Pseudomonotone Operators Research Articles

On quasi-static contact problem with generalized Coulomb friction, normal compliance and damage

In this paper, we study a quasi-static frictional contact problem for a viscoelastic body with damage effect inside the body as well as normal compliance condition and multi-valued friction law on the contact boundary. The considered friction law generalizes Coulomb friction condition into multi-valued setting. The variational–hemi-variational formulation of the problem is derived and arguments of fixed point theory and surjectivity results for pseudo-monotone operators are applied, in order to prove the existence and uniqueness of solution.

Existence Theorems for Elliptic and Evolutionary Variational and Quasi-Variational Inequalities

This paper gives new existence results for elliptic and evolutionary variational and quasi-variational inequalities. Specifically, we give an existence theorem for evolutionary variational inequalities involving different types of pseudo-monotone operators. Another existence result embarks on elliptic variational inequalities driven by maximal monotone operators. We propose a new recessivity assumption that extends all the classical coercivity conditions. We also obtain criteria for solvability of general quasi-variational inequalities treating in a unifying way elliptic and evolutionary problems. Two of the given existence results for evolutionary quasi-variational inequalities rely on Mosco-type continuity properties and Kluge's fixed point theorem for set-valued maps. We also focus on the case of compact constraints in the evolutionary quasi-variational inequalities. Here a relevant feature is that the underlying space is the domain of a linear, maximal monotone operator endowed with the graph norm. Applications are also given.

A sub-supersolution approach for a quasilinear Kirchhoff equation

In this paper, we establish an existence result for a quasilinear Kirchhoff equation, via a sub- and supersolution approach, by using the Minty-Browder’s Theorem for pseudomonotone operators theory.

Convergence analysis of the proximal point algorithm for pseudo-monotone equilibrium problems

In this paper, we study the weak and strong convergence of the proximal point algorithm for equilibrium problems of pseudo-monotone type in Hilbert spaces. We prove the weak convergence of the generated sequence to a common solution of two equilibrium problems and some strong convergence results with additional assumptions on pseudo-monotone bifunctions. Then we study a regularization of Halpern-type and prove the strong convergence of the generated sequence to an equilibrium point of two pseudo-monotone bifunctions without any additional assumption on bifunctions. Finally, some examples of pseudo-monotone bifunctions from pseudo-monotone operators and Nash-Cournot oligopolistic equilibrium models are also presented. Our results extend some similar results in the literature for monotone and pseudo-monotone equilibrium problems and also the related results for variational inequalities associated with monotone and pseudo-monotone operators.

On the finite dimensional approximations of some mixed variational inequalities

We construct the finite-dimensional approximations for mixed variational inequalities with pseudomonotone operators and convex non-differentiable functionals in Banach spaces. Such variational inequalities arise in the mathematical description of the processes of an established filtration and in the problems of determining the equilibrium of soft shells. The convergence of these approximations are investigated.

Existence of solution for a class of nonlocal elliptic problem via sub–supersolution method

We show the existence of solution for some classes of nonlocal problems. Our proof combines the presence of sub and supersolution with the pseudomonotone operators theory.

Weak structural stability of pseudo-monotone equations

The inclusion $\beta(u)\ni h$ in $V'$ is studied, assuming that $V$ is a reflexive Banach space,and that $\beta: V \to {\cal P}(V')$ is a generalized pseudo-monotone operator in the sense of Browder-Hess [MR 0365242].A notion of strict generalized pseudo-monotonicity is also introduced.The above inclusion is here reformulated as a minimization problem for a (nonconvex) functional$V \!\times V'\to \mathbf{R} \cup \{+\infty\}$. A nonlinear topology of weak-type is introduced, and related compactness resultsare proved via De Giorgi's notion of$\Gamma$-convergence.The compactness and the convergence of the family of operators $\beta$provide the (weak) structural stability of the inclusion $\beta(u)\ni h$with respect to variations of $\beta$ and $h$,under the only assumptions that the $\beta$s are equi-coercive andthe $h$s are equi-bounded. These results are then applied to the weak stability of the Cauchy problemfor doubly-nonlinear parabolic inclusions of the form$D_t\partial\varphi(u) + \alpha(u) \ni h$,$\partial\varphi$ being the subdifferential of a convex lower semicontinuous mapping $\varphi$,and $\alpha$ a generalized pseudo-monotone operator.The technique of compactness by strict convexity is also used in the limit procedure.

