scholarly journals Multiple solutions for an inclusion quasilinear problem with nonhomogeneous boundary condition through Orlicz–Sobolev spaces

2017 ◽  
Vol 19 (03) ◽  
pp. 1650031 ◽  
Author(s):  
Rodrigo C. M. Nemer ◽  
Jefferson A. Santos
Keyword(s):  
Sobolev Spaces ◽  
Point Theory ◽  
Neumann Condition ◽  
Generalized Gradient ◽  
Inclusion Problems ◽  
Nonhomogeneous Boundary Condition ◽  
Nonhomogeneous Boundary ◽  
Locally Lipschitz Functional ◽  
Locally Lipschitz ◽  
Quasilinear Problem

In this work, we study multiplicity of nontrivial solution for the following class of differential inclusion problems with nonhomogeneous Neumann condition through Orlicz–Sobolev spaces, [Formula: see text] where [Formula: see text] is a domain, [Formula: see text] and [Formula: see text] is the generalized gradient of [Formula: see text]. The main tools used are Variational Methods for Locally Lipschitz Functional and Critical Point Theory.

Solutions and multiple solutions for quasilinear hemivariational inequalities at resonance

2001 ◽  
Vol 131 (5) ◽  
pp. 1091-1111 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Variational Principle ◽  
Critical Point Theory ◽  
Multiple Solutions ◽  
Existence Theorems ◽  
Point Theory ◽  
Hemivariational Inequalities ◽  
Locally Lipschitz Functions ◽  
Strong Resonance ◽  
At Resonance ◽  
Locally Lipschitz

In this paper we consider quasilinear hemivariational inequalities at resonance. We obtain existence theorems using Landesman-Lazer-type conditions and multiplicity theorems for problems with strong resonance at infinity. Our method of proof is based on the non-smooth critical point theory for locally Lipschitz functions and on a generalized version of the Ekeland variational principle.

Dynamic Programming and a Distributed Parameter Maximum Principle

1968 ◽  
Vol 90 (2) ◽  
pp. 152-156 ◽  
Author(s):  
W. L. Brogan
Keyword(s):  
Dynamic Programming ◽  
Maximum Principle ◽  
Programming Approach ◽  
Distributed Parameter ◽  
Programming Technique ◽  
Nonhomogeneous Boundary Condition ◽  
The Maximum Principle ◽  
Dynamic Programming Approach ◽  
Nonhomogeneous Boundary ◽  
Definition Of

A proof of a distributed parameter maximum principle is given by using dynamic programming. An example problem involving a nonhomogeneous boundary condition is also treated by using the dynamic programming technique and by extending the definition of the differential operator. It is thus demonstrated that for linear systems the dynamic programming approach is just as powerful as the variational approach originally used to derive the maximum principle.

scholarly journals Nonlinear Periodic Systems with thep-Laplacian: Existence and Multiplicity Results

2007 ◽  
Vol 2007 ◽  
pp. 1-23
Author(s):  
Francesca Papalini
Keyword(s):  
Existence Theorems ◽  
Point Theory ◽  
Energy Functional ◽  
Periodic Systems ◽  
Multiplicity Result ◽  
Nonsmooth Critical Point Theory ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Nonsmooth Critical Point ◽  
Locally Lipschitz

We study second-order nonlinear periodic systems driven by the vectorp-Laplacian with a nonsmooth, locally Lipschitz potential function. Under minimal and natural hypotheses on the potential and using variational methods based on the nonsmooth critical point theory, we prove existence theorems and a multiplicity result. We conclude the paper with an existence theorem for the scalar problem, in which the energy functional is indefinite (unbounded from both above and below).

Equivalence of Some Multi-Valued Elliptic Variational Inequalities and Variational-Hemivariational Inequalities

2011 ◽  
Vol 11 (2) ◽  
Author(s):  
Siegfried Carl
Keyword(s):  
Variational Inequalities ◽  
Lipschitz Functions ◽  
Hemivariational Inequalities ◽  
Locally Lipschitz Functions ◽  
Generalized Gradient ◽  
Elliptic Variational Inequalities ◽  
Comparison Results ◽  
Locally Lipschitz ◽  
Additional Regularity ◽  
Definition Of

AbstractFirst, we prove existence and comparison results for multi-valued elliptic variational inequalities involving Clarke’s generalized gradient of some locally Lipschitz functions as multi-valued term. Only by applying the definition of Clarke’s gradient it is well known that any solution of such a multi-valued elliptic variational inequality is also a solution of a corresponding variational-hemivariational inequality. The reverse is known to be true if the locally Lipschitz functions are regular in the sense of Clarke. Without imposing this kind of regularity the equivalence of the two problems under consideration is not clear at all. The main goal of this paper is to show that the equivalence still holds true without any additional regularity, which will fill a gap in the literature. Existence and comparison results for both multi-valued variational inequalities and variational-hemivariational inequalities are the main tools in the proof of the equivalence of these problems.

scholarly journals A continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces

2009 ◽  
Vol 104 (1) ◽  
pp. 132 ◽  
Author(s):  
Mihai Mihailescu ◽  
Vicentiu Radulescu
Keyword(s):  
Continuous Function ◽  
Eigenvalue Problem ◽  
Continuous Spectrum ◽  
Sobolev Spaces ◽  
Differential Operators ◽  
Smooth Boundary ◽  
Nonlinear Eigenvalue Problem ◽  
Sufficient Conditions ◽  
Open Set ◽  
Quasilinear Problem

We study the nonlinear eigenvalue problem $-(\mathrm{div} (a(|\nabla u|)\nabla u)=\lambda|u|^{q(x)-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded open set in ${\mathsf R}^N$ with smooth boundary, $q$ is a continuous function, and $a$ is a nonhomogeneous potential. We establish sufficient conditions on $a$ and $q$ such that the above nonhomogeneous quasilinear problem has continuous families of eigenvalues. The proofs rely on elementary variational arguments. The abstract results of this paper are illustrated by the cases $a(t)=t^{p-2}\log (1+t^r)$ and $a(t)= t^{p-2} [\log (1+t)]^{-1}$.

Sign in / Sign up

Export Citation Format

Share Document