Preserving maximal monotonicity with applications in sum and composition rules

In this paper, we study maximal monotonicity preserving mappings on the Banach space X × X ^*. Indeed, for a maximal monotone set ${M \subset X\times X^*}$ and for a multifunction ${T: X \times X^* \multimap Y \times Y^*}$ , under some sufficient conditions on M and T we show that T(M) is maximal monotone. As two consequences of this result we get sum and composition rules for maximal monotone operators.     

A new iterative algorithm of solution for equilibrium problems, variational inequalities and fixed point problems in a Hilbert space

In this paper, we introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem, the set of solutions of the variational inequality for β-inverse-strongly monotone mappings and the set of fixed points of nonexpansive mappings in a Hilbert space. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. As applications, at the end of paper we utilize our results to study some convergence problem for finding the zeros of maximal monotone operators. Our results are generalizations and extensions of the results of Yao and Liou (Fixed Point Theory Appl. Article ID 384629, 10 p., 2008), Yao et al. (J. Nonlinear Convex Anal. 9(2):239–248, 2008) and Su and Li (Appl. Math. Comput. 181(1):332–341, 2006) and some recent results.     

Regularization and iterative methods for monotone inverse variational inequalities

We consider the monotone inverse variational inequality: find $x\in H$ such that $$\begin{aligned} f(x)\in \Omega , \quad \left\langle \tilde{f}-f(x),x\right\rangle \ge 0, \quad \forall \tilde{f}\in \Omega , \end{aligned}$$ where $\Omega $ is a nonempty closed convex subset of a real Hilbert space $H$ and $f:H\rightarrow H$ is a monotone mapping. A general regularization method for monotone inverse variational inequalities is shown, where the regularizer is a Lipschitz continuous and strongly monotone mapping. Moreover, we also introduce an iterative method as discretization of the regularization method. We prove that both regularized solution and an iterative method converge strongly to a solution of the inverse variational inequality.     

Some regularity theorems for CR mappings

The purpose of this article is to study Lipschitz CR mappings from an h-extendible (or semi-regular) hypersurface in ${\mathbb{C}^n}$ . Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from strongly pseudoconvex domains is also proved.     

Strong convergence of a splitting projection method for the sum of maximal monotone operators

In this paper, a projective-splitting method is proposed for finding a zero of the sum of $n$ maximal monotone operators over a real Hilbert space $\mathcal{H }$ . Without the condition that either $\mathcal{H }$ is finite dimensional or the sum of $n$ operators is maximal monotone, we prove that the sequence generated by the proposed method is strongly convergent to an extended solution for the problem, which is closest to the initial point. The main results presented in this paper generalize and improve some recent results in this topic.     

C 0 and bi-Lipschitz {\mathcal{K}} -equivalence of mappings

In this paper we investigate the classification of mappings up to ${\mathcal{K}}$ -equivalence. We give several results of this type. We study semialgebraic deformations up to semialgebraic C ⁰ ${\mathcal{K}}$ -equivalence and bi-Lipschitz ${\mathcal{K}}$ -equivalence. We give an algebraic criterion for bi-Lipschitz ${\mathcal{K}}$ -triviality in terms of semi-integral closure (Theorem 3.5). We also give a new proof of a result of Nishimura: we show that two germs of smooth mappings ${f, g: \mathbb{R}^n \to \mathbb{R}^n}$ , finitely determined with respect to ${\mathcal{K}}$ -equivalence are C ⁰- ${\mathcal{K}}$ -equivalent if and only if they have the same degree in absolute value.     

Uniformly Quasiregular Maps on the Compactified Heisenberg Group

We show the existence of a non-injective uniformly quasiregular mapping acting on the one-point compactification $\bar{ {\mathbb{H}}}^{1}={\mathbb{H}}^{1}\cup\{\infty\}$ of the Heisenberg group ?¹ equipped with a sub-Riemannian metric. The corresponding statement for arbitrary quasiregular mappings acting on sphere ${\mathbb{S}}^{n} $ was proven by Martin (Conform. Geom. Dyn. 1:24?C27, 1997). Moreover, we construct uniformly quasiregular mappings on $\bar{ {\mathbb{H}}}^{1}$ with large-dimensional branch sets. We prove that for any uniformly quasiregular map g on $\bar{ {\mathbb{H}}}^{1}$ there exists a measurable CR structure ?? which is equivariant under the semigroup ?? generated by g. This is equivalent to the existence of an equivariant horizontal conformal structure.     

