Fixed point theorems of ? convex −ψ concave mixed monotone operators and applications

In this paper, ? convex −ψ concave mixed monotone operators are introduced and some new existence and uniqueness theorems of fixed points for mixed monotone operators with such convexity concavity are obtained. As an application, we give one example to illustrate our results.     

Fixed-point theorems of φ concave-(−ψ) convex mixed monotone operators and applications

In this paper, φ concave-(−ψ) convex operators are introduced and some new existence and uniqueness theorems of fixed points of mixed monotone operators with such concavity and convexity are obtained. Moreover, some applications to nonlinear integral equations on bounded or unbounded regions are given.     

关于α-凸凹混合单调算子的不动点定理

引入了α-凸凹混合单调算子的概念.借助于集值分析的方法,利用锥理论讨论了这类混合单调算子,得到了若干α-凸凹混合单调算子不动点的存在性和唯一性定理.     

Random Mann iteration scheme and random fixed point theorems

In this paper, some concepts such as random monotone operators, random Mann iteration and so on in a separable real Banach space are introduced. Also the existence and uniqueness theorems of random fixed points for random monotone operators satisfying Condition(H) are proved.     

Menger PN-空间中两类混合单调算子新的公共不动点定理

在Menger PN-空间,引入(Co)类压缩型算子半群的有关概念.研究了两类混合单调算子新的公共不动点的存在与唯一性,不要求算子具有任何紧性、凹凸性和连续性,从而获得一些新的结论,改进和推广Banach空间中的有关研究结论.     

The Existence of Positive Almost Periodic Type Solutions for Some Nonlinear Delay Integral Equations

This paper presents the conditions of existence of positive almost periodic type solutions for some nonlinear delay integral equations, by using a fixed point theorem in the mixed monotone operators （Ma, Y.： On a class of mixed monotone operators and a kind of two-point bounded value problem. Indian J. Math., 41（2）, 211-220 （1999]]. Some known results are operators.     

p-Laplacian Inclusions via Fixed Points for Multifunctions in Posets

We consider the Dirichlet problem for elliptic differential inclusions involving the p-Laplacian and governed by multivalued nonlinearities bounded above and below by single-valued monotone functions not necessarily continuous. The aim of this note is to provide existence and comparison results for the problem under consideration without imposing further regularity assumptions. The main tools used in the proof of our results are existence and comparison results for extremal solutions of (single-valued) nonlinear elliptic equations, and an abstract comparison principle for fixed points of multivalued operators in partially ordered sets (posets).     

Nonlinear boundary value problems for second order differential inclusions

In this paper, we study a class of nonlinear value boundary problems for second order differential inclusions with nonlinear perturbations, which satisfy the generalized Hartman condition weaker than that considered in some papers. Using techniques from multivalued analysis, theory of monotone operators and fixed points, we prove the existence of solutions in both “convex” and “nonconvex” cases. Our framework can be incorporated with Dirichlet, Neumann, and mixed boundary problems.     

混合单调算子的不动点存在唯一性定理及其应用

本文首先讨论了一类混合单调算子方程组解的存在唯一性及非对称迭代逼近问题,得到了若干不具有连续性和紧性条件的有关混合单调算子、增算子和减算子的新不动点定理.其次研究了具有a-凹和-a-凸的不具有连续性和紧性条件的混合单调算子的不动点,并得到了一个新结果.最后,我们将所得结果应用于RN上的Hammerstein积分方程之中(参见文[1-12]).     

Newton-Like Dynamics and Forward-Backward Methods for Structured Monotone Inclusions in Hilbert Spaces

In a Hilbert space setting we introduce dynamical systems, which are linked to Newton and Levenberg–Marquardt methods. They are intended to solve, by splitting methods, inclusions governed by structured monotone operators M=A+B, where A is a general maximal monotone operator, and B is monotone and locally Lipschitz continuous. Based on the Minty representation of A as a Lipschitz manifold, we show that these dynamics can be formulated as differential systems, which are relevant to the Cauchy–Lipschitz theorem, and involve separately B and the resolvents of A. In the convex subdifferential case, by using Lyapunov asymptotic analysis, we prove a descent minimizing property and weak convergence to equilibria of the trajectories. Time discretization of these dynamics gives algorithms combining Newton’s method and forward-backward methods for solving structured monotone inclusions.     

