scholarly journals Limit-Point/Limit-Circle Results for Equations with Damping

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
M. Bartušek ◽  
John R. Graef
Keyword(s):  
Differential Equation ◽  
Limit Point ◽  
Second Order ◽  
Limit Circle

The authors study the nonlinear limit-point and limit-circle properties for the second order nonlinear damped differential equation(a(t)|y'|p-1y')'+b(t)|y'|q-1y'+r(t)|y|λ-1y=0, where0<λ≤p≤q,a(t)>0, andr(t)>0. Some examples are given to illustrate the main results.

scholarly journals Limit Circle/Limit Point Criteria for Second-Order Sublinear Differential Equations with Damping Term

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Jing Shao ◽  
Wei Song
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Limit Point ◽  
Second Order ◽  
Damping Term ◽  
Limit Circle ◽  
New Criteria ◽  
Point Type ◽  
Circle Limit

The purpose of the present paper is to establish some new criteria for the classification of the sublinear differential equation as of the nonlinear limit circle type or of the nonlinear limit point type. The criteria presented here generalize some known results in the literature.

scholarly journals Second-Order Differential Operators in the Limit Circle Case

2021 ◽  
Author(s):  
Dmitri R. Yafaev ◽  
◽  
◽  
Keyword(s):  
Differential Equation ◽  
Differential Operators ◽  
Second Order ◽  
Spectral Measures ◽  
Limit Circle ◽  
Jacobi Operators ◽  
Limit Circle Case ◽  
Circle Case ◽  
Stieltjes Transforms

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.

scholarly journals Limit-Point/Limit-Circle Results for Superlinear Damped Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
M. Bartušek ◽  
John R. Graef
Keyword(s):  
Differential Equations ◽  
Limit Point ◽  
Second Order ◽  
Limit Circle

The authors study the nonlinear limit-point and limit-circle properties for second-order nonlinear damped differential equations of the form(a(t)|y'|p-1y')'+b(t)|y'|q-1y'+r(t)|y|λ-1y=0,where0<q≤p≤λ,a(t)>0, andr(t)>0. Examples to illustrate the main results are included.

scholarly journals Limit Circle/Limit Point Criteria for Second-Order Superlinear Differential Equations with a Damping Term

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Lihong Xing ◽  
Wei Song ◽  
Zhengqiang Zhang ◽  
Qiyi Xu
Keyword(s):  
Differential Equations ◽  
Limit Point ◽  
Second Order ◽  
Damping Term ◽  
Limit Circle ◽  
New Criteria ◽  
Point Type ◽  
Circle Limit

The purpose of the present paper is to establish some new criteria for the classifications of superlinear differential equations as being of the nonlinear limit circle type or of the nonlinear limit point type. The criteria presented here generalize some known results in literature.

scholarly journals Extensions of symmetric singular second-order dynamic operators on time scales

Filomat ◽  
2016 ◽  
Vol 30 (6) ◽  
pp. 1475-1484 ◽  
Author(s):  
Bilender Allakhverdiev
Keyword(s):  
Boundary Conditions ◽  
Time Scales ◽  
Limit Point ◽  
Second Order ◽  
Boundary Values ◽  
Infinite Time ◽  
Limit Circle ◽  
Symmetric Operators

A space of boundary values is constructed for minimal symmetric singular second-order dynamic operators on semi-infinite and infinite time scales in limit-point and limit-circle cases. A description of all maximal dissipative, maximal accumulative, selfadjoint, and other extensions of such symmetric operators is given in terms of boundary conditions.

HELP inequalities for limit-circle and regular problems

1991 ◽  
Vol 432 (1886) ◽  
pp. 367-390 ◽  
Keyword(s):  
Differential Equation ◽  
Limit Point ◽  
Strong Limit ◽  
End Points ◽  
Limit Circle ◽  
End Point ◽  
Limit Circle Case ◽  
Limit Point Case ◽  
Regular Problems ◽  
Circle Case

Everitt’s criterion for the validity of the generalized Hardy-Littlewood inequality presupposes that the associated differential equation is singular at one end-point of the interval of definition and is in the strong-limit-point case at the end-point. In this paper we investigate the cases when the differential equation is in the limit-circle case and non-oscillatory at the singular end-point and when both end-points of the interval are regular.

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