scholarly journals ON THE POTENTIALITY OF A CLASS OF OPERATORS RELATIVE TO LOCAL BILINEAR FORMS

2021 ◽  
Vol 7 (1) ◽  
pp. 26
Author(s):  
Svetlana A. Budochkina ◽  
Ekaterina S. Dekhanova
Keyword(s):  
Bilinear Form ◽  
Sufficient Conditions ◽  
Potential Operator ◽  
Bilinear Forms ◽  
Analytical Dynamics ◽  
Order Ordinary Differential Equation ◽  
Nonlinear Functional Analysis ◽  
Nonlinear Functional ◽  
Necessary And Sufficient ◽  
Theoretical Results

The inverse problem of the calculus of variations (IPCV) is solved for a second-order ordinary differential equation with the use of a local bilinear form. We apply methods of analytical dynamics, nonlinear functional analysis, and modern methods for solving the IPCV. In the paper, we obtain necessary and sufficient conditions for a given operator to be potential relative to a local bilinear form, construct the corresponding functional, i.e., found a solution to the IPCV, and define the structure of the considered equation with the potential operator. As a consequence, similar results are obtained when using a nonlocal bilinear form. Theoretical results are illustrated with some examples.

Dynamical Behaviors of a Fractional-Order Predator–Prey Model with Holling Type IV Functional Response and Its Discretization

2019 ◽  
Vol 20 (2) ◽  
pp. 125-136
Author(s):  
A. M. Yousef ◽  
S. Z. Rida ◽  
Y. Gh. Gouda ◽  
A. S. Zaki
Keyword(s):  
Functional Response ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Dynamical Behaviors ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Theoretical Results

AbstractIn this paper, we investigate the dynamical behaviors of a fractional-order predator–prey with Holling type IV functional response and its discretized counterpart. First, we seek the local stability of equilibria for the fractional-order model. Also, the necessary and sufficient conditions of the stability of the discretized model are achieved. Bifurcation types (include transcritical, flip and Neimark–Sacker) and chaos are discussed in the discretized system. Finally, numerical simulations are executed to assure the validity of the obtained theoretical results.

Multiconsensus of first-order multiagent systems with directed topologies

2020 ◽  
Vol 34 (23) ◽  
pp. 2050240
Author(s):  
Xiao-Wen Zhao ◽  
Guangsong Han ◽  
Qiang Lai ◽  
Dandan Yue
Keyword(s):  
Multiagent Systems ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Consensus Problem ◽  
Multiple Agents ◽  
First Order ◽  
The Matrix ◽  
Directed Topologies ◽  
Necessary And Sufficient ◽  
Theoretical Results

The multiconsensus problem of first-order multiagent systems with directed topologies is studied. A novel consensus problem is introduced in multiagent systems — multiconsensus. The states of multiple agents in each subnetwork asymptotically converge to an individual consistent value in the presence of information exchanges among subnetworks. Linear multiconsensus protocols are proposed to solve the multiconsensus problem, and the matrix corresponding to the protocol is designed. Necessary and sufficient conditions are derived based on matrix theory, under which the stationary multiconsensus and dynamic multiconsensus can be reached. Simulations are provided to demonstrate the effectiveness of the theoretical results.

Multivariate Simultaneous Generalized ARCH

1995 ◽  
Vol 11 (1) ◽  
pp. 122-150 ◽  
Author(s):  
Robert F. Engle ◽  
Kenneth F. Kroner
Keyword(s):  
Sufficient Conditions ◽  
Likelihood Estimation ◽  
Positive Definiteness ◽  
Covariance Matrices ◽  
Simultaneous Equations ◽  
Equivalence Relations ◽  
Conditional Covariance ◽  
Arch Models ◽  
Necessary And Sufficient ◽  
Theoretical Results

This paper presents theoretical results on the formulation and estimation of multivariate generalized ARCH models within simultaneous equations systems. A new parameterization of the multivariate ARCH process is proposed, and equivalence relations are discussed for the various ARCH parameterizations. Constraints sufficient to guarantee the positive definiteness of the conditional covariance matrices are developed, and necessary and sufficient conditions for covariance stationarity are presented. Identification and maximum likelihood estimation of the parameters in the simultaneous equations context are also covered.