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.     

Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems

The purpose of this paper is to present some new fixed point theorems for mixed monotone operators with perturbation by using the properties of cones and a fixed point theorem for mixed monotone operators. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.     

An alternative approach to well-posedness of a class of differential inclusions in Hilbert spaces

We show the well-posedness of initial value problems for differential inclusions of a certain type using abstract perturbation results for maximal monotone operators in Hilbert spaces. For this purpose the time derivative is established in an exponentially weighted _L2$L_2$ space. The problem of well-posedness then reduces to show that the sum of two maximal monotone operators in time and space is again maximal monotone. The theory is exemplified by three inclusions describing phenomena in mathematical physics involving hysteresis.     

New existence and uniqueness theorems of positive fixed points for mixed monotone operators with perturbation

In this paper, we obtain some new and general existence and uniqueness theorems of positive fixed points for mixed monotone operators with perturbation, which extend the corresponding results in [Z.T. Zhang, New fixed point theorems of mixed monotone operators and applications, J. Math. Anal. Appl. 204 (1996) 307-319, Theorem 1, Corollaries 1 and 2]. Moreover, some applications to nonlinear integral equations on unbounded region are given.     

Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator S on ?² is skew. We show its domain is a proper subset of the domain of its adjoint S^∗, and −S^∗ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator T on L²[0,1]. We compare the domain of T with the domain of its adjoint T^∗ and show that the skew part of T admits two distinct linear maximal monotone skew extensions. These unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators cannot be significantly weakened, and they are simpler than the example given by Phelps-Simons. Interesting consequences on Fitzpatrick functions for sums of two maximal monotone operators are also given.     

On random fixed point theorems of random monotone operators

In this paper, we investigate the existence of random fixed point for random mixed monotone operators and random increasing (decreasing) operators and obtain some new random fixed point theorems.     

有限域上交错矩阵方程解的计数公式及其q超几何级数表达

设Ｆｑ是ｑ元有限域，这里ｑ是任意一个素数ｐ的方幂。本文利用Ｆｑ上奇异辛几何给出当Ａ，Ｂ分别是阶ｎ秩２ｖ和ｍ秩２ｓ的一般交错矩阵时，Ｆｑ上适合方程ＸＡＸ‘＝Ｂ的秩ｋ的解Ｘ和解Ｘ的个数的明显公式，并用ｑ超几何级数简化表达公式。     

On Bregman-Type Distances for Convex Functions and Maximally Monotone Operators

Given two point to set operators, one of which is maximally monotone, we introduce a new distance in their graphs. This new concept reduces to the classical Bregman distance when both operators are the gradient of a convex function. We study the properties of this new distance and establish its continuity properties. We derive its formula for some particular cases, including the case in which both operators are linear monotone and continuous. We also characterize all bi-functions D for which there exists a convex function h such that D is the Bregman distance induced by h.     

A novel proof of the existence of solutions for a new system of generalized mixed quasi-variational-like inclusions involving (<Emphasis Type="Italic">A</Emphasis>, <Emphasis Type="Italic">η</Emphasis>, <Emphasis Type="Italic">m</Emphasis>)-accretive operators

In this paper, we introduce a new system of generalized mixed quasi-variational-like inclusions with (A, η, m)-accretive operators and relaxed cocoercive mappings. By using the fixed point theorem of Nadler, we prove the existence of solutions for this general system of generalized mixed quasi-variational-like inclusions and its special cases. The results in this paper unify, extend and improve some known results in the literature. The novel proof method is simpler than those iterative algorithm approach for proving the existence of solutions of all classes of system of set-valued variational inclusions in the literature.     

一类混合单调算子的耦合不动点定理及其应用

讨论了一类混合单调算子的耦合不动点定理,并获得了最大最小耦合不动点.作为应用,讨论了Banach空间中含有不连续项的混合单调Volterra型积分方程耦合拟解的存在性问题.