Generalized Vector Equilibrium Problems for Pseudomonotone Multivalued Bifunctions<Superscript><Emphasis Type="Bold">1</Emphasis></Superscript>

In this paper, we apply a new version of the Brézis, Nirenberg, and Stampacchia theorem; we use pseudomonotonicity and some coercivity conditions to establish some existence result for a solution of generalized vector equilibrium problems for multivalued bifunctions. The proper quasiconvexity of multivalued bifunctions is introduced and existence theorems for generalized vector equilibrium problems related to multivalued mappings with the KKM property are obtained. The new results extend and modify various existence theorems for similar problems.     

Collective fixed points and maximal elements with applications to abstract economies

In this paper, we first establish collective fixed points theorems for a family of multivalued maps with or without assuming that the product of these multivalued maps is Φ-condensing. As an application of our collective fixed points theorem, we derive the coincidence theorem for two families of multivalued maps defined on product spaces. Then we give some existence results for maximal elements for a family of L_S-majorized multivalued maps whose product is Φ-condensing. We also prove some existence results for maximal elements for a family of multivalued maps which are not L_S-majorized but their product is Φ-condensing. As applications of our results, some existence results for equilibria of abstract economies are also derived. The results of this paper are more general than those given in the literature.     

Nonlinear elliptic differential equations with multivalued nonlinearities

In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all ℝ. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of ℝ. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).     

Fixed point and maximal element theorems with applications to abstract economies and minimax inequalities

In this paper, we establish some fixed point theorems for a family of multivalued maps under mild conditions. By using our fixed point theorems, we derive some maximal element theorems for a particular family of multivalued maps, namely the Φ-condensing multivalued maps. As applications of our results, we prove some general equilibrium existence theorems in the generalized abstract economies with preference correspondences. Further applications of our results are also given to minimax inequalities for a family of functions.     

Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system

In this paper we construct a dynamical process (in general, multivalued) generated by the set of solutions of an optimal control problem for the three-dimensional Navier-Stokes system. We prove the existence of a pullback attractor for such multivalued process. Also, we establish the existence of a uniform global attractor containing the pullback attractor. Moreover, under the unproved assumption that strong globally defined solutions of the three-dimensional Navier-Stokes system exist, which guaranties the existence of a global attractor for the corresponding multivalued semiflow, we show that the pullback attractor of the process coincides with the global attractor of the semiflow.     

Existence results for hemivariational inequalities with measures

We prove existence results for multivalued quasilinear elliptic problems of hemivariational inequality type with measure data right-hand sides. In case of L ¹-data, we study existence and enclosure behaviors of solutions by an appropriate sub-supersolution approach. The proofs of our results are based on general existence theory for multivalued pseudomonotone operators, and approximation-, truncation-, and special test function techniques.     

Multiple positive solutions for a class of integral inclusions

This paper deals with sufficient conditions for the existence of at least two positive solutions for a class of integral inclusions arising in the traffic theory. To show our main results, we apply a norm-type expansion and compression fixed point theorem for multivalued map due to Agarwal and O’Regan [A note on the existence of multiple fixed points for multivalued maps with applications, J. Differential Equation 160 (2000) 389–403].     

Positive solutions for nonlinear hemivariational inequalities

In this paper we study the existence of positive solutions for nonlinear problems driven by the p-Laplacian or more generally, by multivalued p-Laplacian-like operators. Both problems have a nonsmooth locally Lipschitz potential (hemivariational inequalities). Using variational methods based on the nonsmooth critical point theory, we prove two existence results with the p-Laplacian and multivalued p-Laplacian-like operators.     

Random coincidence theorems and applications

In this article we state conditions under which the existence of deterministic coincidence points of two multivalued random functions implies the existence of random coincidence points. Moreover, two existence results of measurable selections are obtained for the intersection of the multivalued function evaluated at the random coincidence point. Our results extend or improve some theorems in the literature.     

Existence results for evolutionary inclusions and variational–hemivariational inequalities

We consider an abstract first-order evolutionary inclusion in a reflexive Banach space. The inclusion contains the sum of L-pseudomonotone operator and a maximal monotone operator. We provide an existence theorem which is a generalization of former results known in the literature. Next, we apply our result to the case of nonlinear variational–hemivariational inequalities considered in the setting of an evolution triple of spaces. We specify the multivalued operators in the problem and obtain existence results for several classes of variational–hemivariational inequality problems. Finally, we illustrate our existence result and treat a class of quasilinear parabolic problems under nonmonotone and multivalued flux boundary conditions.