, 717-730, (1953) · Zbl 0051.06503 [23] Hartman, P.; Wintner, A., Linear differential equations with completely monotone solutions, Amer. J. math., 76, 199-206, (1954) · Zbl 0055.07903 [24] Hartman, P.; Wintner, A., On non-oscillatory linear differential equations with monotone coefficients, Amer. J. math., 76, 207-219, (1954) · Zbl 0055.07904 [25] Hille, E., Non-oscillation theorems, Trans. amer. math. soc., 64, 234-252, (1948) · Zbl 0031.35402 [26] Haupt, O., Über das asymptotische verhalten der lösungen gewisser linearer gewöhnlieher differentialgleichungen, Math. Z., 48, 289-292, (1942) · JFM 68.0186.02 [27] Kartsatos, A.G., Advanced ordinary differential equations, (1980), Mariner Publishing Company, Inc Tampa, FL · Zbl 0495.34001 [28] Kato, J.; Strauss, A., On the global existence of solutions and Liapunov functions, Ann. mat. pura appl., 77, 4, 303-316, (1967) · Zbl 0155.13802 [29] Kusano, T.; Naito, M.; Usami, H., Asymptotic behavior of a class of second order nonlinear differential equations, Hiroshima math. J., 16, 149-159, (1986) · Zbl 0612.34052 [30] Kusano, T.; Trench, W.F., Global existence theorems for solutions of nonlinear differential equations with prescribed asymptotic behavior, J. London math. soc., 31, 2, 478-486, (1985) · Zbl 0578.34045 [31] Kusano, T.; Trench, W.F., Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary differential equations, Ann. mat. pura appl., 142, 4, 381-392, (1985) · Zbl 0593.34039 [32] Kusano, T.; Trench, W.F., Global existence of non-oscillatory solutions of perturbed general disconjugate equations, Hiroshima math. J., 17, 415-431, (1987) · Zbl 0638.34028 [33] LaSalle, J.P.; Lefschetz, S., Stability by Liapunov’s direct method with applications, (1961), Academic Press New York [34] Meng, F.W., A note on tong paper: the asymptotic behavior of a class of nonlinear differential equations of second order, Proc. amer. math. soc., 108, 383-386, (1990) · Zbl 0699.34053 [35] Moore, R.; Nehari, Z., Non-oscillation theorems for a class of nonlinear differential equations, Trans. amer. math. soc., 93, 30-52, (1959) · Zbl 0089.06902 [36] Nagumo, M., Über die differentialgleichung y″=f(x,y,y′), Proc. phys. math. soc. Japan, 19, 861-866, (1937) · JFM 63.1021.04 [37] Nehari, Z., On a class of nonlinear second-order differential equations, Trans. amer. math. soc., 95, 101-123, (1960) · Zbl 0097.29501 [38] S.P. Rogovchenko, Y.V. Rogovchenko, Asymptotic behavior of certain second order nonlinear differential equations, in Dynamic Systems Appl., to appear. · Zbl 1002.34037 [39] Rogovchenko, Y.V., On the asymptotic behavior of solutions for a class of second order nonlinear differential equations, Collect. math., 49, 113-120, (1998) · Zbl 0936.34037 [40] Rogovchenko, Y.V.; Villari, Gab, Asymptotic behavior of solutions for second order nonlinear autonomous differential equations, Nonlinear differential equations appl., 4, 271-282, (1997) · Zbl 0880.34058 [41] Sansone, G., Equazioni differenziali nel campo reale, (1948), Nicola Zanichelli Bologna · JFM 67.0306.01 [42] Souplet, P., Existence of exceptional growing-up solutions for a class of non-linear second order ordinary differential equations, Asymptotic anal., 11, 185-207, (1995) · Zbl 0849.34024 [43] Strauss, A., A note on a global existence result of R. conti, Boll. un. mat. ital., 22, 3, 434-441, (1967) · Zbl 0155.13801 [44] Tong, J., The asymptotic behavior of a class of nonlinear differential equations of second order, Proc. amer. math. soc., 84, 235-236, (1982) · Zbl 0491.34036 [45] Trench, W.F., On the asymptotic behavior of solutions of second order linear differential equations, Proc. amer. math. soc., 14, 12-14, (1963) · Zbl 0116.29304 [46] Waltman, P., On the asymptotic behavior of solutions of a nonlinear equation, Proc. amer. math. soc., 15, 918-923, (1964) · Zbl 0131.08901 [47] Wong, J.S.W., On second order nonlinear oscillation, Funkcial. ekvac., 11, 207-234, (1968) · Zbl 0157.14802
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.