Surjectivity and fixed points of relaxed dissipative multifunctions. Differential inclusions approach

One from the most important properties of accretive and monotone operators is the existence of zeros and surjectivity. In the paper we introduce relaxed variants of dissipative, accretive and monotone operators. Using essentially the properties of the solution set of appropriate differential inclusions we study the existence of zeros of such operators. As corollaries the existence of fixed points of relaxed contractive and relaxed nonexpansive multifunctions are obtained.     

Metric subregularity and the proximal point method

We examine the linear convergence rates of variants of the proximal point method for finding zeros of maximal monotone operators. We begin by showing how metric subregularity is sufficient for local linear convergence to a zero of a maximal monotone operator. This result is then generalized to obtain convergence rates for the problem of finding a common zero of multiple monotone operators by considering randomized and averaged proximal methods.     

含p(x)-Laplacian算子的Neumann问题解的存在性

利用极大单调算子理论,给出了一类含p(x)-Laplacian算子的Neumann边值问题解的存在的充分条件.讨论用到了Lp(x)(Ω)和W01,p(x)(Ω)空间的理论.     

Penalty schemes with inertial effects for monotone inclusion problems

We introduce a penalty term-based splitting algorithm with inertial effects designed for solving monotone inclusion problems involving the sum of maximally monotone operators and the convex normal cone to the (nonempty) set of zeros of a monotone and Lipschitz continuous operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the monotone inclusion problem, provided a condition expressed via the Fitzpatrick function of the operator describing the underlying set of the normal cone is verified. Under strong monotonicity assumptions we can even show strong nonergodic convergence of the iterates. This approach constitutes the starting point for investigating from a similar perspective monotone inclusion problems involving linear compositions of parallel-sum operators and, further, for the minimization of a complexly structured convex objective function subject to the set of minima of another convex and differentiable function.     

Some Conditions for Maximal Monotonicity of Bifunctions

We present necessary and sufficient conditions for a monotone bifunction to be maximally monotone, based on a recent characterization of maximally monotone operators. These conditions state the existence of solutions to equilibrium problems obtained by perturbing the defining bifunction in a suitable way.     

Existence and extensions of positive linear operators

In this paper we develop some unified methods, based on the technique of the auxiliary sublinear operator, for obtaining extensions of positive linear operators. In the first part, a version of the Mazur-Orlicz theorem for ordered vector spaces is presented and then this theorem is used in diverse applications: decomposition theorems for operators and functionals, minimax theory and extensions of positive linear operators. In the second part, we give a general sufficient condition (an implication between two inequalities) for the existence of a monotone sublinear operator and of a positive linear operator. Some particular cases in which this condition becomes necessary are also studied. Dedicated to Prof. Romulus Cristescu on his 80th birthday     

Most maximally monotone operators have a unique zero and a super-regular resolvent

Maximally monotone operators play important roles in optimization, variational analysis and differential equations. Finding zeros of maximally monotone operators has been a central topic. In a Hilbert space, we show that most resolvents are super-regular, that most maximally monotone operators have a unique zero and that the set of strongly monotone mapping is of the first category although each strongly monotone operator has a unique zero. The results are established by applying the Baire Category Theorem to the space of nonexpansive mappings.     

Existence Results for Variational Inequalities with Surjectivity Consequences Related to Generalized Monotone Operators

We present existence results for variational inequalities given by generalized monotone operators. As a consequence, we deduce the existence of zeros, or even more, the surjectivity of some classes of set-valued operators. We show that by strengthening the continuity assumptions, similar surjectivity results can be obtained without any monotonicity assumption. In the framework of reflexive Banach spaces, we extend a related result due to Inoan and Kolumbán (Nonlinear Anal. 68:47–53, 2008).     

Nonlinear Elliptic Problems of Neumann-Type

In this paper we study a nonlinear elliptic differential equation driven by the p-Laplacian with a multivalued boundary condition of the Neumann type. Using techniques from the theory of maximal monotone operators and a theorem on the range of the sum of monotone operators, we prove the existence of a (strong) solution.     

Nonlinear elliptic problems of Neumann-type

In this paper we study a nonlinear elliptic differential equation driven by thep-Laplacian with a multivalued boundary condition of the Neumann type. Using techniques from the theory of maximal monotone operators and a theorem of the range of the sum of monotone operators, we prove the existence of a (strong) solution.     

