scholarly journals On some extensions of Karamata’s theory and their applications

2006 ◽  
Vol 80 (94) ◽  
pp. 59-96 ◽  
Author(s):  
V.V. Buldygin ◽  
O.I. Klesov ◽  
J.G. Steinebach
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Uniform Convergence ◽  
Stochastic Differential Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Convergence Theorems ◽  
Regularly Varying Functions ◽  
Regularly Varying ◽  
Inverse Functions

This is a survey of the authors? results on the properties and applications of some subclasses of (so-called) O-regularly varying (ORV) functions. In particular, factorization and uniform convergence theorems for Avakumovic-Karamata functions with non-degenerate groups of regular points are presented together with the properties of various other extensions of regularly varying functions. A discussion of equivalent characterizations of such classes of functions is also included as well as that of their (asymptotic) inverse functions. Applications are given concerning the asymptotic behavior of solutions of certain stochastic differential equations.

scholarly journals Self-adjoint differential equations and generalized Karamata functions

2004 ◽  
Vol 129 (29) ◽  
pp. 25-60 ◽  
Author(s):  
Jaroslav Jaros ◽  
Takasi Kusano
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Regularly Varying Functions ◽  
Regularly Varying

Howard and Maric have recently developed nice nonoscillation theorems for the differential equation U" + q(t)y = 0 (*) by means of regularly varying functions in the sense of Karamata. The purpose of this paper is to show that their results can be fully generalized to differential equations of the form, (p(t)y?)? + q(t)y = o (**) by using the notion of generalized Karamata functions, which is needed to comprehend how delicately the asymptotic behavior of solutions of (**) is affected by the function p(t).

scholarly journals An equation with left and right fractional derivatives

2006 ◽  
Vol 80 (94) ◽  
pp. 259-272 ◽  
Author(s):  
B. Stankovic
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Fractional Derivatives ◽  
Asymptotic Behavior Of Solutions ◽  
Regularly Varying Functions ◽  
Tempered Distributions ◽  
End Points ◽  
Left And Right ◽  
Regularly Varying ◽  
The Space L1

We consider an equation with left and right fractional derivatives and with the boundary condition y(0) = lim x?0+ y(x) = 0, y(b) = lim x?b? y(x) = 0 in the space L1(0, b) and in the subspace of tempered distributions. The asymptotic behavior of solutions in the end points 0 and b have been specially analyzed by using Karamata?s regularly varying functions.

scholarly journals On Avakumovic’s theorem for generalized Thomas-Fermi differential equations

2016 ◽  
Vol 99 (113) ◽  
pp. 125-137 ◽  
Author(s):  
Jaroslav Jaros ◽  
Takaŝi Kusano
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Positive Solutions ◽  
Negative Index ◽  
Regularly Varying Function ◽  
Regularly Varying Functions ◽  
Thomas Fermi ◽  
Regularly Varying

For the generalized Thomas-Fermi differential equation (|x?|??1x?)? = q(t)|x|??1x, it is proved that if 1 ? ? < ? and q(t) is a regularly varying function of index ? with ? > ?? ? 1, then all positive solutions that tend to zero as t ? 1 are regularly varying functions of one and the same negative index p and their asymptotic behavior at infinity is governed by the unique definite decay law. Further, an attempt is made to generalize this result to more general quasilinear differential equations of the form (p(t)|x?|??1x?)? = q(t)|x|??1x.

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Regular Variation ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Continuous Function ◽  
Linear Differential

AbstractWe study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

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