scholarly journals Characterization of Self-Adjoint Domains for Two-Interval Even Order Singular C-Symmetric Differential Operators in Direct Sum Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Qinglan Bao ◽  
Xiaoling Hao ◽  
Jiong Sun
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Direct Sum ◽  
Even Order ◽  
Special Case ◽  
Symmetric Differential Operators

This paper is concerned with the characterization of all self-adjoint domains associated with two-interval even order singular C-symmetric differential operators in terms of boundary conditions. The previously known characterizations of Lagrange symmetric differential operators are a special case of this one.

scholarly journals TWO-INTERVAL EVEN-ORDER DIFFERENTIAL OPERATORS IN MODIFIED HILBERT SPACES

2011 ◽  
Vol 85 (2) ◽  
pp. 241-260
Author(s):  
JIANQING SUO ◽  
WANYI WANG
Keyword(s):  
Boundary Conditions ◽  
Hilbert Spaces ◽  
Differential Operators ◽  
Order Equation ◽  
Inner Product ◽  
Coupling Matrix ◽  
Direct Sum ◽  
Even Order

AbstractBy modifying the inner product in the direct sum of the Hilbert spaces associated with each of two underlying intervals on which an even-order equation is defined, we generate self-adjoint realisations for boundary conditions with any real coupling matrix which are much more general than the coupling matrices from the ‘unmodified’ theory.

scholarly journals On the product of self-adjoint Sturm-Liouville differential operators in direct sum spaces

2006 ◽  
Vol 37 (1) ◽  
pp. 77-92 ◽  
Author(s):  
Sobhy El-Sayed Ibrahim
Keyword(s):  
Boundary Conditions ◽  
Finite Number ◽  
Differential Operators ◽  
Second Order ◽  
Regular Case ◽  
Maximal Domain ◽  
Direct Sum ◽  
Sesquilinear Form ◽  
Sturm Liouville

In this paper, the second-order symmetric Sturm-Liouville differential expressions $ \tau_1,\tau_2, \ldots, \tau_n $, with real coefficients on any finite number of intervals are studied in the setting of the direct sum of the $ L_w^2 $-spaces of functions defined on each of the separate intervals. It is shown that the characterization of singular self-adjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximal domain of the product operators, it is an exact parallel of that in the regular case. This characterization is an extension of those obtained in [6], [7], [8], [9], [12], [14] and [15].

Higher order Wirtinger inequalities

1980 ◽  
Vol 85 (1-2) ◽  
pp. 87-110 ◽  
Author(s):  
Kurt Kreith ◽  
Charles A. Swanson
Keyword(s):  
Boundary Conditions ◽  
Quadratic Forms ◽  
Differential Operators ◽  
Conjugate Point ◽  
Wirtinger Inequality ◽  
Even Order ◽  
Linear Differential ◽  
Natural Boundary Conditions ◽  
Derivatives Of ◽  
Admissible Functions

SynopsisWirtinger-type inequalities of order n are inequalities between quadratic forms involving derivatives of order k ≦ n of admissible functions in an interval (a, b). Several methods for establishing these inequalities are investigated, leading to improvements of classical results as well as systematic generation of new ones. A Wirtinger inequality for Hamiltonian systems is obtained in which standard regularity hypotheses are weakened and singular intervals are permitted, and this is employed to generalize standard inequalities for linear differential operators of even order. In particular second order inequalities of Beesack's type are developed, in which the admissible functions satisfy only the null boundary conditions at the endpoints of [a, b] and b does not exceed the first systems conjugate point (a) of a. Another approach is presented involving the standard minimization theory of quadratic forms and the theory of “natural boundary conditions”. Finally, inequalities of order n + k are described in terms of (n, n)-disconjugacy of associated 2nth order differential operators.

scholarly journals On the boundary conditions for products of Sturm-Liouville differential operators

2001 ◽  
Vol 32 (3) ◽  
pp. 187-199
Author(s):  
Sobhy El-Sayed Ibrahim
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Second Order ◽  
Regular Case ◽  
Sesquilinear Form ◽  
Sturm Liouville

In this paper, the second-order symmetric Sturm-Liouville differential expressions $ \tau_1, \tau_2, \ldots, \tau_n $ with real coefficients are considered on the interval $ I = (a,b) $, $ - \infty \le a < b \le \infty $. It is shown that the characterization of singular self-adjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximan domain of the product operators, and is an exact parallel of the regular case. This characterization is an extension of those obtained in [6], [8], [11-12], [14] and [15].

scholarly journals On the domain of selfadjoint extension of the product of Sturm-Liouville differential operators

2003 ◽  
Vol 2003 (11) ◽  
pp. 695-709
Author(s):  
Sobhy El-Sayed Ibrahim
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Second Order ◽  
Regular Case ◽  
Maximal Domain ◽  
Selfadjoint Extension ◽  
Sesquilinear Form ◽  
Sturm Liouville

The second-order symmetric Sturm-Liouville differential expressionsτ1,τ2,…,τnwith real coefficients are considered on the intervalI=(a,b),−∞≤a<b≤∞. It is shown that the characterization of singular selfadjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximal domain of the product operators, and it is an exact parallel of the regular case. This characterization is an extension of those obtained by Everitt and Zettl (1977), Hinton, Krall, and Shaw (1987), Ibrahim (1999), Krall and Zettl (1988), Lee (1975/1976), and Naimark (1968).

Two-Interval Even Order Differential Operators in Direct Sum Spaces

2011 ◽  
Vol 62 (1-2) ◽  
pp. 13-32 ◽  
Author(s):  
Jianqing Suo ◽  
Wanyi Wang
Keyword(s):  
Differential Operators ◽  
Direct Sum ◽  
Even Order
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