History-dependent variational–hemivariational inequalities in contact mechanics

We consider an abstract class of variational–hemivariational inequalities which arise in the study of a large number of mathematical models of contact. The novelty consists in the structure of the inequalities which involve two history-dependent operators and two nondifferentiable functionals, a convex and a nonconvex one. For these inequalities we provide an existence and uniqueness result of the solution. The proof is based on arguments of surjectivity for pseudomonotone operators and fixed point. Then, we consider a viscoelastic problem in which the contact is frictionless and is modeled with a new boundary condition which describes both the instantaneous and the memory effects of the foundation. We prove that this problem leads to a history-dependent variational–hemivariational inequality in which the unknown is the displacement field. We apply our abstract result in order to prove the unique weak solvability of this viscoelastic contact problem.

Numerical Method for Solving Variation Problems in Mathematical Physics

We consider one axisymmetric problem of the equilibrium position of a soft rotation shell. Generalized statement of this problem is formulated in the form of variational inequality with a pseudo-monotone operator in Banach space. To solve this variational inequality, we suggest the iterative method. This method was realized numerically. The numerical experiments made for the model problems confirmed the efficiency of the iterative method.

Existence results for equilibrium problems with applications to evolution equations

In this paper, we consider evolution equations with time-dependent pseudomonotone and quasimonotone operators by a new approach based on equilibrium problems theory. We establish new existence results for equilibrium problems associated to pseudomonotone and quasimonotone bifunctions in the topological sense. The results obtained are then applied to derive existence of solutions for evolution equations. The approach is new and leads to improve and unify most of the results obtained in this direction.

A System of Time-Dependent Hemivariational Inequalities with Volterra Integral Terms

In this paper, we consider a system of time-dependent hemivariational inequalities with Volterra integral terms by using a surjectivity theorem for pseudomonotone operators and the Banach fixed point theorem, rather than the Knaster-Kuratowski-Mazurkiewicz theorems used by many researchers in recent literature for systems of hemivariational inequalities. Under some suitable conditions, the existence and uniqueness result of solution to the problem considered is obtained by proving that a derived vector inclusion problem with Volterra integral term is solvable.

Hemivariational Inequality Approach to Evolutionary Constrained Problems on Star-Shaped Sets

In this paper, we consider a nonconvex evolutionary constrained problem for a star-shaped set. The problem is a generalization of the classical evolution variational inequality of parabolic type. We provide an existence result; the proof is based on the hemivariational inequality approach, a surjectivity theorem for multivalued pseudomonotone operators in reflexive Banach spaces, and a penalization method. The admissible set of constraints is closed and star-shaped with respect to a certain ball; this allows one to use a discontinuity property of the generalized Clarke subdifferential of the distance function. An application of our results to a heat conduction problem with nonconvex constraints is provided.

A priori estimates and existence for elliptic equations with gradient dependent terms

We consider, in a bounded domain $\Omega \subset \R^{N}$, a class of nonlinear elliptic equations in divergence form as $$\left\{\begin{array}{l} \alpha_0 u -\dive (a(x,u,Du))=H(x,u,Du)\mbox{ in }\Omega ,\\ u=0\mbox{ on }\partial\Omega\,,\end{array} \right. $$ where $\alpha_0\geq 0$, the second order part is a coercive, pseudomonotone operator of Leray-Lions type in the Sobolev space $W^{1,p}_0(\Omega)$, $p>1$, and the function $H$ grows at most like $|Du|^q+f(x)$, with $p-1<q<p$.Assuming $f(x)$ to belong to an (optimal) Lebesgue class $L^m$, with $m<\frac Np$, we prove a priori estimates and existence of solutions, discussing several ranges of the exponent $m,q$ and $p$ which include cases of singular data ($L^1$ data or measures). The obtention of a priori estimates is not straightforward because of the super linear character of the first order terms. To this purpose we use a new approach, generalizing the method introduced in our note [29]. We complete the results known in the previous literature where either $q\leq p-1$ or $m\geq \frac Np$.

