We consider a nonlinear Neumann elliptic inclusion with a source (reaction term) consisting of a convex subdifferential plus a multivalued term depending on the gradient. The convex subdifferential incorporates in our framework problems with unilateral constraints (variational inequalities). Using topological methods and the Moreau-Yosida approximations of the subdifferential term, we establish...

Copyright q 2010 Yeong-Cheng Liou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We present an iterative algorithm for finding a common element x * of the set of solutions of a mixed equilibrium problem and the set of a variatio...

In this paper, we explore the inclusion of latent random variables into the hidden state of a recurrent neural network (RNN) by combining the elements of the variational autoencoder. We argue that through the use of high-level latent random variables, the variational RNN (VRNN)1 can model the kind of variability observed in highly structured sequential data such as natural speech. We empiricall...

In this paper, we consider a more general form of variational inclusions, called generalized variational inclusion (for short, GVI). In connection with GVI, we also consider a generalized resolvent equation with H-resolvent operator, called H-resolvent equation (for short, H-RE). We suggest iterative algorithms to compute the approximate solutions of GVI and H-RE. The existence of a unique solu...

A complete variational treatment is provided for a family of spiked-harmonic oscillator Hamiltonians H = − d dx2 + Bx + λ x (B > 0, λ > 0 ), for arbitrary α > 0 . A compact topological proof is presented that the set S = {ψn} of known exact solutions for α = 2 constitutes an orthonormal basis of the Hilbert space L2(0,∞) . Closed-form expressions are derived for the matrix elements of H with re...

In this paper, we introduce a new class of variational inclusions involving three operator. Using the resolvent operator technique, we establish the equivalence between the general variational inclusions and the resolvent equations. We use this alternative equivalent formulation to suggest and analyze some iterative methods for solving the general variational inclusions. We also consider the cr...

We consider first order periodic differential inclusions in R . The presence of a subdifferential term incorporates in our framework differential variational inequalities in R . We establish the existence of extremal periodic solutions and we also obtain existence results for the “convex” and “nonconvex”problems.

The set of solutions of 1.1 is denoted by A B −1 0 . A number of problems arising in structural analysis, mechanics, and economics can be studied in the framework of this kind of variational inclusions; see, for instance, 1–4 . The problem 1.1 includes many problems as special cases. 1 If B ∂φ : H → 2 , where φ : H → R ∪ ∞ is a proper convex lower semicontinuous function and ∂φ is the subdif an...

We investigate the problem of finding a common solution of a general system of variational inequalities, a variational inclusion, and a fixed-point problem of a strictly pseudocontractive mapping in a real Hilbert space. Motivated by Nadezhkina and Takahashi’s hybrid-extragradient method, we propose and analyze new hybrid-extragradient iterative algorithm for finding a common solution. It is pr...

We present some new critical point theorems for nonlinear dynamical systems which are generalizations of Dancš-Hegedüs-Medvegyev’s principle in uniform spaces and metric spaces by applying an abstract maximal element principle established by Lin and Du. We establish some generalizations of Ekeland’s variational principle, Caristi’s common fixed point theorem for multivalued maps, Takahashi’s no...