We establish variational principles for monotone and maximal bifunctions of Brøndsted-Rockafellar type by using our characterization of bifunction’s maximality in reflexive Banach spaces. As applications, we give an existence result of saddle point for convex-concave function and solve an approximate inclusion governed by a maximal monotone operator.

The subject is the applications of the use of quasi-entropy in finite dimensional spaces to many important quantities in quantum information. 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 t...

In this paper, we survey recent progress on the theory of maximally monotone operators in general Banach space. We also extend various of the results and leave some open questions. 2010 Mathematics Subject Classification: Primary 47H05; Secondary 46B10, 47A06, 47B65, 47N10, 90C25

We propose a notion of operator monotonicity for functions of several variables, which extends the well known notion of operator monotonicity for functions of only one variable. The notion is chosen such that a fundamental relationship between operator convexity and operator monotonicity for functions of one variable is extended also to functions of several variables.

We study systems that are monotone in a generalized sense with respect to cones of rank 2. The main result of the paper is the existence of a Poincaré-Bendixson property for some solutions of those systems.

Our goal is to define exponentials of linear operators. We will try to construct etA as a linear operator, where A : D(A)→ X is a general linear operator, not necessarily bounded. Notationally, it seems like we are looking for a solution to μ̇(t) = Aμ(t), μ(0) = μ0, and we would like to write μ(t) = eμ0. It turns out that this will hold once we make sense of the terms. How can we construct etA w...

We introduce an iterative procedure for finding a point in the zero set (a solution to 0 ∈ A(v) and v ∈ C) of an inverse-monotone or inverse strongly-monotone operator A on a nonempty closed convex subset C in a uniformly smooth and uniformly convex Banach space. We establish weak convergence results under suitable assumptions.

We establish representations of a monotone mapping as the sum of a maximal subdifferential mapping and a ‘remainder’ monotone mapping, where the remainder is either skew linear, or more broadly ‘acyclic’, in the sense that it contains no nontrivial subdifferential component. Examples are given of indecomposable and acyclic operators. In particular, we present an explicit nonlinear acyclic opera...

In this paper, we study the existence of monotone positive solutions for the following nonlinear m-point singular boundary value problem with p-Laplacian operator. The main tool is the monotone iterative technique. We obtain not only the existence of positive solutions for the problem, but also establish iterative schemes for approximating solution.

This paper deals with a new system of nonlinear variational inclusion problems involving (A, η)-maximal relaxed monotone and relative (A, η)-maximal monotone mappings in 2-uniformly smooth Banach spaces. Using the generalized resolvent operator technique, the approximation solvability of the proposed problem is investigated. An iterative algorithm is constructed to approximate the solution of t...