The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion

We consider a differential inclusion which is somehow analogous to the classical Duffing's equation with Dirichlet boundary condition. We prove the existence of a solution using two-steps approach. Firstly we consider an auxiliary problem where we substitute $h := \frac{dx} {dt} \in L^2(0,1)$. Next, using the method of pseudomonotone and coercive operators, we prove the existence of a solution to the auxiliary problem. Finally we prove that under suitable assumptions, an iterative scheme converges to the solution of our inclusion, which appears to be unique.

Existence of solutions for a class of singular elliptic systems with convection term

We show the existence of positive solutions for a class of singular elliptic systems with convection term. The approach combines sub and supersolution method with the pseudomonotone operator theory and perturbation arguments involving singular terms.

An extension of the Fitzpatrick theory

In the seminal work [MR 1009594], Fitzpatrick proved that for any maximal monotone operator $\alpha: V\to {\mathcal P}(V')$ ($V$ being a real Banach space) there exists a lower semicontinuous, convex representative function $f_\alpha: V \times V'\to R\cup \{+\infty\}$ such that \begin{eqnarray} f_\alpha(v,v') \ge \langle v',v\rangle \quad\;\forall (v,v'), \qquad\quad f_\alpha(v,v') = \langle v',v\rangle \;\;\Leftrightarrow\;\;\; v'\in \alpha(v). \end{eqnarray} Here we assume that $\alpha_v$ is a maximal monotone operator for any $v\in V$, and extend the Fitzpatrick theory to provide a new variational formulation for either stationary or evolutionary (nonmonotone) inclusions of the form $\alpha_v(v) \ni v'$. For any $v'\in V'$, we prove existence of a solution via the classical minimax theorem of Ky Fan. Applications include stationary and evolutionary pseudo-monotone operators, and variational inequalities.

Variational inequalities for perturbations of maximal monotone operators in reflexive Banach spaces

Let $X$ be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space $X^*,$ and let $K$ be a nonempty, closed and convex subset of $X$ with $0$ in its interior. Let $T$ be maximal monotone and $S$ a possibly unbounded pseudomonotone, or finitely continuous generalized pseudomonotone, or regular generalized pseudomonotone operator with domain $K$. Let $\phi$ be a proper, convex and lower semicontinuous function. New results are given concerning the solvability of perturbed variational inequalities involving the operator $T+S$ and the function $\phi$. The associated range results for nonlinear operators are also given, as well asextensions and/or improvements of known results of Kenmochi, Le, Browder, Browder and Hess, De Figueiredo, Zhou, and others.

Boundary hemivariational inequality problems with doubly nonlinear operators

In this paper, we investigate a class of nonlinear boundary hemivariational inequality problems. Under suitable hypotheses, the existence of solutions is established via rewriting these problems into a class of evolution inclusions and using the discretization method and the theory of pseudomonotone operators. Moreover, the continuous dependence result of the solutions to the initial data is given.

A System of Generalized Variational-Hemivariational Inequalities with Set-Valued Mappings

By using surjectivity theorem of pseudomonotone and coercive operators rather than the KKM theorem and fixed point theorem used in recent literatures, we obtain some conditions under which a system of generalized variational-hemivariational inequalities concerning set-valued mappings, which includes as special cases many problems of hemivariational inequalities studied in recent literatures, is solvable. As an application, we prove an existence theorem of solutions for a system of generalized variational-hemivariational inequalities involving integrals of Clarke's generalized directional derivatives.

Evolution parabolic inequalities for pseudo-monotone operators

In this article, we prove the existence of weak solutions of variational parabolic inequalities with pseudo- monotonicity and nonzero initial data using finite differences in time.