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On the asymptotic integration of nonlinear differential equations. (English) Zbl 1123.34038
This interesting paper starts with an extensive overview of results regarding asymptotic integration of various classes of linear and nonlinear differential equations, mostly of second order, ranging from classical theorems to the most recent contributions in that field. The authors proceed with the discussion of existence of vanishing at infinity positive solutions of the elliptic differential equation
$\Delta u+f\left( x,u\right) +g\left( \left| x\right| \right) x\cdot\nabla u=0.$
This requires the existence of a solution $$u(t)$$ of the so-called comparison equation associated with the original partial differential equation that can be continued to the right and satisfies the condition $\max\left[ u^{\prime}(t),0\right] <\frac{u(t)}{t}\leq m<+\infty$ as $$t\rightarrow+\infty.$$ This fact is established for a general nonlinear differential equation
$u^{\prime\prime}+f\left( t,u,u^{\prime}\right) =0\tag{*}$
in Theorem 2 which also yields the existence of eventually positive asymptotically linear solutions with the property that
$\limsup_{t\rightarrow+\infty}\left[ u\left( t\right) -tu^{\prime}\left( t\right) \right] >0.$
In Section 4, several new results on the existence of asymptotically linear solutions of equations (*) and
$u^{\prime\prime}+f\left( t,u\right) =0$
that extend and complement recent contributions by O. G. Mustafa and Y. V. Rogovchenko [Nonlinear Anal. 51, 339–368 (2002; Zbl 1017.34005); Nonlinear Stud. 13, 155–166 (2006; Zbl 1104.34038)], C. G. Philos et al. [Nonlinear Anal. 59, 1157–1179 (2004; Zbl 1094.34032)], and C. G. Philos and P. C. Tsamatos [Electron. J. Differ. Equ. 79, 1–25 (2005; Zbl 1075.34044)] are presented. In Section 5, a new proof of the important theorem regarding asymptotic integration of an $$n$$-th order homogeneous linear differential equation due to I. M. Sobol’ [Dokl. Akad. Nauk SSSR 61, 219–222 (1948; Zbl 0038.25301)] based on the Banach fixed point theorem is provided. The paper concludes with an impressive and valuable list of 90 references on the subject.

##### MSC:
 34D05 Asymptotic properties of solutions to ordinary differential equations 34E05 Asymptotic expansions of solutions to ordinary differential equations
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##### References:
 [1] Agarwal, R.P.; Grace, S.R.; O’Regan, D., Oscillation theory for second order linear, half-linear, superlinear and sublinear dynamic equations, (2002), Kluwer Academic Publishers Dordrecht · Zbl 1073.34002 [2] Agarwal, R.P.; Mustafa, O.G.; Rogovchenko, Y.V., Existence and asymptotic behavior of solutions of a boundary value problem on an infinite interval, Math. comput. modelling, 41, 135-157, (2005) · Zbl 1083.34021 [3] Agarwal, R.P.; O’Regan, D., Existence theory for single and multiple solutions to singular positone BVP, J. differential equations, 175, 393-414, (2001) · Zbl 0999.34018 [4] Agarwal, R.P.; O’Regan, D., Infinite interval problems for differential, difference and integral equations, (2001), Kluwer Academic Publishers Dordrecht · Zbl 1003.39017 [5] Agarwal, R.P.; O’Regan, D., Upper and lower solutions for singular problems with nonlinear boundary data, Nonlinear differential equations appl., 9, 419-440, (2002) · Zbl 1025.34019 [6] Avramescu, C., Sur l’existence des solutions convergentes de systèmes d’équations différentielles non linéaires, Ann. mat. pura appl., 81, 147-168, (1969) · Zbl 0196.10701 [7] Bellman, R., The stability of solutions of linear differential equations, Duke math. J., 10, 643-647, (1943) · Zbl 0061.18502 [8] Bellman, R., The boundedness of solutions of linear differential equations, Duke math. J., 14, 83-97, (1947) · Zbl 0029.35702 [9] Bellman, R., Stability theory of differential equations, (1953), McGraw-Hill London · Zbl 0052.31505 [10] Bihari, I., A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta math. acad. sci. hung., 7, 81-94, (1956) · Zbl 0070.08201 [11] 