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Karamata functions and differential equations: achievements from the 20th century. (English) Zbl 1429.34003

In his seminal article [Mathematica, Cluj 4, 38–53 (1930; JFM 56.0907.01)], J. Karamata studied positive functions \(r\) on intervals \([a,+\infty)\), \(a>0\), such that for some exponent \(\rho\), one has \(\lim_{x\to\infty}r(tx)/r(x)=t^\rho\) for all \(t>0\), with the aim of generalising one of the Hardy-Littlewood Tauberian theorems (see [J. Korevaar, Tauberian theory. A century of developments. Berlin: Springer (2004; Zbl 1056.40002)]). These functions are called today Karamata or regularly varying functions with regularity index \(\rho\). There also is the opposite concept of rapidly varying functions for which the limit is either \(0\) or \(\infty\) for all \(t>1\); there are generalisations and variants: see, e.g. [V. G. Avakumovic, Bull. Int. Acad. Yougoslave Sci. Beaux-Arts 29/30, 107–117 (1936; Zbl 0015.25003); E. Omey and E. Willekens, J. Lond. Math. Soc., II. Ser. 37, No. 1, 105–118 (1988; Zbl 0612.26001); J. Jaroš and T. Kusano, Bull., Cl. Sci. Math. Nat., Sci. Math. 129, No. 29, 25–60 (2004; Zbl 1091.34521)]. A general reference is [N. H. Bingham et al., Regular variation. Cambridge: Cambridge University Press (1987; Zbl 0617.26001)]. The author describes the rôle of these classes of functions in the subject of differential equations through a historical survey; see also [V. Marić, Regular variation and differential equations. Berlin: Springer (2000; Zbl 0946.34001)].
In §2, the author focuses on nonlinear differential equations: V. G. Avakumović [Acad. Serbe, Bull. Acad. Sci. Math. Natur. A 1, 101–113 (1947); ibid 2, 223–235 (1948; Zbl 0033.27302)] studied the asymptotics of solutions of the following Thomas-Fermi-type equation: \(y''=f(x)y^\lambda\) with boundary conditions \(y(0)=1\) and \(\lim_{x\to\infty}y(x)=0\), where \(f\) is equivalent to a Karamata function of regularity index \(\rho>-2\). This thread of research has been pursued, e.g., by the following researchers: P.-K. Wong [Pac. J. Math. 13, 737–760 (1963; Zbl 0115.07203)], J. L. Geluk [Proc. Am. Math. Soc. 112, No. 2, 429–431 (1991; Zbl 0722.34045)], V. Marić and M. Tomić [Math. Z. 149, 261–266 (1976; Zbl 0316.34054); Publ. Inst. Math., Nouv. Sér. 21(35), 119–129 (1977; Zbl 0359.34045); J. Differ. Equations 35, 36–44 (1980; Zbl 0433.34041)], T. Kusano, J. V. Manojlović and V. Marić [“Complete asymptotic analysis of second-order differential equations of Thomas-Fermi type in the framework of regular variation”, RIMS Kokyuroku 1959, 14–34 (2015), http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1959-02.pdf].
In §3, the author focuses on linear differential equations: E. Omey [Bull. Soc. Math. Belg., Sér. B 33, 207–229 (1981; Zbl 0497.34013)] studied the equation \(y''=\pm f(x)y\) with boundary condition \(\lim_{x\to\infty}y(x)=\infty\) and \(f\) a positive function such that \(\int_1^\infty sf(s)=\infty\) and \(x^2f(x)\) converges to some constant \(c\) at infinity, and proved that \(y\) is a Karamata function with index satisfying \(\rho(\rho-1)=\pm c\) in the following cases: if one chooses the \(+\) sign with \(c\in[0,\infty]\); if one chooses the \(-\) sign with \(c\in[0,1/4]\). The latter case is connected to one of A. A. Friedmann’s cosmological equations [Z. Phys. 10, 377–386 (1922; JFM 48.1031.02)]. This thread of research has been pursued, e.g., by the following researchers: V. Marić and M. Tomić [Zb. Rad., Prir.-Mat. Fak., Univ. Novom Sadu, Ser. Mat. 14, No. 2, 1–11 (1984; Zbl 0597.34060); Publ. Inst. Math., Nouv. Sér. 48(62), 199–207 (1990; Zbl 0722.34007)], H. Howard and V. Marić [Bull. Lond. Math. Soc. 26, No. 4, 373–381 (1994; Zbl 0821.34028); Bull., Cl. Sci. Math. Nat., Sci. Math. 114, No. 22, 85–98 (1997; Zbl 0947.34015)].

MSC:

34-03 History of ordinary differential equations
01A60 History of mathematics in the 20th century
34E05 Asymptotic expansions of solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
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