Let C be a closed convex subset of a real Hilbert space H . Let A be an inverse-strongly monotone mapping of C into H and let B be a maximal monotone operator on H such that the domain of B is included in C . We introduce two iteration schemes of finding a point of (A+B)−10, where (A+B)−10 is the set of zero points of A+B. Then, we prove two strong convergence theorems of Halpern’s type in a Hi...

‎the aim of this paper is to study the convergence of solutions of the‎ ‎following second order difference inclusion‎ ‎begin{equation*}begin{cases}exp^{-1}_{u_i}u_{i+1}+theta_i exp^{-1}_{u_i}u_{i-1} in c_ia(u_i),quad igeqslant 1 u_0=xin m‎, ‎quad‎ ‎underset{igeqslant 0}{sup} d(u_i,x)

Let C be a closed convex subset of a real Hilbert space H. Let T be a supper hybrid mapping of C into H, let A be an inverse strongly monotone mapping of C into H and let B be a maximal monotone operator on H such that the domain of B is included in C. In this paper, we introduce two iterative sequences by hybrid methods of finding a point of F (T )∩ (A+B)−10, where F (T ) is the set of fixed p...

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator S on l is skew. We show its domain is a proper subset of the domain of its adjoint S∗, and −S∗ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the in...

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximal monotone operators provided that Rockafellar’s constraint qualification holds. In this paper, we prove the maximal monotonicity of A+B provided that A and B are maximal monotone operators such that domA ∩ int domB ̸= ∅, A+NdomB is of type (FPV), and domA∩domB ⊆ domB. The proof ...

Monotone operators are of basic importance in optimization as they generalize simultaneously subdifferential operators of convex functions and positive semidefinite (not necessarily symmetric) matrices. In 1970, Asplund studied the additive decomposition of a maximal monotone operator as the sum of a subdifferential operator and an “irreducible” monotone operator. In 2007, Borwein and Wiersma [...

The theory of maximal set-valued monotone mappings provide a powerful framework to the study of convex programming and variational inequalities. Based on the notion of relatively maximal relaxed monotonicity, the approximation solvability of a general class of inclusion problems is discussed, while generalizing most of investigations on weak convergence using the proximal point algorithm in a r...