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 [12] Bitterlich-Willmann, J., Über die asymptoten der lösungen einer differentialgleichung, Monatsh. math. phys., 50, 35-39, (1941) · JFM 67.0315.02 [13] Boas, M.L.; Boas, R.P.; Levinson, N., The growth of solutions of a differential equation, Duke math. J., 9, 847-853, (1942) · Zbl 0061.18605 [14] F. Brauer, Some stability and perturbation problems for differential and integral equations, Monographs. Mat. 25, I.M.P.A., Rio de Janeiro, 1976. [15] Brauer, F.; Wong, J.S.W., On the asymptotic relationships between solutions of two systems of ordinary differential equations, J. differential equations, 6, 527-543, (1969) · Zbl 0185.16601 [16] Caligo, D., Comportamento asintotico degli integrali dell’equazione $$y''(x) + A(x) y(x) = 0$$, nell’ipotesi $$\lim_{x \rightarrow + \infty} A(x) = 0$$, Boll. un. mat. ital., 3, 286-295, (1941) · JFM 67.0315.03 [17] Coffman, C.V.; Wong, J.S.W., Oscillation and nonoscillation theorems for second order ordinary differential equations, Funkc. ekvac., 15, 119-130, (1972) · Zbl 0287.34024 [18] Constantin, A., On the asymptotic behavior of second order nonlinear differential equations, Rend. mat. appl., 7, 627-634, (1993) · Zbl 0808.34050 [19] Constantin, A., Global existence of solutions for perturbed differential equations, Ann. mat. pura appl., 168, 237-299, (1995) [20] Constantin, A., Existence of positive solutions of quasilinear elliptic equations, Bull. austral. math. soc., 54, 147-154, (1996) · Zbl 0878.35040 [21] Constantin, A., Positive solutions of quasilinear elliptic equations, J. math. anal. appl., 213, 334-339, (1997) · Zbl 0891.35033 [22] Constantin, A., On the existence of positive solutions of second order differential equations, Ann. mat. pura appl., 184, 131-138, (2005) · Zbl 1223.34041 [23] Coppel, W.A., Stability and asymptotic behavior of differential equations, (1965), D.C. Heath and Comp. Boston · Zbl 0154.09301 [24] 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 [25] Eastham, M.S.P., The asymptotic solution of linear differential systems. applications of the Levinson theorem, (1989), Clarendon Press Oxford · Zbl 0674.34045 [26] M. Ehrnström, Positive solutions for second-order nonlinear differential equations, Nonlinear Anal. Theor. Methods Appl. 64 (2006) 1608-1620. [27] M. Ehrnström, On radial solutions of certain semi-linear elliptic equations, Nonlinear Anal. Theor. Methods Appl. 64 (2006) 1578-1586. [28] Evans, L.C.; Gariepy, R.F., Measure theory and fine properties of functions, (1992), CRC Press Boca Raton · Zbl 0626.49007 [29] Ghizzetti, A., Un teorema sul comportamento asintotico degli integrali delle equazioni differenziali lineari omogenee, Rend. mat. appl. univ. roma, 8, 28-42, (1949) · Zbl 0039.09501 [30] Golomb, M., Bounds for solutions of nonlinear differential systems, Arch. rational mech. anal., 1, 272-282, (1958) · Zbl 0082.07701 [31] Grammatikopoulos, M.K., Oscillatory and asymptotic behavior of differential equations with deviating arguments, Hiroshima math. J., 6, 31-53, (1976) · Zbl 0324.34071 [32] T.H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. Math. 20 (1918/1919) 292-296. · JFM 47.0399.02 [33] Hale, J.K.; Onuchic, N., On the asymptotic behavior of solutions of a class of differential equations, Contributions differential equations, 2, 61-75, (1963) · Zbl 0135.30002 [34] Hartman, P., Unrestricted solution fields of almost-separable differential equations, Trans. amer. math. soc., 63, 560-580, (1948) · Zbl 0031.39801 [35] Hartman, P., On non-oscillatory linear differential equations of second order, Amer. J. math., 74, 389-400, (1952) · Zbl 0048.06602 [36] Hartman, P.; Wintner, A., On the asignment of asymptotic values for the solutions of linear differential equations of second order, Amer. J. math., 77, 475-483, (1955) · Zbl 0064.33205 [37] Hartman, P., Ordinary differential equations, (1964), Wiley New York · Zbl 0125.32102 [38] Hartman, P., Asymptotic integration of ordinary differential equations, SIAM J. math. anal., 14, 772-779, (1983) · Zbl 0545.34038 [39] Hartman, P.; Onuchic, N., On the asymptotic integration of ordinary differential equations, Pacific J. math., 13, 1193-1207, (1963) · Zbl 0188.46102 [40] Haupt, O., Über lösungen linearer differentialgleichungen mit asymptoten, Math. Z., 48, 212-220, (1942) · JFM 68.0185.03 [41] Haupt, O., Über das asymptotische verhalten der lösungen gewisser linearer gewöhnlicher differentialgleichungen, Math. Z., 48, 289-292, (1942) · JFM 68.0186.02 [42] Headley, V.B., A multidimensional nonlinear Gronwall inequality, J. math. anal. appl., 47, 250-255, (1974) · Zbl 0289.34018 [43] Kelley, J.L., General topology, (1955), Van Nostrand Reinhold Comp. New York · Zbl 0066.16604 [44] Kiguradze, I.T.; Chanturia, T.A., Asymptotic properties of solutions of nonautonomous ordinary differential equations, (1993), Kluwer Academic Publishers Dordrecht · Zbl 0782.34002 [45] 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 [46] Kusano, T.; Trench, W.F., Global existence theorems for solutions of nonlinear differential equations with prescribed asymptotic behavior, J. London math. soc., 31, 478-486, (1985) · Zbl 0578.34045 [47] Kusano, T.; Trench, W.F., Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary differential equations, Ann. mat. pura appl., 142, 381-392, (1985) · Zbl 0593.34039 [48] Kwong, M.K.; Wong, J.S.W., An application of integral inequality to second order nonlinear oscillation, J. differential equations, 46, 63-77, (1982) · Zbl 0503.34021 [49] Lipovan, O., On the asymptotic behavior of the solutions to a class of second order nonlinear differential equations, Glasgow math. J., 45, 179-187, (2003) · Zbl 1037.34041 [50] 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 [51] A.B. Mingarelli, Volterra-Stieltjes integral equations and generalized ordinary differential expressions, Lecture Notes in Mathematics, vol. 989, Springer, Berlin, 1983. · Zbl 0516.45012 [52] Mustafa, O.G., Initial value problem with infinitely many linear-like solutions for a second-order differential equation, Appl. math. lett., 18, 931-934, (2005) · Zbl 1095.34505 [53] Mustafa, O.G., On the existence of solutions with prescribed asymptotic behavior for perturbed nonlinear differential equations of second order, Glasgow math. J., 47, 177-185, (2005) · Zbl 1072.34049 [54] Mustafa, O.G.; Rogovchenko, Y.V., Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations, Nonlinear anal. theor. methods appl., 51, 339-368, (2002) · Zbl 1017.34005 [55] Mustafa, O.G.; Rogovchenko, Y.V., Global existence and asymptotic behavior of solutions of second-order nonlinear differential equations, Funkc. ekvac., 47, 167-186, (2004) · Zbl 1118.34046 [56] Mustafa, O.G.; Rogovchenko, Y.V., Asymptotic behavior of nonoscillatory solutions of second order nonlinear differential equations, Proc. dynam. systems appl., 4, 312-319, (2004) · Zbl 1082.34042 [57] O.G. Mustafa, Y.V. Rogovchenko, Asymptotic integration of a class of nonlinear differential equations, Appl. Math. Lett., in press. · Zbl 1126.34339 [58] O.G. Mustafa, Y.V. Rogovchenko, On asymptotic integration of a nonlinear second-order differential equation, Nonlinear Stud., in press. · Zbl 1104.34038 [59] Naito, M., Integral averages and the asymptotic behavior of solutions of second order ordinary differential equations, J. math. anal. appl., 164, 370-380, (1992) · Zbl 0754.34045 [60] Nohel, J.A., Commentary to chapter I, stability and asymptotic behavior of solutions of ordinary differential equations, in selected papers of norman Levinson, vol. 1, (1998), Birkhäuser Boston [61] Opial, Z., Sur un système d’inégalités intégrales, Ann. polon. math., 3, 200-209, (1957) · Zbl 0078.07302 [62] O’Regan, D., Upper and lower solutions for singular problems arising in the theory of membrane response of a spherical cap, Nonlinear anal. theor. methods appl., 47, 1163-1174, (2001) · Zbl 1042.34523 [63] Orpel, A., On the existence of positive radial solutions for a certain class of elliptic BVPs, J. math. anal. appl., 299, 690-702, (2004) · Zbl 1172.35394 [64] Pachpatte, B.G., On some integral inequalities