scholarly journals A Comparison of a Priori Estimates of the Solutions of a Linear Fractional System with Distributed Delays and Application to the Stability Analysis

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 75
Author(s):  
Hristo Kiskinov ◽  
Magdalena Veselinova ◽  
Ekaterina Madamlieva ◽  
Andrey Zahariev
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Integral Representations ◽  
Distributed Delays ◽  
Finite Time Stability ◽  
Gronwall’S Inequality ◽  
Priori Estimates ◽  
The Stability ◽  
Traditional Approaches

In this article, we consider a retarded linear fractional differential system with distributed delays and Caputo type derivatives of incommensurate orders. For this system, several a priori estimates for the solutions, applying the two traditional approaches—by the use of the Gronwall’s inequality and by the use of integral representations of the solutions are obtained. As application of the obtained estimates, different sufficient conditions which guaranty finite-time stability of the solutions are established. A comparison of the obtained different conditions in respect to the used estimates and norms is made.

scholarly journals Existence and global stability of periodic solution for delayed discrete high-order Hopfield-type neural networks

2005 ◽  
Vol 2005 (3) ◽  
pp. 281-297 ◽  
Author(s):  
Hong Xiang ◽  
Ke-Ming Yan ◽  
Bai-Yan Wang
Keyword(s):  
Neural Networks ◽  
Periodic Solution ◽  
Global Stability ◽  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
High Order ◽  
Priori Estimates

By using coincidence degree theory as well as a priori estimates and Lyapunov functional, we study the existence and global stability of periodic solution for discrete delayed high-order Hopfield-type neural networks. We obtain some easily verifiable sufficient conditions to ensure that there exists a unique periodic solution, and all theirs solutions converge to such a periodic solution.

On Approximations to Solutions of Nonlinear Integral Equations of the Urysohn Type

1973 ◽  
Vol 16 (1) ◽  
pp. 137-141
Author(s):  
K. A. Zischka
Keyword(s):  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Successive Approximations ◽  
Uniqueness Of Solutions ◽  
Nonlinear Integral Equations ◽  
Priori Estimates ◽  
The Given ◽  
Uniqueness Of A Solution

This note will derive a priori estimates of the errors due to replacing the given integral operator A by a similar operator A* of the same type when successive approximations are applied to the integral equation φ=Aφ.The existence and uniqueness of solutions to this equation follow easily by applying a well known fixed point theorem in a Banach space to the above mapping [1, 2]. Moreover, sufficient conditions for the existence and uniqueness of a solution to Urysohn's equation are stated explicitly in a note by the author [3].

On the Strong Stability of Operator-Difference Schemes in Time-Integral Norms

2001 ◽  
Vol 1 (1) ◽  
pp. 72-85 ◽  
Author(s):  
Boško S. Jovanović ◽  
Piotr P. Matus
Keyword(s):  
Hilbert Spaces ◽  
A Priori Estimates ◽  
A Priori ◽  
Strong Stability ◽  
Difference Schemes ◽  
Initial Condition ◽  
Time Integral ◽  
Right Hand ◽  
Priori Estimates ◽  
The Stability

Abstract In this paper we investigate the stability of two-level operator-difference schemes in Hilbert spaces under perturbations of operators, the initial condition and right hand side of the equation. A priori estimates of the error are obtained in time- integral norms under some natural assumptions on the perturbations of the operators.

STABILITY AND CONVERGENCE OF TWO-LEVEL DIFFERENCE SCHEMES IN INTEGRAL WITH RESPECT TO TIME NORMS

1998 ◽  
Vol 08 (06) ◽  
pp. 1055-1070 ◽  
Author(s):  
ALEXANDER A. SAMARSKII ◽  
PETR P. MATUS ◽  
PETR N. VABISHCHEVICH
Keyword(s):  
Hilbert Spaces ◽  
A Priori Estimates ◽  
A Priori ◽  
Mathematical Physics ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Priori Estimates ◽  
Linear Problems ◽  
The Difference ◽  
The Stability

Nowadays the general theory of operator-difference schemes with operators acting in Hilbert spaces has been created for investigating the stability of the difference schemes that approximate linear problems of mathematical physics. In most cases a priori estimates which are uniform with respect to the t norms are usually considered. In the investigation of accuracy for evolutionary problems, special attention should be given to estimation of the difference solution in grid analogs of integral with respect to the time norms. In this paper a priori estimates in such norms have been obtained for two-level operator-difference schemes. Use of that estimates is illustrated by convergence investigation for schemes with weights for parabolic equation with the solution belonging to [Formula: see text].

scholarly journals On the Sirs Epidemic Model with Free Boundaries

2021 ◽  
Vol 66 (1) ◽  
pp. 21-47
Author(s):  
J. O. Takhirov ◽  
◽  
Z. K. Djumanazarova ◽  
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Global Solvability ◽  
Reproductive Number ◽  
Free Boundaries ◽  
Sirs Epidemic Model ◽  
Priori Estimates ◽  
Initial Area