The Liouville theorem for conformal mappings on Carnot groups with Goursat-Darboux distribution

Under minimal assumptions on smoothness we prove the Liouville theorem on conformal mappings for one infinite series of Carnot groups \(\mathbb{J}^k\) with sub-Riemannian metric with Goursat-Darboux distribution, k ≥ 2: each mapping with 1-bounded distortion of a connected domain U on \(\mathbb{J}^k\) is equal to the restriction to U of the action of an element of the finite-dimensional group of 1-quasiconformal smooth mappings.     

An iterative algorithm for common fixed points for nonexpansive semigroups and strictly pseudo-contractive mappings with optimization problems

In this paper, we introduce an iterative algorithm for finding a common element of the set of solutions of a system of mixed equilibrium problems, the set of solutions of a variational inclusion problems for inverse strongly monotone mappings, the set of common fixed points for nonexpansive semigroups and the set of common fixed points for an infinite family of strictly pseudo-contractive mappings in Hilbert spaces. Furthermore, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions which solves some optimization problems. Our results extend and improve the recent results of Chang et al. (Appl Math Comput 216:51–60, 2010), Hao (Appl Math Comput 217(7):3000–3010, 2010), Jaiboon and Kumam (Nonlinear Anal 73:1180–1202, 2010) and many others.     

A sharp lower bound of Burkholder’s functional for K-quasiconformal mappings and its applications

In this paper, for \(K\) -quasiconformal mappings of a bounded domain into the complex plane, we build a sharp lower bound of Burkholder’s functional. As an application, we give two explicit and sharp lower bounds of Burkholder’s integrals for two subclasses of \(K\) -quasiconformal mappings, respectively. As the second application, we obtain a sharp upper bound of the \(L^p\) -integral of \(\sqrt{J_f}\) for certain \(K\) -quasiconformal mappings.     

Pluriharmonic mappings and linearly connected domains in ℂ n

In this paper we obtain certain sufficient conditions for the univalence of pluriharmonic mappings defined in the unit ball \(\mathbb{B}^n \) of ? ⁿ . The results are generalizations of conditions of Chuaqui and Hernández that relate the univalence of planar harmonic mappings with linearly connected domains, and show how such domains can play a role in questions regarding injectivity in higher dimensions. In addition, we extend recent work of Hernández and Martín on a shear type construction for planar harmonic mappings, by adapting the concept of stable univalence to pluriharmonic mappings of the unit ball \(\mathbb{B}^n \) into ? ⁿ .     

Traces of monotone Sobolev functions

In this paper we prove that ifu: ${\mathbb{B}}^n \to {\mathbb{R}}$ , where ${\mathbb{B}}^n $ is the unit ball in ? ⁿ , is a monotone function in the Sobolev space W^1·p ( ${\mathbb{B}}^n $ ), andn ? 1 <p ≤n, thenu has nontangential limits at all the points of $\partial {\mathbb{B}}^n $ except possibly on a set ofp-capacity zero. The key ingredient in the proof is an extension of a classical theorem of Lindelöf to monotone functions in W^1·p ( ${\mathbb{B}}^n $ ),n ? 1 <p ≤n.     

Univalence criterion and quasiconformal extension of holomorphic mappings

In this paper we are concerned with solutions, in particular with the univalent solutions, of the Loewner differential equation associated with non-normalized subordination chains on the Euclidean unit ball B in ${\mathbb{C}^n}$ . We also give applications to univalence conditions and quasiconformal extensions to ${\mathbb{C}^n}$ of holomorphic mappings on B. Finally we consider the asymptotical case of these results. The results in this paper are complete generalizations to higher dimensions of well known results due to Becker. They improve and extend previous sufficient conditions for univalence and quasiconformal extension to ${\mathbb{C}^n}$ of holomorphic mappings on B.     

