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Szilárd László

Publications and source records attributed to Szilárd László.

9 recordsLinked to original sources
arXiv2017

On Perturbed Weak Vector Equilibrium Problems under new Semi-continuities

In this paper we introduce a new semicontinuity notion, which is weaker than upper semicontinuity, and assures the closedness of the sets $G(y)=\{x\in K: f(x,y)\not\in -\inte C\}.$ Furhter, this semicontinuity is also closed under addition. These two properties make our new semicontinuity applicable in situations where other semicontinuities, like quasi upper semicontinuity or order upper semicontinuity, fail. The above emphasized properties are some key tools in order to provide new sufficient conditions that ensure the existence of the solution of a perturbed weak vector equilibrium problem in Hausdorff topological vector spaces ordered by a cone. Further, we introduce a dual problem and we provide conditions that assure that every solution of the dual problem is also a solution of the perturbed weak vector equilibrium problem.

math.FA↗
arXiv2017

A Primal-Dual Approach of Weak Vector Equilibrium Problems

In this paper we provide some new sufficient conditions that ensure the existence of the solution of a weak vector equilibrium problem in Hausdorff topological vector spaces ordered by a cone. Further, we introduce a dual problem and we provide conditions that assure the solution set of the original problem and its dual coincide. We show that many known problems from the literature can be treated in our primal-dual model. We provide several coercivity conditions in order to obtain solution existence of the primal-dual problems without compactness assumption. We pay a special attention to the case when the base space is a reflexive Banach space. We apply the results obtained to perturbed vector equilibrium problems.

math.FA↗
arXiv2016

Some Minimax Results on Dense Sets and Dense Families of Functionals in C(K) and B(K)

In this paper, by an example we show that the general minimax results of Fan and Sion cannot be extended on usual dense sets. Nevertheless, we obtain some new minimax results on a special type of dense set that we call self-segment-dense. We apply our results in order to obtain denseness of some family of functionals in function spaces.

math.FA↗
arXiv2014

An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions

We propose a forward-backward proximal-type algorithm with inertial/memory effects for minimizing the sum of a nonsmooth function with a smooth one in the nonconvex setting. The sequence of iterates generated by the algorithm converges to a critical point of the objective function provided an appropriate regularization of the objective satisfies the Kurdyka-Łojasiewicz inequality, which is for instance fulfilled for semi-algebraic functions. We illustrate the theoretical results by considering two numerical experiments: the first one concerns the ability of recovering the local optimal solutions of nonconvex optimization problems, while the second one refers to the restoration of a noisy blurred image.

math.OC↗
arXiv2014

Densely defined equilibrium problems

In the present work we deal with set-valued equilibrium problems for which we provide sufficient conditions for the existence of a solution. The conditions that we consider are imposed not on the whole domain, but rather on a self segment-dense subset of it, a special type of dense subset. As an application, we obtain a generalized Debreu-Gale-Nikaido-type theorem, with a considerably weakened Walras law in its hypothesis. Further, we consider a non-cooperative n-person game and prove the existence of a Nash equilibrium, under assumptions that are less restrictive than the classical ones.

math.FA↗
arXiv2013

Existence of solutions of inverted variational inequalities

In this paper we introduce two new generalized variational inequalities, and we give some existence results of the solutions for these variational inequalities involving operators belonging to a recently introduced class of operators. We show by examples, that our results fail outside of this class. Further, we establish a result that may be viewed as a generalization of Minty's theorem, that is, we show that under some circumstances the set of solutions of these variational inequalities coincide. We also show that the operators, involved in these variational inequalities, to belong to the above mentioned class, is essential in obtaining this result. As application, we show that Brouwer's fixed point theorem is an easy consequence of our results.

math.FA↗
arXiv2013

Injectivity and univalence of complex functions via monotonicity

In this paper we provide sufficient conditions that ensure the monotonicity, respectively the global injectivity of an operator. Further, some new analytical conditions that assure the injectivity/univalence of a complex function of one complex variable are obtained. We also show that some classical results, such as Alexander-Noshiro-Warschawski and Wolff theorem or Mocanu theorem, are easy consequences of our results.

math.FA↗
arXiv2013

Generalized monotone operators on dense sets

In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized convexity of a real valued function, which is obtained out of its local counterpart on some dense sets.

math.FA↗