scholarly journals A saddle point type solution for a system of operator equations

2021 ◽  
pp. 1-12
Author(s):  
Piotr Kowalski
Keyword(s):  
Saddle Point ◽  
Minimax Theorem ◽  
Dirichlet Problems ◽  
Type Solution ◽  
Operator Equations ◽  
Potential Operators ◽  
System Of Operator Equations ◽  
Ky Fan ◽  
Point Type

Let Ω⊂Rn n>1 and let p,q≥2. We consider the system of nonlinear Dirichlet problems equation* brace(Au)(x)=Nu′(x,u(x),v(x)),x∈Ω,r-(Bv)(x)=Nv′(x,u(x),v(x)),x∈Ω,ru(x)=0,x∈∂Ω,rv(x)=0,x∈∂Ω,endequation* where N:R×R→R is C1 and is partially convex-concave and A:W01,p(Ω)→(W01,p(Ω))* B:W01,p(Ω)→(W01,p(Ω))* are monotone and potential operators. The solvability of this system is reached via the Ky–Fan minimax theorem.

scholarly journals New Preconditioners for Nonsymmetric Saddle Point Systems with Singular (1,1) Block

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Qingbing Liu
Keyword(s):  
Saddle Point ◽  
Krylov Subspace ◽  
Optimal Parameter ◽  
Spectral Characteristics ◽  
Ill Conditioning ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods ◽  
Saddle Point Matrix ◽  
Large Linear Systems ◽  
Point Type

We investigate the solution of large linear systems of saddle point type with singular (1,1) block by preconditioned iterative methods and consider two parameterized block triangular preconditioners used with Krylov subspace methods which have the attractive property of improved eigenvalue clustering with increased ill-conditioning of the (1,1) block of the saddle point matrix, including the choice of the parameter. Meanwhile, we analyze the spectral characteristics of two preconditioners and give the optimal parameter in practice. Numerical experiments that validate the analysis are presented.

scholarly journals Differential transcendence of solutions of systems of linear differential equations based on total reduction of the system

2020 ◽  
pp. 24-24
Author(s):  
Ivana Jovovic
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Complex Field ◽  
System Matrix ◽  
Operator Equations ◽  
Total Reduction ◽  
Commuting Operators ◽  
Constant Coefficients ◽  
Linear Differential ◽  
System Of Operator Equations

In this paper we consider total reduction of the nonhomogeneous linear system of operator equations with constant coefficients and commuting operators. The totally reduced system obtained in this manner is completely decoupled. All equations of the system differ only in the variables and in the nonhomogeneous terms. The homogeneous parts are obtained using the generalized characteristic polynomial of the system matrix. We also indicate how this technique may be used to examine differential transcendence of the solution of the linear system of the differential equations with constant coefficients over the complex field and meromorphic free terms.

To the definition of operators , which are performed IN Kaluzhnin’s graph-schems with parafractal characteristics

2020 ◽  
pp. 98-104
Author(s):  
V.M. Simonov
Keyword(s):  
Standard Procedure ◽  
Principal Result ◽  
Operator Equations ◽  
Definition Of ◽  
System Of Operator Equations

The regular formula of operator, which is performed in given Kaluzhnin’s graph-scheme with parafractal characteristics, can be definded by two procedures. The standard procedure is based on the solution of system of operator equations, which is given birth by this graph-scheme. The modified procedure is based on the solution of several lesser-scale systems of operator equations, which are given birth by parafractals. The modified procedure is simpler than the standard one, but it is not evident identity of the results of both procedures. The principal result of this article is the theorem about this identity.

Multivariate version of slant Toeplitz operators on the Lebesgue space

2021 ◽  
pp. 2150152
Author(s):  
Shesh Kumar Pandey ◽  
Gopal Datt
Keyword(s):  
Toeplitz Operator ◽  
Lebesgue Space ◽  
Spectral Properties ◽  
Toeplitz Operators ◽  
Operator Equations ◽  
System Of Operator Equations

The paper introduces the [Formula: see text]th-order slant Toeplitz operator on the Lebesgue space of [Formula: see text]-torus, where [Formula: see text] such that [Formula: see text] for all [Formula: see text]. It investigates certain properties of [Formula: see text]th-order slant Toeplitz operators on the Lebesgue space [Formula: see text]. The paper deals with a system of operator equations, characterizing the [Formula: see text]th-order slant Toeplitz operators. At the end, we discuss certain spectral properties of the considered operator.

scholarly journals On minimax inequalities on spaces having certain contractible subsets

1993 ◽  
Vol 47 (1) ◽  
pp. 25-40 ◽  
Author(s):  
Sehie Park
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Minimax Theorem ◽  
Saddle Point Theorem ◽  
Minimax Inequalities ◽  
Von Neumann ◽  
Matching Theorem ◽  
Quasi Variational Inequality ◽  
Ky Fan

The concept of a convex space is extended to an H-space; that is, a space having certain family of contractible subsets. For such spaces the KKM type theorems, the Fan-Browder fixed point theorem, the Ky Fan type matching theorem, and minimax inequalities are given. Moreover, applications to a von Neumann-Sion type minimax theorem, a saddle point theorem, a quasi-variational inequality, and a Kakutani type fixed point theorem are obtained.

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