Generalized monotone bifunctions and equilibrium problems

Using quasimonotone and pseudomonotone bifunctions, we derive existence results for the following equilibrium problem: given a closed and convex subsetK of a real topological vector space, find such that for allyK. In addtion, we study the solution set and the uniquencess of a solution. The paper generalizes results obtained recently for variational inequalities.     

Atomicity and boundedness of monotone Puiseux monoids

In this paper, we study the atomic structure of Puiseux monoids generated by monotone sequences. To understand this atomic structure, it is often useful to know whether the monoid has a bounded generating set. We provide necessary and sufficient conditions for the atomicity and boundedness to be transferred from a monotone Puiseux monoid to all its submonoids. Finally, we present two special subfamilies of monotone Puiseux monoids and fully classify their atomic structure.     

Graph-distance convergence and uniform local boundedness of monotone mappings

In this article we study graph-distance convergence of monotone operators. First, we prove a property that has been an open problem up to now: the limit of a sequence of graph-distance convergent maximal monotone operators in a Hilbert space is a maximal monotone operator. Next, we show that a sequence of maximal monotone operators converging in the same sense in a reflexive Banach space is uniformly locally bounded around any point from the interior of the domain of the limit mapping. The result is an extension of a similar one from finite dimensions. As an application we give a simplified condition for the stability (under graph-distance convergence) of the sum of maximal monotone mappings in Hilbert spaces.

     

Existence and boundedness of solutions to maximal monotone inclusion problem

In Hilbert spaces, the inclusion problem with an arbitrary maximal monotone operator is considered. We prove that the nonemptiness of the solution set of the inclusion problem is equivalent to a coercivity condition. Moreover, a sufficient and necessary condition for the boundedness of the solution set is obtained.     

Some remarks on a boundedness assumption for monotone dynamical systems

We study the consequences of the assumption that all forward orbits are bounded for monotone dynamical systems. In particular, it turns out that this assumption has more implications than is immediately apparent.

     

Equivalence of communication and projective boundedness properties for monotone and homogeneous functions

This work concerns the eigenvalue problem for a monotone and homogeneous self-mapping f of a finite-dimensional positive cone. A communication criterion is formulated such that it is equivalent to the projective boundedness of the upper eigenspaces associated with f, a property that yields the existence of a nonlinear eigenvalue. Using the idea of dual function, a similar result is obtained for lower eigenspaces.     

Local boundedness of solutions to quasilinear elliptic systems

The mathematical analysis to achieve everywhere regularity in the interior of weak solutions to nonlinear elliptic systems usually starts from their local boundedness. Having in mind De Giorgi’s counterexamples, some structure conditions must be imposed to treat systems of partial differential equations. On the contrary, in the scalar case of a general elliptic single equation a well established theory of regularity exists. In this paper we propose a unified approach to local boundedness of weak solutions to a class of quasilinear elliptic systems, with a structure condition inspired by Ladyzhenskaya–Ural’tseva’s work for linear systems, as well as valid for the general scalar case. Our growth assumptions on the nonlinear quantities involved are new and general enough to include anisotropic systems with sharp exponents and the p, q-growth case.     

Local boundedness of minimizers in a limit case

We prove the local boundedness of minimizers of a functional with anisotropic polynomial growth. The result here obtained is optimal if compared with previously know counterexamples. This work has been performed as a part of a National Research Project, supported by MPI (40%, 1987).     

Local-global properties of bifunctions

Representations of composite systems, such as bilinear programming, models of consumer/producer behavior, and sensitivity problems involve bifunctions (functions of two vector arguments). Such bifunctions are typically convex, pseudoconvex, or quasiconvex in each of their arguments, but not jointly convex, pseudoconvex, or quasiconvex. These functions do not in general possess the strong local-global property, namely, that every stationary point is a global minimum. In this paper, we define conditions that ensure that a bifunction possesses only a global minimum. In exploring this question, we use P-convexity and pseudo P-convexity, which are classes of bifunctions that generalize quasiconvexity and pseudoconvexity.     

