It is known that the proximal point algorithm converges weakly to a zero of a maximal monotone operator, but it fails to converge strongly. Then, in (Math. Program. 87:189-202, 2000), Solodov and Svaiter introduced the new proximal-type algorithm to generate a strongly convergent sequence and established a convergence property for the algorithm in Hilbert spaces. Further, Kamimura and Takahashi...

We propose a modification of the classical extragradient and proximal point algorithms for finding a zero of a maximal monotone operator in a Hilbert space. At each iteration of the method, an approximate extragradient-type step is performed using information obtained from an approximate solution of a proximal point subproblem. The algorithm is of a hybrid type, as it combines steps of the extr...

If L is a selfdual Lagrangian L on a reflexive phase space X ×X∗, then the vector field x → ∂̄L(x) := {p ∈ X∗; (p, x) ∈ ∂L(x, p)} is maximal monotone. Conversely, any maximal monotone operator T on X is derived from such a potential on phase space, that is there exists a selfdual Lagrangian L on X ×X∗ (i.e, L∗(p, x) = L(x, p)) such that T = ∂̄L. This solution to problems raised by Fitzpatrick can...

We study systems of nonlinear partial differential equations parabolic type, in which the elliptic operator is replaced by first-order divergence acting on a flux function, related to spatial gradient unknown through an additional implicit equation. This setting, broad enough terms applications, significantly expands paradigm problems. Formulating four conditions concerning form equation, we fi...

We study the convergence of maximal monotone operators with the help of representations by convex functions. In particular, we prove the convergence of a sequence of sums of maximal monotone operators under a general qualification condition of the Attouch–Brezis type.

The classical degree function constructed earlier for pseudomonotone mappings has been used to develop a broader degree theory of classical type for the sum of a maximal monotone map from a reflexive Banach space to its dual together with a bounded pseudomonotone map. The proof uses the generalized Yosida approximation of the maximal monotone mapping.

In this paper, we present a few recent existence results via variational approach for the Cauchy problem dy dt (t) + A(t)y(t) ∋ f(t), y(0) = y0, t ∈ [0, T ], where A(t) : V → V ′ is a nonlinear maximal monotone operator of subgradient type in a dual pair (V, V ′) of reflexive Banach spaces. In this case, the above Cauchy problem reduces to a convex optimization problem via Brezis–Ekeland device...

with respect to the Euclidian norm, where F :Rn →Rn is a nonlinear operator. Section 2 is devoted to a brief description of asynchronous parallel algorithm. In Section 3, we prove the main result concerning the convergence of the general algorithm in the synchronous case to a fixed point of a nonlinear operator fromRn to Rn. A particular case of this algorithm (algorithm of Jacobi) is applied i...

In this paper a reduction and equivalence theorems for the boundedness of the composition of a quasilinear operator T with the Hardy and Copson operators in weighted Lebesgue spaces are proved. New equivalence theorems are obtained for the operator T to be bounded in weighted Lebesgue spaces restricted to the cones of monotone functions, which allow to change the cone of non-decreasing function...

Correspondence: zhangml@henu. edu.cn School of Mathematics and Information Sciences, Henan University, Kaifeng 475000, China Abstract In this study, Mann-type iterative process is considered for finding a common element in the fixed point set of strict pseudocontractions and in the zero point set of the operator which is the sum of inverse stronglymonotone operators and maximal monotone operato...