Karamata functions and differential equations: achievements from the 20th century
In his seminal article [Mathematica, Cluj 4, 38--53 (1930; JFM 56.0907.01)], \textit{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 [\textit{J. Korevaar}, Tauberian theory. A century of developments. Berlin: Springer (2004; Zbl 1056.40002)]). These functions are called today \textit{Karamata\/} or \textit{regularly varying\/} functions with \textit{regularity index~\(\rho\)}. There also is the opposite concept of \textit{rapidly varying\/} functions for which the limit is either~\(0\) or~\(\infty\) for all~\(t>1\); there are generalisations and variants: see, e.g. [\textit{V. G. Avakumovic}, Bull. Int. Acad. Yougoslave Sci. Beaux-Arts 29/30, 107--117 (1936; Zbl 0015.25003); \textit{E. Omey} and \textit{E. Willekens}, J. Lond. Math. Soc., II. Ser. 37, No. 1, 105--118 (1988; Zbl 0612.26001); \textit{J. Jaroš} and \textit{T. Kusano}, Bull., Cl. Sci. Math. Nat., Sci. Math. 129, No. 29, 25--60 (2004; Zbl 1091.34521)]. A general reference is [\textit{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 [\textit{V. Marić}, Regular variation and differential equations. Berlin: Springer (2000; Zbl 0946.34001)]. In \S2, the author focuses on nonlinear differential equations: \textit{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: \textit{P.-K. Wong} [Pac. J. Math. 13, 737--760 (1963; Zbl 0115.07203)], \textit{J. L. Geluk} [Proc. Am. Math. Soc. 112, No. 2, 429--431 (1991; Zbl 0722.34045)], \textit{V. Marić} and \textit{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)], \textit{T. Kusano}, \textit{J. V. Manojlović} and \textit{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), \url{http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1959-02.pdf}]. In \S3, the author focuses on linear differential equations: \textit{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 \textit{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: \textit{V. Marić} and \textit{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)], \textit{H. Howard} and \textit{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)].
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