Local convergence of interior-point algorithms for degenerate monotone LCP

Most asymptotic convergence analysis of interior-point algorithms for monotone linear complementarity problems assumes that the problem is nondegenerate, that is, the solution set contains a strictly complementary solution. We investigate the behavior of these algorithms when this assumption is removed.The work of this author was based on research supported by the National Science Foundation under grant DDM-9109404 and the Office of Naval Research under grant N00014-93-1-0234.The work of this author was based on research supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Local boundedness of minimizers of integrals of the calculus of variations

In this paper we prove the local boundedness of minimizers of integral functionals with non-standard growth conditions.     

Vector variational inequalities with cone-pseudomonotone bifunctions

In this article, two types of cone-pseudomonotone bifunctions have been introduced and the weaker form of pseudomonotonicity is used to establish an existence theorem for a Stampacchia-kind vector variational inequality problem given in terms of bifunctions. For a vector optimization problem, the necessary and sufficient optimality conditions in terms of an associated vector variational inequality problem have been established using a generalized form of cone pseudoconvexity of objective function.     

Local boundedness of variational solutions to evolutionary problems with non-standard growth

We prove the local boundedness of variational solutions and parabolic minimizers to evolutionary problems, where the integrand f is convex and satisfies a non-standard p, q-growth condition with

$$1 < p \leq q \leq p \tfrac{n+2}{n}.$$

A function \({u\colon \Omega_T := \Omega \times (0,T) \to \mathbb{R}}\) is called parabolic minimizer if it satisfies the minimality condition

$$\int_{\Omega_T} u \cdot \partial_t \varphi +f(x, Du) {\rm d} z \leq \int_{\Omega_T} f(x, Du + D \varphi) {\rm d}z$$

for every \({\varphi \in C^\infty_0(\Omega_T)}\). Moreover, we will show local boundedness for parabolic minimizers, if f satisfies an anisotropic growth condition.

     

On cyclic and n-cyclic monotonicity of bifunctions

In the recent literature, the connection between maximal monotone operators and the Fitzpatrick function is investigated. Subsequently, this relation has been extended to maximal monotone bifunctions and their Fitzpatrick transform. In this paper we generalize some of these results to maximal \(n\) -cyclically monotone and maximal cyclically monotone bifunctions, by introducing and studying the Fitzpatrick transforms of order \(n\) or infinite order for bifunctions.     

Proximal point algorithm for infinite pseudo-monotone bifunctions

In this paper, first we study the weak convergence of the proximal point algorithm for an infinite family of equilibrium problems of pseudo-monotone type in Hilbert spaces. Then with additional conditions on the bifunctions, we prove the strong convergence for the family to a common equilibrium point. We also study a regularization of Halpern type and prove the strong convergence of the generated sequence to an equilibrium point of the family of infinite pseudo-monotone bifunctions without any additional assumptions on the bifunctions. A concrete example of a family of pseudo-monotone bifunctions is also presented.     

On maximal monotonicity of bifunctions on Hadamard manifolds

We study some conditions for a monotone bifunction to be maximally monotone by using a corresponding vector field associated to the bifunction and vice versa. This approach allows us to establish existence of solutions to equilibrium problems in Hadamard manifolds obtained by perturbing the equilibrium bifunction.     

Local boundedness of solutions to non-local parabolic equations modeled on the fractional p-Laplacian

We state and prove estimates for the local boundedness of subsolutions of non-local, possibly degenerate, parabolic integro-differential equations of the form

${?}_{t}u(x,t)+\text{P.V.}\underset{{\mathbb{R}}^n}{\int }K(x,y,t)|u(x,t)?u(y,t){|}^{p?2}(u(x,t)?u(y,t))dy=0,$

$(x,t)\in {\mathbb{R}}^n\times \mathbb{R}$, where P.V. means in the principle value sense, $p\in (1,\infty )$ and the kernel obeys $K(x,y,t)\approx |x?y{|}^{n+ps}$ for some $s\in (0,1)$, uniformly in $(x,y,t)\in {\mathbb{R}}^n\times {\mathbb{R}}^n\times \mathbb{R}$.