Halpern projection methods for solving pseudomonotone multivalued variational inequalities in Hilbert spaces

In this paper, we introduce new approximate projection and proximal algorithms for solving multivalued variational inequalities involving pseudomonotone and Lipschitz continuous multivalued cost mappings in a real Hilbert space. The first proposed algorithm combines the approximate projection method with the Halpern iteration technique. The second one is an extension of the Halpern projection method to variational inequalities by using proximal operators. The strongly convergent theorems are established under standard assumptions imposed on cost mappings. Finally we introduce a new and interesting example to the multivalued cost mapping, and show its pseudomontone and Lipschitz continuous properties. We also present some numerical experiments to illustrate the behavior of the proposed algorithms.

     

On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints

In this paper we establish some implicit function theorems for a class of locally Lipschitz set-valued maps and then apply them to investigate some questions concerning the stability of optimization problems with inclusion constraints. In consequence we have an extension of some of the corresponding results of Robinson, Aubin, and others.     

On bounding the effective conductivity of isotropic composite materials

In this paper inequalities for the effective conductivity of isotropic composite materials are derived. These inequalities depend on several coefficients characterizing the microstructure of composites. The obtained coefficients can be exactly calculated for models of a two-component aggregate of multisized, coated ellipsoidal inclusions, packed to fill all space. As a result, new bounds for effective conductivity, considerably narrower than those of Hashin-Shtrikman, are established for such models of composite materials.     

On the rate of convergence for central limit theorems of sojourn times of Gaussian fields

The aim of this paper is to control the rate of convergence for central limit theorems of sojourn times of Gaussian fields in both cases: the fixed and the moving level. Our main tools are the Malliavin calculus and the Stein method, developed by Nualart, Peccati and Nourdin. We also extend some results of Berman to the multidimensional case.     

New Scalarizing Approach to the Stability Analysis in Parametric Generalized Ky Fan Inequality Problems

This paper gives sufficient conditions for the upper and lower semicontinuities of the solution mapping of a parametric mixed generalized Ky Fan inequality problem. We use a new scalarizing approach quite different from traditional linear scalarization approaches which, in the framework of the stability analysis of solution mappings of equilibrium problems, were useful only for weak vector equilibrium problems and only under some convexity and strict monotonicity assumptions. The main tools of our approach are provided by two generalized versions of the nonlinear scalarization function of Gerstewitz. Our stability results are new and are obtained by a unified technique. An example is given to show that our results can be applied, while some corresponding earlier results cannot.     

A backward parabolic equation with a time-dependent coefficient: Regularization and error estimates

We consider the problem of determining the temperature u(x,t)$u(x,t)$, for (x,t)∈[0,π]×[0,T)$(x,t)\in [0,\pi ]\times [0,T)$ in the parabolic equation with a time-dependent coefficient. This problem is severely ill-posed, i.e., the solution (if it exists) does not depend continuously on the given data. In this paper, we use a modified method for regularizing the problem and derive an optimal stability estimation. A numerical experiment is presented for illustrating the estimate.     

Orbits in a stochastic Goodwin–Lotka–Volterra model

This paper examines the cycling behavior of a deterministic and a stochastic version of the economic interpretation of the Lotka–Volterra model, the Goodwin model. We provide a characterization of orbits in the deterministic highly non-linear model. We then study a stochastic version, with Brownian noise introduced via a heterogeneous productivity factor. Existence conditions for a solution to the system are provided. We prove that the system produces cycles around a unique equilibrium point in finite time for general volatility levels, using stochastic Lyapunov techniques for recurrent domains. Numerical insights are provided.     

A Categorical Approach to Groupoid Frobenius Algebras

In this paper, we show that $\mathcal{G}$ -Frobenius algebras (for $\mathcal{G}$ a finite groupoid) correspond to a particular class of Frobenius objects in the representation category of $D(k[\mathcal{G}])$ , where $D(k[\mathcal{G}])$ is the Drinfeld double of the quantum groupoid $k[\mathcal{G}]$ (Nikshych et al. 2000).