scholarly journals On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Dinh-Ke Tran ◽  
Nhu-Thang Nguyen
Keyword(s):  
Evolution Equations ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
Global Solvability ◽  
Kernel Functions ◽  
Stability Of Solutions ◽  
Nonlinear Perturbations ◽  
Completely Positive ◽  
Positive Kernel ◽  
Nonlocal Evolution Equations

<p style='text-indent:20px;'>We study a class of nonlocal partial differential equations with nonlinear perturbations, which is a general model for some equations arose from fluid dynamics. Our aim is to analyze some sufficient conditions ensuring the global solvability, regularity and stability of solutions. Our analysis is based on the theory of completely positive kernel functions, local estimates and a new Gronwall type inequality.</p>

scholarly journals Nonlinear semigroups and the existence and stability of solutions of semilinear nonautonomous evolution equations

1996 ◽  
Vol 1 (4) ◽  
pp. 351-380 ◽  
Author(s):  
Bernd Aulbach ◽  
Nguyen Van Minh
Keyword(s):  
Evolution Equations ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Nonlinear Operators ◽  
Nonlinear Semigroups ◽  
Evolution Semigroup ◽  
Existence And Stability ◽  
Functional Differential

This paper is concerned with the existence and stability of solutions of a class of semilinear nonautonomous evolution equations. A procedure is discussed which associates to each nonautonomous equation the so-called evolution semigroup of (possibly nonlinear) operators. Sufficient conditions for the existence and stability of solutions and the existence of periodic oscillations are given in terms of the accretiveness of the corresponding infinitesimal generator. Furthermore, through the existence of integral manifolds for abstract evolutionary processes we obtain a reduction principle for stability questions of mild solutions. The results are applied to a class of partial functional differential equations.

INTEGRAL SOLUTIONS TO A CLASS OF NONLOCAL EVOLUTION EQUATIONS

2010 ◽  
Vol 12 (06) ◽  
pp. 1031-1054 ◽  
Author(s):  
JESÚS GARCÍA-FALSET ◽  
SIMEON REICH
Keyword(s):  
Banach Space ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Sufficient Conditions ◽  
Accretive Operator ◽  
Integral Solution ◽  
Nonlinear Evolution Equations ◽  
Nonlocal Evolution Equations ◽  
Integral Solutions

We study the existence of integral solutions to a class of nonlinear evolution equations of the form [Formula: see text] where A : D(A) ⊆ X → 2X is an m-accretive operator on a Banach space X, and f : [0, T] × X → X and [Formula: see text] are given functions. We obtain sufficient conditions for this problem to have a unique integral solution.

scholarly journals Blow-Up Phenomena for Certain Nonlocal Evolution Equations and Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Evolution Equation ◽  
Positive Solutions ◽  
Evolution Equations ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Duality Argument ◽  
Nonlocal Evolution Equation ◽  
Global Nonexistence ◽  
Nonlocal Evolution Equations ◽  
Evolution Systems

We provide sufficient conditions for the nonexistence of global positive solutions to the nonlocal evolution equationutt(x,t)=(J ∗ u-u)(x,t)+up(x,t), (x,t)∈RN×(0,∞),(u(x,0),ut(x,0))=(u0(x),u1(x)),x∈RN,whereJ:RN→R+,p>1, and(u0,u1)∈Lloc1(RN;R+)×Lloc1(RN;R+). Next, we deal with global nonexistence for certain nonlocal evolution systems. Our method of proof is based on a duality argument.

scholarly journals Synchronization of Coupled Networks with Uncertainties

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Yi Zuo ◽  
Xinsong Yang
Keyword(s):  
Lyapunov Stability ◽  
Stability Theory ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Adaptive Controller ◽  
Nonlinear Perturbations ◽  
Coupled Networks ◽  
Barbalat Lemma ◽  
Asymptotic Synchronization ◽  
Theoretical Results

Asymptotic synchronization for a class of coupled networks with nondelayed and delayed couplings is investigated. A distinct feature of the network is that all the dynamical nodes are affected by uncertain nonlinear nonidentical perturbations. In order to synchronize the network onto a given isolate trajectory, a novel adaptive controller is designed to overcome the effects of the nonidentical uncertain nonlinear perturbations. The designed controller has better robustness than classical adaptive controller, since it can realize the synchronization goal whether the nodes have these perturbations or not. Based on the Lyapunov stability theory and the Barbalat lemma, sufficient conditions guaranteeing the asymptotic synchronization of the coupled network are derived. Two examples with numerical simulations are given to illustrate the effectiveness of the theoretical results. Simulations also demonstrate that our adaptive controller has better robustness than existing ones.

