Related papers: Resolvent Compositions for Positive Linear Operato…
In the renormalisation analysis of critical phenomena in quasi-periodic systems, a fundamental role is often played by fixed points of functional recurrences of the form \begin{equation*} f_{n}(x) = \sum_{i=1}^\ell a_i(x) f_{n_i}…
A common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions. A classical example is the introduction of least squares solutions via the normal equations associated with a…
Projection operators are important in Analysis, Optimization and Algorithm. It is well known that these operators are firmly nonexpansive. In this paper, we provide an exact result that sharpens this well-known result. We develop the theory…
In this paper, we investigate various properties (e.g., nonexistence, asymptotic behavior, uniqueness and integral representation formula) of positive solutions to nonlinear tri-harmonic equations in $\mathbb{R}^{n}$ ($n\geq2$) and…
We investigate compositions of a positive integer with a fixed number of parts, when there are several types of each natural number. These compositions produce new relationships among binomial coefficients, Catalan numbers, and numbers of…
We propose stochastic splitting algorithms for solving large-scale composite inclusion problems involving monotone and linear operators. They activate at each iteration blocks of randomly selected resolvents of monotone operators and,…
We study the resolvent norm of a certain class of closed linear operators on a Hilbert space, including unbounded operators with compact resolvent. It is shown that for any point in the resolvent set there exist directions in which the norm…
Motivated by structures that appear in deep neural networks, we investigate nonlinear composite models alternating proximity and affine operators defined on different spaces. We first show that a wide range of activation operators used in…
Fundamental properties of unbounded composition operators in $L^2$-spaces are studied. Characterizations of normal and quasinormal composition operators are provided. Formally normal composition operators are shown to be normal. Composition…
Some concepts, such as non-compactness measure and condensing operators, defined on metric spaces are extended to uniform spaces. Such extensions allow us to locate, in the context of uniform spaces, some classical results existing in…
The convergence of a new general variable metric algorithm based on compositions of averaged operators is established. Applications to monotone operator splitting are presented.
We in this paper study the nonexpansive operators equipped with arbitrary metric and investigate the connections between firm nonexpansiveness, cocoerciveness and averagedness. The convergence of the associated fixed-point iterations is…
We give some new estimates for the norm and essential norm of a weighted composition operator on the Bloch space. As corollaries, we obtain some new characterizations of the boundedness and compactness of a weighted composition operator on…
Building up on classical linear formulations, we posit that a broad class of problems in signal synthesis and in signal recovery are reducible to the basic task of finding a point in a closed convex subset of a Hilbert space that satisfies…
In this article, we study the convergence of algorithms for solving monotone inclusions in the presence of adjoint mismatch. The adjoint mismatch arises when the adjoint of a linear operator is replaced by an approximation, due to…
The correspondence between the class of nonexpansive mappings and the class of maximally monotone operators via the reflected resolvents of the latter has played an instrumental role in the convergence analysis of the splitting methods.…
In this paper we establish a multivariable non-commutative generalization of L\"owner's classical theorem from 1934 characterizing operator monotone functions as real functions admitting analytic continuation mapping the upper complex…
In this paper, new 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. The usefulness of the obtained results is illustrated…
In this paper we analyze several new methods for solving nonconvex optimization problems with the objective function formed as a sum of two terms: one is nonconvex and smooth, and another is convex but simple and its structure is known.…
It is well known that increasing functions do not preserve operator order in general; nor do decreasing functions reverse operator order. However, operator monotone increasing or operator monotone decreasing do. In this article, we employ a…