Solutions of Fourth-Order Self-Conjugate Problems. Oscillation Properties

2010 ◽  
Keyword(s):  
Fourth Order ◽  
Oscillation Properties

scholarly journals Growth and oscillation properties of solutions of a fourth order linear difference equation

1985 ◽  
Vol 26 (3) ◽  
pp. 310-328 ◽  
Author(s):  
John W. Hooker ◽  
William T. Patula
Keyword(s):  
Difference Equation ◽  
Fourth Order ◽  
Initial Conditions ◽  
Linear Difference Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Order Difference ◽  
Order Linear ◽  
Linearly Independent ◽  
Separation Of Zeros ◽  
Oscillation Properties

AbstractFor the fourth-order linear difference equation Δ4un−2 = bn un, with bn > 0 for all n, generalized zeros are defined, following Hartman [5], and two theorems are proved concerning separation of zeros of linearly independent solutions. Some preliminary results deal with non-oscillation and asymptotic behavior of solutions of this equation for various types of initial conditions. Finally, recessive solutions are defined, and results are obtained analogous to known results for recessive solutions of second-order difference equations.

Lower Bounds for the Essential Spectrum of Fourth-Order Differential Operators

1969 ◽  
Vol 21 ◽  
pp. 460-465
Author(s):  
Kurt Kreith
Keyword(s):  
Lower Bound ◽  
Essential Spectrum ◽  
Lower Bounds ◽  
Fourth Order ◽  
Differential Operators ◽  
Intimate Connection ◽  
Sturm Liouville ◽  
Image Position ◽  
Oscillation Properties ◽  
Liouville Operators

In this paper, we seek to determine the greatest lower bound of the essential spectrum of self-adjoint singular differential operators of the form1where 0 ≦ x < ∞. In the event that this bound is + ∞, our results will yield criteria for the discreteness of the spectrum of (1).Such bounds have been established by Friedrichs (3) for Sturm-Liouville operators of the formand our techniques will be closely related to those of (3). However, instead of studying the solutions of2directly, we shall exploit the intimate connection between the infimum of the essential spectrum of (1) and the oscillation properties of (2).

scholarly journals On Oscillation Properties of Self-Adjoint Boundary Value Problems of Fourth Order

2021 ◽  
Author(s):  
A. A. Vladimirov ◽  
A. A. Shkalikov
Keyword(s):  
Boundary Value Problems ◽  
Fourth Order ◽  
Boundary Value ◽  
Nontrivial Solutions ◽  
New Approach ◽  
Oscillatory Problem ◽  
Derivatives Of ◽  
Inertia Index ◽  
Oscillation Properties

Abstract The connection between the number of internal zeros of nontrivial solutions to fourth-order self-adjoint boundary value problems and the inertia index of these problems is studied. We specify the types of problems for which such a connection can be established. In addition, we specify the types of problems for which a connection between the inertia index and the number of internal zeros of the derivatives of nontrivial solutions can be established. Examples demonstrating the effectiveness of the proposed new approach to an oscillatory problem are considered.

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