Hidden maximal monotonicity in evolutionary variational-hemivariational inequalities

In this paper, we propose a new methodology to study evolutionary variational-hemivariational inequalities based on the theory of evolution equations governed by maximal monotone operators. More precisely, the proposed approach, based on a hidden maximal monotonicity, is used to explore the well-posedness for a class of evolutionary variational-hemivariational inequalities involving history-dependent operators and related problems with periodic and antiperiodic boundary conditions. The applicability of our theoretical results is illustrated through applications to a fractional evolution inclusion and a dynamic semipermeability problem.

A New Approximation Method for Finding Zeros of Maximal Monotone Operators

One of the major problems in the theory of maximal monotone operators is to find a point in the solution set Zer( ), set of zeros of maximal monotone mapping . The problem of finding a zero of a maximal monotone in real Hilbert space has been investigated by many researchers. Rockafellar considered the proximal point algorithm and proved the weak convergence of this algorithm with the maximal monotone operator. Güler gave an example showing that Rockafellar’s proximal point algorithm does not converge strongly in an infinite-dimensional Hilbert space. In this paper, we consider an explicit method that is strong convergence in an infinite-dimensional Hilbert space and a simple variant of the hybrid steepest-descent method, introduced by Yamada. The strong convergence of this method is proved under some mild conditions. Finally, we give an application for the optimization problem and present some numerical experiments to illustrate the effectiveness of the proposed algorithm.

Generalized relaxed inertial method with regularization for solving split feasibility problems in real Hilbert spaces

In this paper, we propose a new modified relaxed inertial regularization method for finding a common solution of a generalized split feasibility problem, the zeros of sum of maximal monotone operators, and fixed point problem of two nonlinear mappings in real Hilbert spaces. We prove that the proposed method converges strongly to a minimum-norm solution of the aforementioned problems without using the conventional two cases approach. In addition, we apply our convergence results to the classical variational inequality and equilibrium problems, and present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other existing methods in the literature. The results obtained in this paper extend, generalize and improve several results in this direction.

Variational approach to nonlinear stochastic differential equations in Hilbert spaces

Here we survey a few functional methods to existence theory for infinite dimensional stochastic differential equations of the form dX+A(t)X(t)=B(t,X(t))dW(t), X(0)=X_0, where A(t) is a non\-linear maximal monotone operator in a variational couple (V,V'). The emphasis is put on a new approach of the classical existence result of N. Krylov and B. Rozovski on existence for the infinite dimensional stochastic differential equations which is given here via the theory of nonlinear maximal monotone operators in Banach spaces. A variational approach to this problem is also developed.

New Convergence Theorems for Maximal Monotone Operators in Banach Spaces

The purpose of this paper is to introduce a new hybrid iterative scheme for resolvents of maximal monotone operators in Banach spaces by using the notion of generalized fprojection. Next, we apply this result to the convex minimization and variational inequality problems in Banach spaces. The results presented in this paper improve and extend important recent results in the literature.

The sum theorem for maximal monotone operators in reflexive Banach spaces revisited

The goal of this note is to present a new shorter proof for the maximal monotonicity of the Minkowski sum of two maximal monotone multi-valued operators defined in a reflexive Banach space under the classical interiority condition involving their domains.