On Stability of Solutions of Certain Differential Equations of the Third Order

1967 ◽  
Vol 10 (5) ◽  
pp. 681-688 ◽  
Author(s):  
B.S. Lalli
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Third Order ◽  
The Third ◽  
Image Position

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation1.1using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.

scholarly journals An Invariance Property of Predictors in Kernel-Induced Hypothesis Spaces

2006 ◽  
Vol 18 (4) ◽  
pp. 749-759 ◽  
Author(s):  
Nicola Ancona ◽  
Sebastiano Stramaglia
Keyword(s):  
Time Series ◽  
Radial Basis Function ◽  
Basis Function ◽  
Sufficient Conditions ◽  
Kernel Functions ◽  
Invariance Property ◽  
Risk Minimization ◽  
Independent Variables ◽  
Input Variables ◽  
New Variables

We consider kernel-based learning methods for regression and analyze what happens to the risk minimizer when new variables, statistically independent of input and target variables, are added to the set of input variables. This problem arises, for example, in the detection of causality relations between two time series. We find that the risk minimizer remains unchanged if we constrain the risk minimization to hypothesis spaces induced by suitable kernel functions. We show that not all kernel-induced hypothesis spaces enjoy this property. We present sufficient conditions ensuring that the risk minimizer does not change and show that they hold for inhomogeneous polynomial and gaussian radial basis function kernels. We also provide examples of kernel-induced hypothesis spaces whose risk minimizer changes if independent variables are added as input.

Attractivity for fractional evolution equations with almost sectorial operators

2018 ◽  
Vol 21 (3) ◽  
pp. 786-800 ◽  
Author(s):  
Yong Zhou
Keyword(s):  
Evolution Equations ◽  
Sufficient Conditions ◽  
Cauchy Problems ◽  
Fractional Evolution Equations ◽  
Integer Order ◽  
Sectorial Operators ◽  
Almost Sectorial Operators ◽  
Globally Attractive

Abstract In this paper, we initiate the question of the attractivity of solutions for fractional evolution equations with almost sectorial operators. We establish sufficient conditions for the existence of globally attractive solutions for the Cauchy problems in cases that semigroup is compact as well as noncompact. Our results essentially reveal certain characteristics of solutions for fractional evolution equations, which are not possessed by integer order evolution equations.

Oscillation of Semilinear Elliptic Inequalities by Riccati Transformations

1980 ◽  
Vol 32 (4) ◽  
pp. 908-923 ◽  
Author(s):  
E. S. Noussair ◽  
C. A. Swanson
Keyword(s):  
Exterior Domain ◽  
Nonlinear Term ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
Second Derivative ◽  
Semilinear Elliptic ◽  
Generalized Riccati Transformation ◽  
The Matrix ◽  
Elliptic Inequality ◽  
Elliptic Inequalities

A generalized Riccati transformation will be utilized to derive a Riccati-type inequality (3) associated with a semilinear elliptic inequality yL(y; x) ≦ 0 possessing a positive solution y in an exterior domain in Euclidean n-space. On the basis of (3), general sufficient conditions for the elliptic inequality to be oscillatory are developed in § 3. The matrix of coefficients of the second derivative terms in L(y;x) (i.e. (Aij) in (1)) is not restricted in any way beyond the usual ellipticity hypothesis (iv) below, and thereby one of the difficulties mentioned in [9] and inherent in the method there is resolved. Furthermore, the nonlinear term B﹛x, y) in (1) is not required to be one-signed.

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