Embeddings of weighted Sobolev spaces into spaces of continuous functions

1992 ◽  
Vol 439 (1906) ◽  
pp. 279-296 ◽  
Keyword(s):  
Weighted Sobolev Spaces ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Hölder Continuous ◽  
Compact Embeddings ◽  
Spaces Of Continuous Functions ◽  
Hölder Continuous Functions ◽  
Continuous And Compact Embeddings ◽  
Necessary And Sufficient ◽  
Theoretical Results

We give sufficient conditions and necessary conditions (which in some cases are both necessary and sufficient) for continuous and compact embeddings of the weighted Sobolev space W 1,p ( Ω ;v 0 v 1 )into spaces of weighted continuous and Holder continuous functions. The theoretical results are illustrated by several examples.

scholarly journals Computer-controlled variable-structure systems

1992 ◽  
Vol 34 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Xinghuo Yu ◽  
Renfrey B. Potts
Keyword(s):  
Computer Control ◽  
Sufficient Conditions ◽  
Variable Structure ◽  
Dimensional System ◽  
Sliding Modes ◽  
Variable Structure Systems ◽  
Computer Controlled ◽  
Two Dimensional System ◽  
Necessary And Sufficient ◽  
Theoretical Results

AbstractA theory is developed for the computer control of variable-structure systems, using periodic zero-order-hold sampling. A simple two-dimensional system is first analysed, and necessary and sufficient conditions for the occurrence of pseudo-sliding modes are discussed. The method is then applied to a discrete model of a cylindrical robot. The theoretical results are illustrated by computer simulations.

scholarly journals Convergence on Gauss-Seidel iterative methods for linear systems with general H-matrices

2015 ◽  
Vol 30 ◽  
pp. 843-870 ◽  
Author(s):  
Cheng-yi Zhang ◽  
Dan Ye ◽  
Cong-Lei Zhong ◽  
SHUANGHUA SHUANGHUA
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Sufficient Conditions ◽  
Numerical Linear Algebra ◽  
Convergence Results ◽  
Diagonally Dominant Matrices ◽  
Sufficient Conditions For Convergence ◽  
Diagonally Dominant ◽  
Necessary And Sufficient ◽  
Theoretical Results

It is well known that as a famous type of iterative methods in numerical linear algebra, Gauss-Seidel iterative methods are convergent for linear systems with strictly or irreducibly diagonally dominant matrices, invertible H−matrices (generalized strictly diagonally dominant matrices) and Hermitian positive definite matrices. But, the same is not necessarily true for linear systems with non-strictly diagonally dominant matrices and general H−matrices. This paper firstly proposes some necessary and sufficient conditions for convergence on Gauss-Seidel iterative methods to establish several new theoretical results on linear systems with nonstrictly diagonally dominant matrices and general H−matrices. Then, the convergence results on preconditioned Gauss-Seidel (PGS) iterative methods for general H−matrices are presented. Finally, some numerical examples are given to demonstrate the results obtained in this paper.

scholarly journals Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 912
Author(s):  
Nikolai Sidorov ◽  
Denis Sidorov ◽  
Aliona Dreglea
Keyword(s):  
Integral Equations ◽  
Nonlinear Operator ◽  
Sufficient Conditions ◽  
Branch Points ◽  
Operator Equations ◽  
Nonlinear Operator Equations ◽  
Nonlinear Functional ◽  
Branching Points ◽  
Nonstandard Models ◽  
Necessary And Sufficient

The necessary and sufficient conditions of existence of the nonlinear operator equations’ branches of solutions in the neighbourhood of branching points are derived. The approach is based on the reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, integral equations, spectral theory based on index of Kronecker-Poincaré, Morse-Conley index, power geometry and other methods are employed. Proposed methodology enables justification of the theorems on existence of bifurcation points and bifurcation sets in the nonstandard models. Formulated theorems are constructive. For a certain smoothness of the nonlinear operator, the asymptotic behaviour of the solutions is analysed in the neighbourhood of the branch points and uniformly converging iterative schemes with a choice of the uniformization parameter enables the comprehensive analysis of the problems details. General theorems and effectiveness of the proposed methods are illustrated on the nonlinear integral equations.

