scholarly journals On the Maximal Monotonicity of the Sum of a Maximal Monotone Linear Relation and the Subdifferential Operator of a Sublinear Function

2012 ◽  
pp. 19-26 ◽  
Author(s):  
Heinz Bauschke ◽  
Xianfu Wang ◽  
Liangjin Yao
Keyword(s):  
Linear Relation ◽  
Maximal Monotone ◽  
Subdifferential Operator ◽  
Maximal Monotonicity ◽  
Sublinear Function

scholarly journals The sum theorem for maximal monotone operators in reflexive Banach spaces revisited

2021 ◽  
Vol 66 (1) ◽  
pp. 123-126
Author(s):  
Mircea D. Voisei
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Reflexive Banach Space ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Minkowski Sum ◽  
Reflexive Banach Spaces ◽  
Maximal Monotonicity ◽  
Sum Theorem

The goal of this note is to present a new shorter proof for the maximal monotonicity of the Minkowski sum of two maximal monotone multi-valued operators defined in a reflexive Banach space under the classical interiority condition involving their domains.

scholarly journals On the maximal monotonicity and the range of the sum of nonlinear maximal monotone operators

1991 ◽  
Vol 34 (1) ◽  
pp. 143-153 ◽  
Author(s):  
J. R. L. Webb ◽  
Weiyu Zhao
Keyword(s):  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Maximal Monotonicity ◽  
Multivalued Operators ◽  
Nonlinear Maximal Monotone ◽  
Nondecreasing Functions

Conditions are given on two maximal monotone (multivalued) operators A and B which ensure that A + B is also maximal. One condition used is that ∥Bx∥≦k(∥x∥)Ax| +d|(A + B)x| + c(∥x∥) for every x∈D(A)⊆D(B), where 0≦k(r)<1, and c(r)≧0 are nondecreasing functions, and 0≦d≦1 is a constant. Here, for a set C, |C| denotes inf{∥y∥:y∈C}. This extends the well known result which has d = 0 (and is used in the proof here). The second part of the paper uses similar hypotheses to give conditions under which the range of the sum, R(A + B), has the same interior and same closure as the sum of the ranges, R(A) + R(B).

scholarly journals Hidden maximal monotonicity in evolutionary variational-hemivariational inequalities

2021 ◽  
Vol 26 (6) ◽  
pp. 1144-1165
Author(s):  
Emilio Vilches ◽  
Shengda Zeng
Keyword(s):  
Evolution Equations ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Hemivariational Inequalities ◽  
Well Posedness ◽  
Evolution Inclusion ◽  
Theory Of Evolution ◽  
Maximal Monotonicity ◽  
Theoretical Results

In this paper, we propose a new methodology to study evolutionary variational-hemivariational inequalities based on the theory of evolution equations governed by maximal monotone operators. More precisely, the proposed approach, based on a hidden maximal monotonicity, is used to explore the well-posedness for a class of evolutionary variational-hemivariational inequalities involving history-dependent operators and related problems with periodic and antiperiodic boundary conditions. The applicability of our theoretical results is illustrated through applications to a fractional evolution inclusion and a dynamic semipermeability problem.

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 $n_+(A)=n_-(A)$.

scholarly journals Fractional Order of Evolution Inclusion Coupled with a Time and State Dependent Maximal Monotone Operator

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1395
Author(s):  
Charles Castaing ◽  
Christiane Godet-Thobie ◽  
Le Xuan Truong
Keyword(s):  
Fractional Order ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Separable Hilbert Space ◽  
Maximal Monotone ◽  
Fractional Flow ◽  
Order Problem ◽  
Absolutely Continuous ◽  
State Dependent ◽  
Fractional Differential

This paper is devoted to the study of evolution problems involving fractional flow and time and state dependent maximal monotone operator which is absolutely continuous in variation with respect to the Vladimirov’s pseudo distance. In a first part, we solve a second order problem and give an application to sweeping process. In a second part, we study a class of fractional order problem driven by a time and state dependent maximal monotone operator with a Lipschitz perturbation in a separable Hilbert space. In the last part, we establish a Filippov theorem and a relaxation variant for fractional differential inclusion in a separable Banach space. In every part, some variants and applications are presented.

scholarly journals Topological degree and application to a parabolic variational inequality problem

2001 ◽  
Vol 25 (4) ◽  
pp. 273-287 ◽  
Author(s):  
A. Addou ◽  
B. Mermri
Keyword(s):  
Variational Inequality ◽  
Topological Degree ◽  
Variational Inequality Problem ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Parabolic Variational Inequality ◽  
Inequality Problem ◽  
Monotone Map ◽  
New Formulation

We are interested in constructing a topological degree for operators of the formF=L+A+S, whereLis a linear densely defined maximal monotone map,Ais a bounded maximal monotone operators, andSis a bounded demicontinuous map of class(S+)with respect to the domain ofL. By means of this topological degree we prove an existence result that will be applied to give a new formulation of a parabolic variational inequality problem.

scholarly journals Prospective study of dynamic whole-body 68Ga-DOTATOC-PET/CT acquisition in patients with well-differentiated neuroendocrine tumors

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Philippe Thuillier ◽  
David Bourhis ◽  
Jean Philippe Metges ◽  
Romain Le Pennec ◽  
Karim Amrane ◽  
...  
Keyword(s):  
Neuroendocrine Tumors ◽  
Linear Relation ◽  
Whole Body ◽  
Pet Ct ◽  
Non Linear ◽  
Well Differentiated ◽  
Analog Systems ◽  
Pet System ◽  
Dynamic Acquisition ◽  
Linear Regressions

AbstractTo present the feasibility of a dynamic whole-body (DWB) 68Ga-DOTATOC-PET/CT acquisition in patients with well-differentiated neuroendocrine tumors (WD-NETs). Sixty-one patients who underwent a DWB 68Ga-DOTATOC-PET/CT for a histologically proven/highly suspected WD-NET were prospectively included. The acquisition consisted in single-bed dynamic acquisition centered on the heart, followed by the DWB and static acquisitions. For liver, spleen and tumor (1–5/patient), Ki values (in ml/min/100 ml) were calculated according to Patlak's analysis and tumor-to-liver (TLR-Ki) and tumor-to-spleen ratios (TSR-Ki) were recorded. Ki-based parameters were compared to static parameters (SUVmax/SUVmean, TLR/TSRmean, according to liver/spleen SUVmean), in the whole-cohort and according to the PET system (analog/digital). A correlation analysis between SUVmean/Ki was performed using linear and non-linear regressions. Ki-liver was not influenced by the PET system used, unlike SUVmax/SUVmean. The regression analysis showed a non-linear relation between Ki/SUVmean (R2 = 0.55,0.68 and 0.71 for liver, spleen and tumor uptake, respectively) and a linear relation between TLRmean/TLR-Ki (R2 = 0.75). These results were not affected by the PET system, on the contrary of the relation between TSRmean/TSR-Ki (R2 = 0.94 and 0.73 using linear and non-linear regressions in digital and analog systems, respectively). Our study is the first showing the feasibility of a DWB 68Ga-DOTATOC-PET/CT acquisition in WD-NETs.

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