We investigate an epidemic non-linear reaction-diffusion system with two free boundaries. A free boundary is introduced to describe the expanding front of the infectious environment. A priori estimates of the required functions are established, which are necessary for the correctness and global solvability of the problem. We get sufficient conditions for the spread or disappearance of the disease. It has been proven that with a base reproductive number the disease disappears in the long term if the initial values and the initial area are sufficiently small.

scholarly journals On a Priori Estimates of Solutions of Systems of Higher Order Nonlinear Functional-Differential Inequalities

2007 ◽  
Vol 14 (3) ◽  
pp. 445-456
Author(s):  
Ivan Kiguradze ◽  
Nino Partsvania
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Higher Order ◽  
Infinite Interval ◽  
Differential Inequalities ◽  
Nonlinear Functional ◽  
Estimates Of Solutions ◽  
Priori Estimates ◽  
Functional Differential ◽  
The Stability

Abstract Systems of higher order nonlinear functional-differential inequalities arising in the theory of boundary value problems as well as in the stability theory are considered on a finite and an infinite interval. On a finite interval, a priori estimates are obtained for solutions of these systems satisfying boundary conditions of periodic type, and on an infinite interval, a priori estimates are established for the derivatives of bounded solutions.

Stability of Difference Schemes for Nonlinear Time-dependent Problems

2003 ◽  
Vol 3 (2) ◽  
pp. 313-329 ◽  
Author(s):  
Piter Matus
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Quasilinear Parabolic Equations ◽  
Basic Point ◽  
Nonlinear Part ◽  
Priori Estimates ◽  
The Difference ◽  
The Stability

AbstractIn the present paper, a priori estimates of the stability in the sense of the initial data of the difference schemes approximating quasilinear parabolic equations and nonlinear transfer equation have been obtained. The basic point is connected with the necessity of estimating all derivatives entering into the nonlinear part of the difference equations. These estimates have been proved without any assumptions about the properties of the differential equations and depend only on the behavior of the initial and boundary conditions. As distinct from linear problems, the obtained estimates of stability in the general case exist only for the finite instant of time t 6 t0 connected with the fact that the solution of the Riccati equation becomes infinite. is already associated with the behavior of the second derivative of the initial function and coincides with the time of the exact solution destruction (heat localization in the peaking regime). A close relation between the stability and convergence of the difference scheme solution is given. Thus, not only a priori estimates for stability have been established, but it is also shown that the obtained conditions permit exact determination of the time of destruction of the solution of the initial boundary value problem for the original nonlinear differential equation in partial derivatives. In the present paper, concrete examples confirming the theoretical conclusions are given.

DISCONTINUOUS/CONTINUOUS LEAST-SQUARES FINITE ELEMENT METHODS FOR ELLIPTIC PROBLEMS

2005 ◽  
Vol 15 (06) ◽  
pp. 825-842 ◽  
Author(s):  
RICKARD E. BENSOW ◽  
MATS G. LARSON
Keyword(s):  
Finite Element ◽  
Least Squares ◽  
Finite Element Methods ◽  
A Priori Estimates ◽  
A Priori ◽  
Solution Space ◽  
First Order ◽  
Poisson Problem ◽  
Priori Estimates ◽  
The Stability

Least-squares finite element methods (LSFEM) are useful for first-order systems, where they avoid the stability consideration of mixed methods and problems with constraints, like the div-curl problem. However, LSFEM typically suffer from requirements on the solution to be very regular. This rules out, e.g., applications posed on nonconvex domains. In this paper we study a least-squares formulation where the discrete space is enriched by discontinuous elements in the vicinity of singularities. The weighting on the interelement terms are chosen to give correct regularity of the solution space and thus making computation of less regular problems possible. We apply this technique to the first-order Poisson problem, show coercivity and a priori estimates, and present numerical results in 3D.

scholarly journals Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)-D Protter-Morawetz Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
Nedyu Popivanov ◽  
Todor Popov ◽  
Allen Tesdall
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Generalized Solution ◽  
Generalized Solutions ◽  
Singular Solutions ◽  
Priori Estimates ◽  
Characteristic Cone ◽  
Equation Boundary ◽  
Necessary And Sufficient

For the four-dimensional nonhomogeneous wave equation boundary value problems that are multidimensional analogues of Darboux problems in the plane are studied. It is known that for smooth right-hand side functions the unique generalized solution may have a strong power-type singularity at only one point. This singularity is isolated at the vertexOof the boundary light characteristic cone and does not propagate along the bicharacteristics. The present paper describes asymptotic expansions of the generalized solutions in negative powers of the distance toO. Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for the singular solutions are obtained.

Global and Asymptotic Stability of Operator-Difference Schemes

2004 ◽  
Vol 4 (2) ◽  
pp. 192-205 ◽  
Author(s):  
Boško S Jovanović
Keyword(s):  
Hilbert Space ◽  
Asymptotic Stability ◽  
A Priori Estimates ◽  
A Priori ◽  
Difference Schemes ◽  
Priori Estimates ◽  
The Stability

Abstract The stability of linear two- and three-level operator difference schemes in a Hilbert space has been investigated. A few a priori estimates of the global and asymptotic stability in various energy norms have been obtained.

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