similar to Bellman-bihari inequalities, J. math. anal. appl., 49, 794-802, (1975) · Zbl 0305.26009 [65] Philos, C.G., Asymptotic behavior of a class of nonoscillatory solutions of differential equations with deviating arguments, Math. slovaca, 33, 409-428, (1983) · Zbl 0523.34069 [66] Philos, C.G.; Purnaras, I.K.; Tsamatos, P.C., Asymptotic to polynomials solutions for nonlinear differential equations, Nonlinear anal. theor. methods appl., 59, 1157-1179, (2004) · Zbl 1094.34032 [67] Philos, C.G.; Tsamatos, P.C., Solutions approaching polynomials at infinity to nonlinear ordinary differential equations, Electron. J. differential equations, 79, 1-25, (2005) · Zbl 1075.34044 [68] Rogovchenko, S.P.; Rogovchenko, Y.V., Asymptotics of solutions for a class of second order nonlinear differential equations, Univ. iagel. acta math., 36, 157-164, (1998) · Zbl 1002.34037 [69] Rogovchenko, S.P.; Rogovchenko, Y.V., Asymptotic behavior of solutions of second order nonlinear differential equations, Portugal. math., 57, 17-33, (2000) · Zbl 0955.34035 [70] Rogovchenko, S.P.; Rogovchenko, Y.V., Asymptotic behavior of certain second order nonlinear differential equations, Dynam. systems appl., 10, 185-200, (2001) · Zbl 0997.34037 [71] 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 [72] Rogovchenko, Y.V.; Villari, G., Asymptotic behavior of solutions for second order nonlinear autonomous differential equations, Nonlinear differential equations appl., 4, 271-307, (1997) · Zbl 0880.34058 [73] Seifert, G., Global asymptotic behavior of solutions of positively damped Liénard equations, Ann. polon. math., 51, 283-290, (1990) · Zbl 0721.34064 [74] Sobol’, I.M., On the asymptotic behavior of the solutions of linear differential equations, Dokl. akad. nauk SSSR, 61, 219-222, (1948), (in Russian) [75] Sobol’, I.M., On Riccati equations and the reduction to them of linear equations of second order, Dokl. akad. nauk SSSR, 65, 257-278, (1949), (in Russian) [76] Sobol’, I.M., Limiting solution of Riccati’s equation and its application to investigation of solutions of a linear differential equation of second order, Moskov. GoS. univ. Uć. zap. mat., 155, 195-205, (1952), (Russian) [77] 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 [78] Trench, W.F., Asymptotic behavior of solutions of $$\mathit{Lu} = g(t, u, \ldots, u^{(k - 1)})$$, J. differential equations, 11, 38-48, (1972) · Zbl 0235.34083 [79] Trench, W.F., Systems of differential equations subject to mild integral conditions, Proc. amer. math. soc., 87, 263-270, (1983) · Zbl 0514.34031 [80] Trench, W.F., Global solutions of nonlinear perturbations of linear differential equations, Wssiaa, 1, 543-557, (1992) · Zbl 0832.34034 [81] Viswanatham, B., A generalization of Bellman’s lemma, Proc. amer. math. soc., 14, 15-18, (1963) · Zbl 0118.08001 [82] Wahlén, E., Positive solutions of second-order differential equations, Nonlinear anal. theor. methods appl., 58, 359-366, (2004) · Zbl 1052.34043 [83] Waltman, P., On the asymptotic behavior of solutions of a nonlinear equation, Proc. amer. math. soc., 15, 918-923, (1964) · Zbl 0131.08901 [84] Weyl, H., Comment on the preceding paper, Amer. J. math., 68, 7-12, (1946) · Zbl 0061.19707 [85] Wilkins, J.E., On the growth of solutions of linear differential equations, Bull. amer. math. soc., 50, 388-394, (1944) · Zbl 0061.18701 [86] Wintner, A., Comments on “flat” oscillations of low frequency, Duke math. J., 24, 365-366, (1957) · Zbl 0079.11002 [87] Wong, J.S.W., On two theorems of waltman, SIAM J. appl. math., 14, 724-728, (1966) · Zbl 0145.33702 [88] Wong, J.S.W., On second order nonlinear oscillation, Funkc. ekvac., 11, 207-234, (1968) · Zbl 0157.14802 [89] Yin, Z., Monotone positive solutions of second-order nonlinear differential equations, Nonlinear anal. theor. methods appl., 54, 391-403, (2003) · Zbl 1034.34045 [90] Zhao, Z., Positive solutions of nonlinear second order ordinary differential equations, Proc. amer. math. soc., 121, 465-469, (1994) · Zbl 0802.34026
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