scholarly journals On the positive solutions of a higher order functional differential equation with a discontinuity

1982 ◽  
Vol 5 (2) ◽  
pp. 263-273 ◽  
Author(s):  
John R. Graef ◽  
Paul W. Spikes ◽  
Myron K. Grammatikopoulos
Keyword(s):  
Differential Equation ◽  
Functional Differential Equation ◽  
Bounded Solution ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Bounded Solutions ◽  
Nonlinear Functional ◽  
Functional Differential ◽  
Necessary And Sufficient ◽  
Special Case

Then-th order nonlinear functional differential equation[r(t)x(n−υ)(t)](υ)=f(t,x(g(t)))is considered; necessary and sufficient conditions are given for this equation to have: (i) a positive bounded solutionx(t)→B>0ast→∞; and (ii) all positive bounded solutions converging to0ast→∞. Other results on the asymptotic behavior of solutions are also given. The conditions imposed are such that the equation with a discontinuity[r(t)x(n−υ)(t)](υ)=q(t)x−λ,   λ>0is included as a special case.

scholarly journals Strong Tolerance and Strong Universality of Interval Eigenvectors in a Max-Łukasiewicz Algebra

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1504
Author(s):  
Martin Gavalec ◽  
Zuzana Němcová ◽  
Ján Plavka
Keyword(s):  
Set Theory ◽  
Binary Operation ◽  
Sufficient Conditions ◽  
Discrete Events ◽  
Linear Combinations ◽  
Main Method ◽  
Max Algebra ◽  
Discrete Events Systems ◽  
Necessary And Sufficient ◽  
Theoretical Results

The Łukasiewicz conjunction (sometimes also considered to be a logic of absolute comparison), which is used in multivalued logic and in fuzzy set theory, is one of the most important t-norms. In combination with the binary operation ‘maximum’, the Łukasiewicz t-norm forms the basis for the so-called max-Łuk algebra, with applications to the investigation of systems working in discrete steps (discrete events systems; DES, in short). Similar algebras describing the work of DES’s are based on other pairs of operations, such as max-min algebra, max-plus algebra, or max-T algebra (with a given t-norm, T). The investigation of the steady states in a DES leads to the study of the eigenvectors of the transition matrix in the corresponding max-algebra. In real systems, the input values are usually taken to be in some interval. Various types of interval eigenvectors of interval matrices in max-min and max-plus algebras have been described. This paper is oriented to the investigation of strong, strongly tolerable, and strongly universal interval eigenvectors in a max-Łuk algebra. The main method used in this paper is based on max-Ł linear combinations of matrices and vectors. Necessary and sufficient conditions for the recognition of strong, strongly tolerable, and strongly universal eigenvectors have been found. The theoretical results are illustrated by numerical examples.

scholarly journals Two-Weight Norm Estimates for Multilinear Fractional Integrals in Classical Lebesgue Spaces

2015 ◽  
Vol 18 (5) ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Mieczysław Mastyło ◽  
Alexander Meskhi
Keyword(s):  
Riesz Potential ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
Fractional Integrals ◽  
Potential Operator ◽  
Norm Estimates ◽  
Riesz Potential Operator ◽  
Weight Problems ◽  
Multilinear Fractional Integrals ◽  
Necessary And Sufficient

AbstractWe derive criteria governing two-weight estimates for multilinear fractional integrals and appropriate maximal functions. The two and one weight problems for multi(sub)linear strong fractional maximal operators are also studied; in particular, we derive necessary and sufficient conditions guaranteeing the trace type inequality for this operator. We also establish the Fefferman-Stein type inequality, and obtain one-weight criteria when a weight function is of product type. As a consequence, appropriate results for multilinear Riesz potential operator with product kernels follow.

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