We provide and analyze a based average for two convex functions, based on a kernel function. It covers several known averages such as the arithmetic average, epigraphical average, and the proximal average. When applied to the Fitzpatrick function and the conjugate of Fitzpatrick function associated with a monotone operator, our average produces an autoconjugate (also known as selfdual Lagrangia...

An iterative scheme for solving ill-posed nonlinear operator equations with monotone operators is introduced and studied in this paper. A discrete version of the Dynamical Systems Method (DSM) algorithm for stable solution of ill-posed operator equations with monotone operators is proposed and its convergence is proved. A discrepancy principle is proposed and justified. A priori and a posterior...

In a 1999 paper, Bercovici and Pata showed that a natural bijection between the classically, free and Boolean infinitely divisible measures held at the level of limit theorems of triangular arrays. This result was extended to include monotone convolution by the authors in [AW14]. In recent years, operator-valued versions of free, Boolean and monotone probability have also been developed. Belins...

We consider the learning algorithms under general source condition with the polynomial decay of the eigenvalues of the integral operator in vector-valued function setting. We discuss the upper convergence rates of Tikhonov regularizer under general source condition corresponding to increasing monotone index function. The convergence issues are studied for general regularization schemes by using...

Any maximal monotone operator can be characterized by a convex function. The family of such convex functions is invariant under a transformation connected with the Fenchel-Legendre conjugation. We prove that there exist a convex representation of the operator which is a fixed point of this conjugation. 2000 Mathematics Subject Classification: 47H05 keywords: maximal monotone operators, conjugat...

In this paper, we introduce and study fuzzy variational-like inclusion, fuzzy resolvent equation and $H(cdot,cdot)$-$phi$-$eta$-accretive operator in real  uniformly smooth Banach spaces. It is established that fuzzy variational-like inclusion is equivalent to a fixed point problem as well as to a fuzzy resolvent equation. This equivalence is used to define an iterative algorithm for solving fu...

The notion of monotonic independence, introduced by N. Muraki, is considered in a more general frame, similar to the construction of operatorvalued free probability. The paper presents constructions for maps with similar properties to the H and K transforms from the literature, semi inner-product bimodule analogues for the monotone and weakly monotone product of Hilbert spaces, an ad-hoc versio...

In this paper, a new algorithm for solving a class of variational inclusions involving H-monotone operators is considered in Hilbert spaces. We investigate a general iterative algorithm, which consists of a resolvent operator technique step followed by a suitable projection step. We prove the convergence of the algorithm for a maximal monotone operator without Lipschitz continuity. These result...

The notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex, lower semicontinuous, and proper function and any (not necessarily symmetric) continuous linear positive operator. It was recently discovered that most fundamental results on maximal monotone operators allow simpler proofs utilizing Fitzpatrick functions. In this pa...