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 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.

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.

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.

scholarly journals On the asymptotic behavior of solutions of certain third-order nonlinear differential equations

2005 ◽  
Vol 2005 (1) ◽  
pp. 29-35 ◽  
Author(s):  
Cemil Tunç
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Third Order ◽  
The Third ◽  
All Solutions

We establish sufficient conditions under which all solutions of the third-order nonlinear differential equation x ⃛+ψ(x,x˙,x¨)x¨+f(x,x˙)=p(t,x,x˙,x¨) are bounded and converge to zero as t→∞.

scholarly journals Asymptotic Behavior of Solutions of Even-Order Advanced Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Omar Bazighifan ◽  
Hijaz Ahmad
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Second Order ◽  
Asymptotic Behavior Of Solutions ◽  
Qualitative Behavior ◽  
Riccati Transformation ◽  
Even Order ◽  
Second Order Equations

In this paper, we establish the qualitative behavior of the even-order advanced differential equation a υ y κ − 1 υ β ′ + ∑ i = 1 j q i υ g y η i υ = 0 ,   υ ≥ υ 0 . The results obtained are based on the Riccati transformation and the theory of comparison with first- and second-order equations. This new theorem complements and improves a number of results reported in the literature. Two examples are presented to demonstrate the main results.

Logarithmic growth filtrations for -modules over the bounded Robba ring

2021 ◽  
Vol 157 (6) ◽  
pp. 1265-1301
Author(s):  
Shun Ohkubo
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Unit Disc ◽  
Open Unit ◽  
Asymptotic Behavior Of Solutions ◽  
Newton Polygons ◽  
Open Unit Disc ◽  
Hypergeometric Differential Equation ◽  
Logarithmic Growth

In the 1970s, Dwork defined the logarithmic growth (log-growth for short) filtrations for $p$ -adic differential equations $Dx=0$ on the $p$ -adic open unit disc $|t|<1$ , which measure the asymptotic behavior of solutions $x$ as $|t|\to 1^{-}$ . Then, Dwork calculated the log-growth filtration for $p$ -adic Gaussian hypergeometric differential equation. In the late 2000s, Chiarellotto and Tsuzuki proposed a fundamental conjecture on the log-growth filtrations for $(\varphi ,\nabla )$ -modules over $K[\![t]\!]_0$ , which can be regarded as a generalization of Dwork's calculation. In this paper, we prove a generalization of the conjecture to $(\varphi ,\nabla )$ -modules over the bounded Robba ring. As an application, we prove a generalization of Dwork's conjecture proposed by Chiarellotto and Tsuzuki on the specialization property for log-growth Newton polygons.

scholarly journals Asymptotic behavior of solutions of nonlinear functional differential equations

1994 ◽  
Vol 17 (4) ◽  
pp. 703-712
Author(s):  
Jong Soo Jung ◽  
Jong Yeoul Park ◽  
Hong Jae Kang
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Maximal Monotone Operator ◽  
Functional Differential Equations ◽  
Maximal Monotone ◽  
Asymptotic Behavior Of Solutions ◽  
Nonlinear Functional ◽  
Functional Differential

Using the properties of almost nonexpansive curves introduced by B. Djafari Rouhani, we study the asymptotic behavior of solutions of nonlinear functional differential equationdu(t)/dt+Au(t)+G(u)(t)?f(t), whereAis a maximal monotone operator in a Hilbert spaceH,f?L1(0,8:H)andG:C([0,8):D(A)¯)?L1(0,8:H)is a given mapping.

scholarly journals Existence and asymptotic behavior of intermediate type of positive solutions of fourth-order nonlinear differential equations

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4185-4211
Author(s):  
Katarina Djordjevic ◽  
Jelena Manojlovic
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Fourth Order ◽  
Asymptotic Form ◽  
Nonlinear Differential Equations ◽  
Quasilinear Differential Equation ◽  
Intermediate Type ◽  
Integral Conditions ◽  
Regularly Varying Functions ◽  
Regularly Varying

Under the assumptions that p and q are regularly varying functions satisfying conditions ??a t/p(t)1/? dt < ? and ??a (t/p(t))1/? dt = ? existence and asymptotic form of regularly varying intermediate solutions are studied for a fourth-order quasilinear differential equation (p(t)jx??(t)|?-1 x??(t))?? + q(t)|x(t)|?-1 x(t) = 0, ? > ? > 0. It is shown that under certain integral conditions there exist two types of intermediate solutions which according to their asymptotic behavior is to be divided into six mutual distinctive classes, while asymptotic behavior of each member of any of these classes is governed by a unique explicit law.

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