We explore the solvability of a general system of nonlinear relaxed cocoercive variational inequality (SNVI) problems based on a new projection system for the direct product of two nonempty closed and convex subsets of real Hilbert spaces. 2000 Mathematics subject classification: primary 49J40, 65B05; secondary 47H20.

We investigate the asymptotic behavior of a stochastic version of the forward-backward splitting algorithm for finding a zero of the sum of a maximally monotone set-valued operator and a cocoercive operator in Hilbert spaces. Our general setting features stochastic approximations of the cocoercive operator and stochastic perturbations in the evaluation of the resolvents of the set-valued operat...

We deal with monotone inclusion problems of the form 0 ∈ Ax+Dx+NC(x) in real Hilbert spaces, where A is a maximally monotone operator, D a cocoercive operator and C the nonempty set of zeros of another cocoercive operator. We propose a forwardbackward penalty algorithm for solving this problem which extends the one proposed by H. Attouch, M.-O. Czarnecki and J. Peypouquet in [3]. The condition ...

Recommended by Saburou Saitoh We consider a new algorithm for a generalized system for relaxed cocoercive nonlinear inequalities involving three different operators in Hilbert spaces by the convergence of projection methods. Our results include the previous results as special cases extend and improve the main results of R.

In this paper, we introduce a general iterative scheme to investigate the problem of finding a common element of the fixed point set of a strict pseudocontraction and the solution set of a variational inequality problem for a relaxed cocoercive mapping by viscosity approximate methods. Strong convergence theorems are established in a real Hilbert space.

We begin by considering second order dynamical systems of the from ẍ(t) + Γ(ẋ(t)) + λ(t)B(x(t)) = 0, where Γ : H → H is an elliptic bounded self-adjoint linear operator defined on a real Hilbert space H, B : H → H is a cocoercive operator and λ : [0,+∞)→ [0,+∞) is a relaxation function depending on time. We show the existence and uniqueness of strong global solutions in the framework of the Cau...