Friedrichs extension of operators defined by even order Sturm–Liouville equations on time scales

2012 ◽  
Vol 218 (22) ◽  
pp. 10829-10842 ◽  
Author(s):  
Petr Zemánek ◽  
Petr Hasil
Keyword(s):  
Time Scales ◽  
Friedrichs Extension ◽  
Even Order ◽  
Sturm Liouville ◽  
Extension Of Operators

On the Friedrichs extension of semi-bounded difference operators

1994 ◽  
Vol 116 (1) ◽  
pp. 167-177 ◽  
Author(s):  
M. Benammar ◽  
W. D. Evans
Keyword(s):  
Liouville Equation ◽  
Liouville Operator ◽  
Difference Operators ◽  
Dirichlet Integral ◽  
Friedrichs Extension ◽  
Sturm Liouville

In [5] Kalf obtained a characterization of the Friedrichs extension TF of a general semi-bounded Sturm–Liouville operator T, the only assumptions made on the coefficients being those necessary for T to be defined. The domain D(TF) of TF was described in terms of ‘weighted’ Dirichiet integrals involving the principal and non-principal solutions of an associated non-oscillatory Sturm–Liouville equation. Conditions which ensure that members of D(TF) have a finite Dirichlet integral were subsequently given by Rosenberger in [7].

scholarly journals DUAL EQUATION AND INVERSE PROBLEM FOR AN INDEFINITE STURM–LIOUVILLE PROBLEM WITH M TURNING POINTS OF EVEN ORDER

2012 ◽  
Vol 17 (5) ◽  
pp. 618-629
Author(s):  
Hamidreza Marasi ◽  
Aliasghar Jodayree Akbarfam
Keyword(s):  
Inverse Problem ◽  
Infinite Product ◽  
Turning Points ◽  
Liouville Problem ◽  
Product Representation ◽  
Dual Equation ◽  
Number Of Zeros ◽  
Even Order ◽  
Sturm Liouville ◽  
Infinite Product Representation

In this paper the differential equation y″ + (ρ 2 φ 2 (x) –q(x))y = 0 is considered on a finite interval I, say I = [0, 1], where q is a positive sufficiently smooth function and ρ 2 is a real parameter. Also, [0, 1] contains a finite number of zeros of φ 2 , the so called turning points, 0 < x 1 < x 2 < … < x m < 1. First we obtain the infinite product representation of the solution. The product representation, satisfies in the original equation. As a result the associated dual equation is derived and then we proceed with the solution of the inverse problem.

Sturm-Liouville eigenvalue problems on time scales

1999 ◽  
Vol 99 (2-3) ◽  
pp. 153-166 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Martin Bohner ◽  
Patricia J.Y. Wong
Keyword(s):  
Time Scales ◽  
Eigenvalue Problems ◽  
Sturm Liouville

Variable Change for Sturm-Liouville Differential Expressions on Time Scales

2003 ◽  
Vol 9 (1) ◽  
pp. 93-107
Author(s):  
Calvin Ahlbrandt ◽  
Martin Bohner ◽  
Tammy Voepel
Keyword(s):  
Time Scales ◽  
Sturm Liouville
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