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.

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.

On strongly decreasing solutions of cyclic systems of second-order nonlinear differential equations

2015 ◽  
Vol 145 (5) ◽  
pp. 1007-1028 ◽  
Author(s):  
Jaroslav Jaroš ◽  
Kusano Takaŝi
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Nonlinear Differential Equations ◽  
Radial Symmetry ◽  
Second Order ◽  
Accurate Information ◽  
Regularly Varying Functions ◽  
Cyclic System ◽  
Regularly Varying ◽  
Image Position

The n-dimensional cyclic system of second-order nonlinear differential equationsis analysed in the framework of regular variation. Under the assumption that αi and βi are positive constants such that α1 … αn > β1 … βn and pi and qi are regularly varying functions, it is shown that the situation in which the system possesses decreasing regularly varying solutions of negative indices can be completely characterized, and moreover that the asymptotic behaviour of such solutions is governed by a unique formula describing their order of decay precisely. Examples are presented to demonstrate that the main results for the system can be applied effectively to some classes of partial differential equations with radial symmetry to provide new accurate information about the existence and the asymptotic behaviour of their radial positive strongly decreasing solutions.

scholarly journals Euler case for a general fourth-order differential equation

2004 ◽  
Vol 2004 (51) ◽  
pp. 2705-2717
Author(s):  
A. S. A. Al-Hammadi
Keyword(s):  
Differential Equation ◽  
General Formula ◽  
Fourth Order ◽  
Asymptotic Form ◽  
Order Differential Equation ◽  
Order Equation ◽  
Fourth Order Equation ◽  
Fourth Order Differential Equation

We deal with an Euler case for a general fourth-order equation and under this case, we obtain the general formula for the asymptotic form of the solutions.

scholarly journals Even Order Half-Linear Differential Equations with Regularly Varying Coefficients

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1236
Author(s):  
Vojtěch Růžička
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Type Equation ◽  
Real Numbers ◽  
Regularly Varying Functions ◽  
Varying Coefficients ◽  
Even Order ◽  
Regularly Varying ◽  
Nonoscillation Criterion ◽  
Linear Differential

We establish nonoscillation criterion for the even order half-linear differential equation (−1)nfn(t)Φx(n)(n)+∑l=1n(−1)n−lβn−lfn−l(t)Φx(n−l)(n−l)=0, where β0,β1,…,βn−1 are real numbers, n∈N, Φ(s)=sp−1sgns for s∈R, p∈(1,∞) and fn−l is a regularly varying (at infinity) function of the index α−lp for l=0,1,…,n and α∈R. This equation can be understood as a generalization of the even order Euler type half-linear differential equation. We obtain this Euler type equation by rewriting the equation above as follows: the terms fn(t) and fn−l(t) are replaced by the tα and tα−lp, respectively. Unlike in other texts dealing with the Euler type equation, in this article an approach based on the theory of regularly varying functions is used. We establish a nonoscillation criterion by utilizing the variational technique.

The asymptotic form of the Titchmarsh–Weyl coefficient for a fourth order differential equation

1984 ◽  
Vol 97 ◽  
pp. 109-123
Author(s):  
D. B. Hinton ◽  
J. K. Shaw
Keyword(s):  
Differential Equation ◽  
Asymptotic Formula ◽  
Real Axis ◽  
Fourth Order ◽  
Asymptotic Form ◽  
Order Differential Equation ◽  
Order Equation ◽  
Fourth Order Equation ◽  
The Matrix ◽  
Fourth Order Differential Equation

SynopsisThis paper considers the asymptotic form, as λ tends to infinity in sectors omitting the real axis, of the matrix Titchmarsh-Weyl coefficient M(λ) for the fourth order equation y(4) + q(x)y = λy, where q(x) is real and locally absolutely integrable. By letting M0(λ) denote the m-coefficient for the Fourier case y(4) = λy, the asymptotic formula M(λ) = M0(λ) + 0(1) is established.

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 On the nonexistence of Lp solutions of certain nonlinear differential equations

1967 ◽  
Vol 8 (2) ◽  
pp. 133-138 ◽  
Author(s):  
Thomas G. Hallam
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Lp Solutions

The asymptotic behavior of the solutions of ordinary nonlinear differential equations will be considered here. The growth of the solutions of a differential equation will be discussed by establishing criteria to determine when the differential equation does not possess a solution that is an element of the space Lp(0, ∞)(p ≧ 1).

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