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 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 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.

BLOW-UP ANALYSIS FOR SOLUTIONS OF THE LAPLACIAN EQUATION WITH EXPONENTIAL NEUMANN BOUNDARY CONDITION IN DIMENSION TWO

2006 ◽  
Vol 08 (06) ◽  
pp. 737-761 ◽  
Author(s):  
YU-XIA GUO ◽  
JIA-QUAN LIU
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Existence Theorem ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Neumann Boundary ◽  
Asymptotic Behavior Of Solutions ◽  
Laplacian Equation ◽  
Blow Up Analysis

We consider the asymptotic behavior of solutions of the Laplacian equation with exponential Neumann boundary condition in dimension two. As an application, we prove an existence theorem of nonminimum solutions.

Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives

2021 ◽  
Vol 24 (2) ◽  
pp. 483-508
Author(s):  
Mohammed D. Kassim ◽  
Nasser-eddine Tatar
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Logarithmic Function ◽  
Fractional Differential ◽  
Derivatives Of ◽  
L’Hôpital’S Rule

Abstract The asymptotic behaviour of solutions in an appropriate space is discussed for a fractional problem involving Hadamard left-sided fractional derivatives of different orders. Reasonable sufficient conditions are determined ensuring that solutions of fractional differential equations with nonlinear right hand sides approach a logarithmic function as time goes to infinity. This generalizes and extends earlier results on integer order differential equations to the fractional case. Our approach is based on appropriate desingularization techniques and generalized versions of Gronwall-Bellman inequality. It relies also on a kind of Hadamard fractional version of l'Hopital’s rule which we prove here.

scholarly journals Single blow-up solutions for a slightly subcritical biharmonic equation

2006 ◽  
Vol 2006 ◽  
pp. 1-20 ◽  
Author(s):  
Khalil El Mehdi
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Critical Exponent ◽  
Critical Point ◽  
Bounded Domain ◽  
Biharmonic Equation ◽  
Blow Up ◽  
Asymptotic Behavior Of Solutions ◽  
Smooth Bounded Domain ◽  
Nondegenerate Critical Point

We consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent (Pε):∆2u=u9−ε,u>0inΩandu=∆u=0on∂Ω, whereΩis a smooth bounded domain inℝ5,ε>0. We study the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev quotient asεgoes to zero. We show that such solutions concentrate around a pointx0∈Ωasε→0, moreoverx0is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical pointx0of the Robin's function, there exist solutions of (Pε) concentrating aroundx0asε→0.

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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