PHASE FUNCTIONS AND CENTRAL DISPERSIONS IN THE THEORY OF LINEAR FUNCTIONAL EQUATIONS

The terms introduced in this paper, like phase function, conjugate numbers induced by a phase function, fundamental numbers and fundamental orbits, fundamental central dispersion of the phase function and central dispersions of higher orders, were studied by Borůvka and Neuman in connection with pro...

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Bibliographic Details
Published in:The Rocky Mountain journal of mathematics 2005-01, Vol.35 (2), p.603-625
Main Author: LAITOCHOVÁ, JITKA
Format: Article
Language:English
Subjects:
34A30
39B20
central dispersion
Composite functions
conjugate number
Conjugate points
Continuous functions
Determining group
Difference equations
Differential equations
functional equation
Inverse functions
Mathematical functions
Mathematical monotonicity
Oscillation
phase function
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Summary:The terms introduced in this paper, like phase function, conjugate numbers induced by a phase function, fundamental numbers and fundamental orbits, fundamental central dispersion of the phase function and central dispersions of higher orders, were studied by Borůvka and Neuman in connection with properties of solutions of linear differential equations. A new direction is taken in this paper in order to remove the explicit dependence of the ideas upon differential equations. The theory presented here begins by defining anew the terms aforementioned, based only on properties of continuous functions, rather than by means of solutions to differential equations. The central idea for the generalized definitions is in a cyclic group of continuous functions, which effectively replaces the differential equation, giving a new direction to the original ideas of Borůvka and Neuman. The direction is similar to, but different than, the unrestricted n-parameter family theory introduced by Hartman, which generalizes solution properties of nth order linear differential equations to an abstract setting devoid of differential equations. Some applications are given for using phase function ideas to solve certain linear functional equations of higher order and special linear difference equations with constant coefficients. These examples do not have an underlying differential equation and therefore the Borůvka-Neuman theory does not apply.
ISSN:0035-7596
1945-3795
DOI:10.1216/rmjm/1181069749