The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar’s constraint qualification holds, which is called the “sum problem”. In this paper, we establish the maximal monotonicity of A+ B provided that A and B are maximally monotone operators such that domA ∩ int domB 6= ∅,...

in this paper, we improve some recent coupled fixed point resultsfor single-valued operators in the framework of ordered $b$-metricspaces established by bota et al. [m-f. bota, a. petrusel, g.petrusel and b. samet, coupled fixed point theorems forsingle-valued operators in b-metric spaces, fixed point theoryappl. (2015) 2015:231]. also, we prove that perov-type fixed pointtheorem in ordered gen...

In this paper, we introduce some adaptive methods for solving variational inequalities with relatively strongly monotone operators. Firstly, focus on the modification of recently proposed, in smooth case [18], numerical method generalized (with Hölder condition) saddle point problem, which has convergence rate estimates similar to accelerated methods. We provide motivation such an approach and ...

We prove strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemirelatively nonexpansive mapping in a Banach space by using monotone hybrid iteration method. By using these results, we obtain new convergence results for resolvents of maximal monotone operators and hemirelatively nonexpansive mappings in a Ban...

We introduce non-autonomous continuous dynamical systems which are linked to the Newton and Levenberg-Marquardt methods. They aim at solving inclusions governed by maximal monotone operators in Hilbert spaces. Relying on the Minty representation of maximal monotone operators as lipschitzian manifolds, we show that these dynamics can be formulated as first-order in time differential systems, whi...

In this note we provide regularity conditions of closedness type which guarantee some surjectivity results concerning the sum of two maximal monotone operators by using representative functions. The first regularity condition we give guarantees the surjectivity of the monotone operator S(· + p) + T (·), where p ∈ X and S and T are maximal monotone operators on the reflexive Banach space X . The...

In any Banach space a monotone operator with a norm-to-weak upper semicontinuous multivalued selection on an open set D is singlevalued and normto-norm upper semicontinuous at the points of a dense Gδ subset of D. Monotone operators — and especially a special case of them, subdifferentials of convex functions — play an important role in various parts of nonlinear analysis. One of the often inve...

4 We consider an extension of Gödel logic by a unary operator that enables the addition of nonnegative reals to truth-values. Although its propositional fragment has a simple proof system, first-order validity is Π2-hard. We explain the close connection to Scarpellini’s result on Π2hardness of Łukasiewicz’s logic.

The subject is the overview of the use of quasi-entropy in finite dimensional spaces. Operator monotone functions and relative modular operators are used. The origin is the relative entropy, and the f -divergence, monotone metrics, covariance and the χ2-divergence are the most important particular cases. The extension of monotone metrics to those with two parameters is a new concept. Monotone m...