Self-adjoint boundary-value problems on time-scales

Fordyce A. Davidson, Bryan P. Rynne

Research output: Contribution to journalArticle

Abstract

In this paper we consider a second order, Sturm-Liouville-type boundary-value operator of the form Lu := ?[pu?]? + qu, on an arbitrary, bounded time-scale double-struck T, for suitable functions p, q, together with suitable boundary conditions. We show that, with a suitable choice of domain, this operator can be formulated in the Hilbert space L 2(double-struck T?), in such a way that the resulting operator is self-adjoint, with compact resolvent (here, 'self-adjoint' means in the standard functional analytic meaning of this term). Previous discussions of operators of this, and similar, form have described them as 'self-adjoint', but have not demonstrated self-adjointness in the standard functional analytic sense. ©2007 Texas State University.

Original languageEnglish
Pages (from-to)1-10
Number of pages10
JournalElectronic Journal of Differential Equations
Volume2007
Publication statusPublished - 12 Dec 2007

Fingerprint

Time Scales
Boundary Value Problem
Operator
Self-adjointness
Sturm-Liouville
Resolvent
Boundary Value
Hilbert space
Boundary conditions
Arbitrary
Term
Standards
Form

Keywords

  • Boundary-value problem
  • Self-adjoint linear operators
  • Sobolev spaces
  • Time-scales

Cite this

@article{cf5ee2ebf0ee49d38f997d482eba3dc4,
title = "Self-adjoint boundary-value problems on time-scales",
abstract = "In this paper we consider a second order, Sturm-Liouville-type boundary-value operator of the form Lu := ?[pu?]? + qu, on an arbitrary, bounded time-scale double-struck T, for suitable functions p, q, together with suitable boundary conditions. We show that, with a suitable choice of domain, this operator can be formulated in the Hilbert space L 2(double-struck T?), in such a way that the resulting operator is self-adjoint, with compact resolvent (here, 'self-adjoint' means in the standard functional analytic meaning of this term). Previous discussions of operators of this, and similar, form have described them as 'self-adjoint', but have not demonstrated self-adjointness in the standard functional analytic sense. {\circledC}2007 Texas State University.",
keywords = "Boundary-value problem, Self-adjoint linear operators, Sobolev spaces, Time-scales",
author = "Davidson, {Fordyce A.} and Rynne, {Bryan P.}",
year = "2007",
month = "12",
day = "12",
language = "English",
volume = "2007",
pages = "1--10",
journal = "Electronic Journal of Differential Equations",
issn = "1072-6691",
publisher = "Texas State University - San Marcos",

}

Self-adjoint boundary-value problems on time-scales. / Davidson, Fordyce A.; Rynne, Bryan P.

In: Electronic Journal of Differential Equations, Vol. 2007, 12.12.2007, p. 1-10.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Self-adjoint boundary-value problems on time-scales

AU - Davidson, Fordyce A.

AU - Rynne, Bryan P.

PY - 2007/12/12

Y1 - 2007/12/12

N2 - In this paper we consider a second order, Sturm-Liouville-type boundary-value operator of the form Lu := ?[pu?]? + qu, on an arbitrary, bounded time-scale double-struck T, for suitable functions p, q, together with suitable boundary conditions. We show that, with a suitable choice of domain, this operator can be formulated in the Hilbert space L 2(double-struck T?), in such a way that the resulting operator is self-adjoint, with compact resolvent (here, 'self-adjoint' means in the standard functional analytic meaning of this term). Previous discussions of operators of this, and similar, form have described them as 'self-adjoint', but have not demonstrated self-adjointness in the standard functional analytic sense. ©2007 Texas State University.

AB - In this paper we consider a second order, Sturm-Liouville-type boundary-value operator of the form Lu := ?[pu?]? + qu, on an arbitrary, bounded time-scale double-struck T, for suitable functions p, q, together with suitable boundary conditions. We show that, with a suitable choice of domain, this operator can be formulated in the Hilbert space L 2(double-struck T?), in such a way that the resulting operator is self-adjoint, with compact resolvent (here, 'self-adjoint' means in the standard functional analytic meaning of this term). Previous discussions of operators of this, and similar, form have described them as 'self-adjoint', but have not demonstrated self-adjointness in the standard functional analytic sense. ©2007 Texas State University.

KW - Boundary-value problem

KW - Self-adjoint linear operators

KW - Sobolev spaces

KW - Time-scales

UR - http://www.scopus.com/inward/record.url?scp=37449001494&partnerID=8YFLogxK

M3 - Article

VL - 2007

SP - 1

EP - 10

JO - Electronic Journal of Differential Equations

JF - Electronic Journal of Differential Equations

SN - 1072-6691

ER -