scholarly journals A note on the stability and the approximation of solutions for a Dirichlet problem with p(x)-Laplacian

2007 ◽  
Vol 49 (1) ◽  
pp. 75-83
Author(s):  
Marek Galewski
Keyword(s):  
Dirichlet Problem ◽  
Growth Conditions ◽  
Stability Of Solutions ◽  
Dirichlet Problems ◽  
Secondary 35B35 ◽  
Local Growth ◽  
Keywords And Phrases ◽  
2000 Mathematics Subject Classification ◽  
The Stability ◽  
Stability Results

We show the stability results and Galerkin-type approximations of solutions for a family of Dirichlet problems with nonlinearity satisfying some local growth conditions. 2000 Mathematics subject classification: primary 35A15; secondary 35B35, 65N30. Keywords and phrases: p(x)-Laplacian, duality, variational method, stability of solutions, Galerkin-type approximations.

scholarly journals On the stability of solutions for thep(x)-Laplacian equation and some applications to optimisation problems with state constraints

2006 ◽  
Vol 48 (2) ◽  
pp. 245-257 ◽  
Author(s):  
Elżbieta Galewska ◽  
Marek Galewski
Keyword(s):  
Control Problem ◽  
Continuous Dependence ◽  
State Constraints ◽  
Optimal Solution ◽  
Growth Conditions ◽  
Functional Parameter ◽  
Stability Of Solutions ◽  
Dirichlet Problems ◽  
Laplacian Equation ◽  
The Stability

AbstractWe consider the stability of solutions for a family of Dirichlet problems with (p, q)-growth conditions. We apply the results obtained to show continuous dependence on a functional parameter and the existence of an optimal solution in a control problem with state constraints governed by thep(x)-Laplacian equation.

scholarly journals On the Nonlinear Dirichlet Problem with P(x)-Laplacian

2007 ◽  
Vol 75 (3) ◽  
pp. 381-395 ◽  
Author(s):  
Marek Galewski ◽  
Marek Płócienniczak
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Bounded Domain ◽  
Growth Conditions ◽  
Stability Of Solutions ◽  
Dirichlet Problems ◽  
Nonlinear Dirichlet Problem ◽  
Dual Variational Method ◽  
Existence And Stability ◽  
Laplacian Operators

Using a dual variational method which we develop, we show the existence and stability of solutions for a family of Dirichlet problems k = 0, 1,… in a bounded domain in ℝN and with the nonlinearity satisfying some general growth conditions. The assumptions put on v are satisfied by p(x)-Laplacian operators.

On the Dirichlet problem with nonconvex nonlinearity

2010 ◽  
Vol 47 (2) ◽  
pp. 190-199
Author(s):  
Marek Galewski
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Growth Conditions ◽  
Existence Results ◽  
Dirichlet Problems ◽  
Local Growth ◽  
Dual Variational Method ◽  
Nonlinear Eigenvalue

We provide existence results for 2 m order Dirichlet problems with nonconvex nonlinearity which satisfies general local growth conditions. In doing so we construct a dual variational method. Problem considered relates to the problem of nonlinear eigenvalue.

Stability of Solutions of Ordinary Differential Equations with Respect to a Closed Set

1968 ◽  
Vol 20 ◽  
pp. 720-726
Author(s):  
T. G. Hallam ◽  
V. Komkov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Compact Set ◽  
Stability Of Solutions ◽  
Closed Set ◽  
The Stability ◽  
Stability Results

The stability of the solutions of an ordinary differential equation will be discussed here. The purpose of this note is to compare the stability results which are valid with respect to a compact set and the stability results valid with respect to an unbounded set. The stability of sets is a generalization of stability in the sense of Liapunov and has been discussed by LaSalle (5; 6), LaSalle and Lefschetz (7, p. 58), and Yoshizawa (8; 9; 10).

scholarly journals Existence and nonexistence of radial solutions of the Dirichlet problem for a class of general k-Hessian equations

2018 ◽  
Vol 23 (4) ◽  
pp. 475-492 ◽  
Author(s):  
Jianxin He ◽  
Xinguang Zhang ◽  
Lishan Liu ◽  
Yonghong Wu
Keyword(s):  
Fixed Point ◽  
Dirichlet Problem ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Growth Conditions ◽  
Radial Solutions ◽  
Local Growth ◽  
Hessian Equations ◽  
Existence And Nonexistence ◽  
The Fixed Point Theorem

In this paper, we establish the existence and nonexistence of radial solutions of the Dirichlet problem for a class of general k-Hessian equations in a ball. Under some suitable local growth conditions for nonlinearity, several new results are obtained by using the fixed-point theorem.

Dual Variational Method for a Fourth Order Dirichlet Problem

2008 ◽  
Vol 15 (4) ◽  
pp. 653-664
Author(s):  
Marek Galewski
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Fourth Order ◽  
Functional Parameter ◽  
Dirichlet Problems ◽  
Dual Variational Method ◽  
Existence And Stability ◽  
Dual Action ◽  
Primal And Dual ◽  
Stability Results

Abstract We obtain the existence and stability results for a fourth order Dirichlet problems with nonlinearity being convex in a certain interval. A dual variational method is introduced, which relies on investigating the primal and dual action functionals on certain subsets of their domains. The dependence on a functional parameter for a fourth order Dirichlet problem is considered as a consequence of stability.

scholarly journals On Existence and Stability of Solutions to Elliptic Systems with Generalised Growth

2007 ◽  
Vol 76 (3) ◽  
pp. 453-470
Author(s):  
Marek Galewski ◽  
Marek Plócienniczak
Keyword(s):  
Variational Approach ◽  
Elliptic Systems ◽  
Growth Conditions ◽  
System Of Equations ◽  
Stability Of Solutions ◽  
Local Growth ◽  
Laplace Operators ◽  
Functional Parameters ◽  
Existence And Stability ◽  
Dual Action

We are concerned with existence and stability of solutions for system of equations with generalised p(x) and m(x)—Laplace operators and where the nonlinearity satisfies some local growth conditions. We provide a variational approach that is based on investigation of the primal and the dual action functionals. As a consequence we consider the dependence of the the system on functional parameters.

On a fourth order Dirichlet Problem

2010 ◽  
Vol 17 (3) ◽  
pp. 495-509
Author(s):  
Marek Galewski ◽  
Joanna Smejda
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Fourth Order ◽  
Existence Of Solutions ◽  
Concave Function ◽  
Growth Conditions ◽  
Dirichlet Problems ◽  
Convex Part ◽  
Dual Variational Method ◽  
Derivatives Of

Abstract We consider by a dual variational method the existence of solutions to certain fourth order Dirichlet problems with nonlinearities corresponding to the derivatives of a sum of a convex and a concave function. The growth conditions are imposed only on the convex part.

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