Visco-penalization of the sum of two monotone operators

A new type of approximating curve for finding a particular zero of the sum of two maximal monotone operators in a Hilbert space is investigated. This curve consists of the zeros of perturbed problems in which one operator is replaced with its Yosida approximation and a viscosity term is added. As the perturbation vanishes, the curve is shown to converge to the zero of the sum that solves a particular strictly monotone variational inequality. As an off-spring of this result, we obtain an approximating curve for finding a particular zero of the sum of several maximal monotone operators. Applications to convex optimization are discussed.     

Classification theorems for operators preserving zeros in a strip

We characterize all linear operators which preserve certain spaces of entire functions whose zeros lie in a closed strip. Necessary and sufficient conditions are obtained for the related problem with real entire functions, and some classical theorems of de Bruijn and Pólya are extended. Specifically, we reveal new differential operators which map real entire functions whose zeros lie in a strip to real entire functions whose zeros lie in a narrower strip; this is one of the properties that characterize a “strong universal factor” as defined by de Bruijn. Using elementary methods, we prove a theorem of de Bruijn and extend a theorem of de Bruijn and Ilieff which states a sufficient condition for a function to have a Fourier transform with only real zeros.     

A proximal point method for the sum of maximal monotone operators

In this paper, we concentrate on the maximal inclusion problem of locating the zeros of the sum of maximal monotone operators in the framework of proximal point method. Such problems arise widely in several applied mathematical fields such as signal and image processing. We define two new maximal monotone operators and characterize the solutions of the considered problem via the zeros of the new operators. The maximal monotonicity and resolvent of both of the defined operators are proved and calculated, respectively. The traditional proximal point algorithm can be therefore applied to the considered maximal inclusion problem, and the convergence is ensured. Furthermore, by exploring the relationship between the proposed method and the generalized forward‐backward splitting algorithm, we point out that this algorithm is essentially the proximal point algorithm when the operator corresponding to the forward step is the zero operator. Copyright © 2013 John Wiley & Sons, Ltd.     

单调及保凸二次样条插值

本文将集中研究保形二次样条插值.假设序列X={x_i}和Y={y_i}(i=0,1,2,…,N)满足x_(i-1)相似文献   

混合单调算子的两点拉伸型不动点定理

本文首次提出了混合单调算子不动点的两点拉伸型条件.同时,利用锥映象的不动点指数理论建立了一类特殊的两点拉伸型混合单调算子的不动点存在性定理,并将所得结论应用于带有超线性项的积分方程与微分方程上,得到了新的结论.因而在本质上推进了混合单调算子不动点问题的研究.     

Maximal Abstract Monotonicity and Generalized Fenchel’s Conjugation Formulas

In this paper, we present a generalization of a theory of monotone operators in the framework of abstract convexity. We show that how generalized Fenchel’s conjugation formulas can be used to obtain some results on maximal abstract monotonicity. We give a necessary and sufficient condition for maximality of abstract monotone operators representable by abstract convex functions by using an additivity constraint qualification.     

Hyponormal Toeplitz operators and zeros of polynomials

The problem of hyponormality for Toeplitz operators with (trigo- nometric) polynomial symbols is studied. We give a necessary and sufficient condition using the zeros of the analytic polynomial induced by the Fourier coefficients of the symbol.

     

Nonlinear boundary value problems for second order differential equations with causal operators

In this paper we deal with second order differential equations with causal operators. To obtain sufficient conditions for existence of solutions we use a monotone iterative method. We investigate both differential equations and differential inequalities. An example illustrates the results obtained.     

Entropy solutions for nonlinear elliptic anisotropic problems with homogeneous Neumann boundary condition

This study is about a nonlinear anisotropic problem with homogeneous Neumann boundary condition. We first prove, by using the technic of monotone operators in Banach spaces, the existence of weak solution, and by approximation methods, we achieve a result of existence and uniqueness of entropy solution     

A general algorithm for multiple-sets split feasibility problem involving resolvents and Bregman mappings

In this paper, by using Bregman distance, we introduce a new iterative process involving products of resolvents of maximal monotone operators for approximating a common element of the set of common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings and the solution set of the multiple-sets split feasibility problem and common zeros of maximal monotone operators. We derive a strong convergence theorem of the proposed iterative algorithm under appropriate situations. Finally, we mention several corollaries and two applications of our algorithm.