Oscillation criteria of Nehari-type for Sturm–Liouville operators and elliptic differential operators of second order and the lower spectrum

1988 ◽  
Vol 109 (1-2) ◽  
pp. 127-144 ◽  
Author(s):  
F. Fiedler
Keyword(s):  
Differential Equation ◽  
Differential Operators ◽  
Second Order ◽  
Oscillation Criteria ◽  
Elliptic Differential Equations ◽  
Ordinary Differential Operators ◽  
Elliptic Differential Operators ◽  
Sturm Liouville ◽  
Lower Spectrum ◽  
Liouville Operators

SynopsisSufficient oscillation criteria of Nehari-type are established for the differential equation −uʺ(t) + q(t)u(t) = 0, 0<t<∞, with and without sign restrictions on q(t), respectively. These results are extended to Sturm-Liouville equations and elliptic differential equations of second order.In Section 7 we present conclusions for the lower spectrum of elliptic differential operators and also for the discreteness of the spectrum of certain ordinary differential operators of second order.

The Expansion Theorems for Sturm-Liouville Operators with an Involution Perturbation

2021 ◽  
Author(s):  
Abdizhahan Sarsenbi
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Second Order ◽  
Order Differential Operator ◽  
Spectral Expansions ◽  
Sturm Liouville ◽  
Complex Valued ◽  
Liouville Operators

In this work, we studied the Green&rsquo;s functions of the second order differential operators with involution. Uniform equiconvergence of spectral expansions related to the second-order differential operators with involution is obtained. Basicity of eigenfunctions of the second-order differential operator operator with complex-valued coefficient is established.

scholarly journals Oscillatory behavior of a second order nonlinear advanced differential equation with mixed neutral terms

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hongwei Shi ◽  
Yuzhen Bai
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Oscillation Criteria ◽  
Order Nonlinear Differential Equation

AbstractIn this paper, we present several new oscillation criteria for a second order nonlinear differential equation with mixed neutral terms of the form $$ \bigl(r(t) \bigl(z'(t)\bigr)^{\alpha }\bigr)'+q(t)x^{\beta } \bigl(\sigma (t)\bigr)=0,\quad t\geq t_{0}, $$(r(t)(z′(t))α)′+q(t)xβ(σ(t))=0,t≥t0, where $z(t)=x(t)+p_{1}(t)x(\tau (t))+p_{2}(t)x(\lambda (t))$z(t)=x(t)+p1(t)x(τ(t))+p2(t)x(λ(t)) and α, β are ratios of two positive odd integers. Our results improve and complement some well-known results which were published recently in the literature. Two examples are given to illustrate the efficiency of our results.

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 Oscillation Criteria for Second Order Neutral Differential Equations

1996 ◽  
Vol 48 (4) ◽  
pp. 871-886 ◽  
Author(s):  
Horng-Jaan Li ◽  
Wei-Ling Liu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Second Order ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
Neutral Differential Equations ◽  
Neutral Delay Differential Equation ◽  
Image Position ◽  
Delay Differential

AbstractSome oscillation criteria are given for the second order neutral delay differential equationwhere τ and σ are nonnegative constants, . These results generalize and improve some known results about both neutral and delay differential equations.

scholarly journals Interval oscillation criteria for certain forced second-order differential equations

2012 ◽  
Vol 28 (2) ◽  
pp. 337-344
Author(s):  
ERCAN TUNC ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Second Order ◽  
Half Line ◽  
Oscillation Criteria ◽  
Second Order Differential Equations ◽  
Interval Oscillation

By using generalized Riccati transformations and an inequality due to Hardy et al., several new interval oscillation criteria are established for the nonlinear damped differential equation... The new interval oscillation criteria are different from most known ones in the sense they are based on the information only on a sequence of subintervals of [t0, ∞), rather than on the whole half-line. Our results improve and extend the known some results in the literature.

Spectral Theory for the Differential Equation Lu= λ Mu

1958 ◽  
Vol 10 ◽  
pp. 431-446 ◽  
Author(s):  
Fred Brauer
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Spectral Theory ◽  
Real Line ◽  
Differential Operators ◽  
Identity Operator ◽  
The Real ◽  
Ordinary Differential Operators ◽  
Extensive Bibliography

Let L and M be linear ordinary differential operators defined on an interval I, not necessarily bounded, of the real line. We wish to consider the expansion of arbitrary functions in eigenfunctions of the differential equation Lu = λMu on I. The case where M is the identity operator and L has a self-adjoint realization as an operator in the Hilbert space L 2(I) has been treated in various ways by several authors; an extensive bibliography may be found in (4) or (8).

V.— On the Spectrum of Second Order Partial Differential Operators.

1970 ◽  
Vol 69 (1) ◽  
pp. 77-84
Author(s):  
K. J. Brown ◽  
I. M. Michael
Keyword(s):  
Differential Equations ◽  
Spectral Properties ◽  
Differential Operators ◽  
Second Order ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Ordinary Differential Operators

SynopsisIn a recent paper, Jyoti Chaudhuri and W. N. Everitt linked the spectral properties of certain second order ordinary differential operators with the analytic properties of the solutions of the corresponding differential equations. This paper considers similar properties of the spectrum of the corresponding partial differential operators.

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