Local boundedness of monotone bifunctions

We consider bifunctions ${F : C\times C\rightarrow \mathbb{R}}$ where C is an arbitrary subset of a Banach space. We show that under weak assumptions, monotone bifunctions are locally bounded in the interior of their domain. As an immediate corollary, we obtain the corresponding property for monotone operators. Also, we show that in contrast to maximal monotone operators, monotone bifunctions (maximal or not maximal) can also be locally bounded at the boundary of their domain; in fact, this is always the case whenever C is a locally polyhedral subset of ${\mathbb{R}^{n}}$ and F(x, ·) is quasiconvex and lower semicontinuous.     

A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let ?? >?0 and let A be an ??-inverse-strongly monotone mapping of C into H and let B be a maximal monotone operator on H. Let F be a maximal monotone operator on H such that the domain of F is included in C. Let 0?< k?<?1 and let g be a k-contraction of H into itself. Let V be a ${\overline{\gamma}}$ -strongly monotone and L-Lipschitzian continuous operator with ${\overline{\gamma} >0 }$ and L >?0. Take ${\mu, \gamma \in \mathbb R}$ as follows: $${0 < \mu < \frac{2\overline{\gamma}}{L^2}, \quad 0 < \gamma < \frac{\overline{\gamma}-\frac{L^2 \mu}{2}}{k}.}$$ In this paper, under the assumption ${(A+B)^{-1}0 \cap F^{-1}0 \neq \emptyset}$ , we prove a strong convergence theorem for finding a point ${z_0\in (A+B)^{-1}0\cap F^{-1}0}$ which is a unique solution of the hierarchical variational inequality $${\langle (V-\gamma g)z_0, q-z_0 \rangle \geq 0, \quad \forall q\in (A+B)^{-1}0 \cap F^{-1}0.}$$ Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space which are useful in nonlinear analysis and optimization.     

On additivity of mappings on measurable functions

We prove the additivity of regular l-additive mappings T: → [0,+∞] of a hereditary cone in the space of measurable functions on a measure space. Some examples are constructed of non-d-additive l-additive mappings T. The monotonicity of l-additive mappings T: → [0,+∞] is established. The examples are constructed of nonmonotone d-additive mappings T. Let (S, +) be a commutative cancellation semigroup. Given a mapping T: → S, we prove the equivalence of additivity and l-additivity. It is shown that a strongly regular 2-homogeneous l-subadditive mapping T is subadditive. All results are new even in case = L _∞ ⁺ .     

Saturating Sperner Families

A family ${\mathcal{F} \subseteq 2^{[n]}}$ saturates the monotone decreasing property ${\mathcal{P}}$ if ${\mathcal{F}}$ satisfies ${\mathcal{P}}$ and one cannot add any set to ${\mathcal{F}}$ such that property ${\mathcal{P}}$ is still satisfied by the resulting family. We address the problem of finding the minimum size of a family saturating the k-Sperner property and the minimum size of a family that saturates the Sperner property and that consists only of l-sets and (l + 1)-sets.     

Sufficient Criteria for \varepsilon Starlike Mappings in a Complex Banach Space

Several sufficient conditions for $\varepsilon $ starlike mappings on the unit ball $B$ in a complex Banach space are provided. From these, we may construct many concrete $\varepsilon $ starlike mappings on $B$ . Furthermore, several growth results associated with these sufficient conditions are also provided.     

Harmonic mappings in Bergman spaces

In this paper, we investigate the properties of mappings in harmonic Bergman spaces. First, we discuss the coefficient estimate, the Schwarz-Pick Lemma and the Landau-Bloch theorem for mappings in harmonic Bergman spaces in the unit disk $\mathbb D $ of $\mathbb C $ . Our results are generalizations of the corresponding ones in Chen et al. (Proc Am Math Soc 128:3231–3240, 2000), Chen et al. (J Math Anal Appl 373:102–110, 2011), Chen et al. (Ann Acad Sci Fenn Math 36:567–576, 2011). Then, we study the Schwarz-Pick Lemma and the Landau-Bloch theorem for mappings in harmonic Bergman spaces in the unit ball $\mathbb B ^{n}$ of $\mathbb C ^{n}$ . The obtained results are generalizations of the corresponding ones in Chen and Gauthier (Proc Am Math Soc 139:583–595 2011). At last, we get a characterization for mappings in harmonic Bergman spaces on $\mathbb B ^{n}$ in terms of their complex gradients.