The Spectrum of a Transport Operator with Abstract Boundary Conditions

2013 ◽  
Vol 15 (2) ◽  
pp. 109
Author(s):  
Hong-xing WU ◽  
Sheng-hua WANG
Keyword(s):  
Boundary Conditions ◽  
Transport Operator ◽  
Abstract Boundary

On compressions of self-adjoint extensions of a symmetric linear relation with unequal deficiency indices

2019 ◽  
Vol 16 (4) ◽  
pp. 567-587
Author(s):  
Vadim Mogilevskii
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Linear Relation ◽  
Deficiency Indices ◽  
Space Extension ◽  
Limit Value ◽  
Abstract Boundary ◽  
Symmetric Linear Relation ◽  
Symmetric Operators

Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with unequal deficiency indices $n_-A <n_+(A)$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an (exit space) extension of $A$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of extensions $\wt A=\wt A^*$. Our main result is a description of compressions $C (\wt A)$ by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter $\tau(\l)$ from the Krein formula for generalized resolvents. We describe also all extensions $\wt A=\wt A^*$ of $A$ with the maximal symmetric compression $C (\wt A)$ and all extensions $\wt A=\wt A^*$ of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators $A$ with equal deficiency indices $n_+(A)=n_-(A)$.

Fixed point theorems for block operator matrix and an application to a structured problem under boundary conditions of Rotenberg’s model type

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
Afif Amar ◽  
Aref Jeribi ◽  
Bilel Krichen
Keyword(s):  
Boundary Conditions ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Coupled System ◽  
Operator Matrix ◽  
Model Type ◽  
Block Operator Matrix ◽  
Growing Cell ◽  
Block Operator ◽  
Abstract Boundary

AbstractIn this manuscript, we introduce and study the existence of solutions for a coupled system of differential equations under abstract boundary conditions of Rotenberg’s model type, this last arises in growing cell populations. The entries of block operator matrix associated to this system are nonlinear and act on the Banach space X p:= L p([0, 1] × [a, b]; dµ dv), where 0 ≤ a < b < ∞; 1 < p < ∞.

Differential Operators with Abstract Boundary Conditions

1978 ◽  
Vol 30 (02) ◽  
pp. 262-288 ◽  
Author(s):  
R. C. Brown
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Vector Space ◽  
Differential Operators ◽  
Dimensional Vector ◽  
Absolutely Continuous ◽  
Abstract Boundary ◽  
Image Position ◽  
Vector Valued Functions ◽  
Vector Valued

Suppose F is a topological vector space. Let ACm ≡ ACm[a, b] be the absolutely continuous m-dimensional vector valued functions y on the compact interval [a, b] with essentially bounded components. Consider the boundary value problem where A0, A are respectively... operator with range in F.

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