A generalized Fitzhugh–Nagumo equation

In this paper we investigate mappings of the classical Fitzhugh–Nagumo equation to a generalized Fitzhugh–Nagumo equation. These mappings are invertible and transform the solutions of the classical Fitzhugh–Nagumo equation into solutions of the generalized Fitzhugh–Nagumo equation considered here. These mappings are found by considering the Lie point symmetries admitted by the classical Fitzhugh–Nagumo equation and the generalized Fitzhugh–Nagumo equation considered here. A particular example of a generalized Fitzhugh–Nagumo equation that satisfies the boundary conditions of the classical Fitzhugh–Nagumo equation is considered. Numerical solutions of the generalized Fitzhugh–Nagumo equation that do not satisfy the boundary conditions of the classical Fitzhugh–Nagumo equation are obtained by implementing the Method of Lines.     

Non-stretch mappings for a sharp estimate of the Beurling–Ahlfors operator

In this paper we identify certain classes of non-stretch mappings that enjoy a sharp estimate of the Beurling–Ahlfors operator. We first make use of a property of subharmonic functions to prove that the Bañuelos–Wang conjecture and the Iwaniec conjecture are true for a class of mappings that satisfy a quasilinear conjugate Beltrami equation. By utilizing the principal solutions of Beltrami equations, we further explicitly construct some classes of non-stretch mappings for which the Bañuelos–Wang conjecture and the Iwaniec conjecture are true.     

Harmonic mappings with hereditary starlikeness

We study a hereditary starlikeness property for planar harmonic mappings on a disk and on an annulus. While such a property is a common trait of conformal mappings, it may be absent in harmonic mappings. It turns out that a sufficient condition for a harmonic mapping f to possess this hereditary property is to have a harmonic argument — a striking feature of conformal mappings that does not extend to all harmonic mappings.     

Avkhadiev–Becker type <Emphasis Type="Italic">p</Emphasis>-valence conditions for harmonic mappings of a disc

We obtain Avkhadiev–Becker type p-valence conditions for locally univalent harmonic mappings defined in the unit disc.     

Propagation of Smallness and the Uniqueness of Solutions to Some Elliptic Equations in the Plane

In this paper we prove theorems on propagation of smallness and the uniqueness of solutions to some elliptic equations in the plane. We start with analogues of these theorems for harmonic functions and use their quasiinvariance under quasiconformal mappings as well as the connection of considered equations with such mappings.     

Properties of the Mappings That Are Close to the Harmonic Mappings. II

We continue studying the mappings that are close to the harmonic mappings (-quasiharmonic mappings with small). This study originates with the previous articles of the author. The results of the article include a theorem on connection between the notion of -quasiharmonic mapping and the solutions to Beltrami systems, an analog to the arithmetic mean property of harmonic functions for -quasiharmonic mappings, a theorem on stability in the Poisson formula for harmonic mappings in the ball, and a theorem on the local smoothing of -quasiharmonic mappings with small which preserves proximity to the harmonic mappings.     

单叶调和映射的线性极值问题

利用线性极值问题有解的必要条件，研究了从单位圆盘到自身的单叶调和映射的傅立叶系数，得到其确界估计。     

Linear Connectivity,Schwarz–Pick Lemma and Univalency Criteria for Planar Harmonic Mapping

In this paper, we first establish a Schwarz–Pick lemma for higher-order derivatives of planar harmonic mappings, and apply it to obtain univalency criteria. Then we discuss distortion theorems,Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images.     

Covering and distortion theorems for planar harmonic univalent mappings

In this paper, we investigate Clunie and Sheil-Small’s covering theorems for sense-preserving planar harmonic univalent mappings defined in the unit disk. Our results significantly improve the earlier known result. Also, we obtain a distortion theorem for fully starlike harmonic mappings in the unit disk.     

New constructions of twistor lifts for harmonic maps

We show that given a harmonic map φ from a Riemann surface to a classical compact simply connected inner symmetric space, there is a J ₂-holomorphic twistor lift of φ (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae in terms of holomorphic data which describes their extended solutions. In particular, this gives explicit formulae for the twistor lifts of all harmonic maps of finite uniton number from a surface to the above symmetric spaces.     

An iterative method of solution for equilibrium and optimization problems

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the variational inequality for an inverse-strongly monotone mapping and the set of solutions of an equilibrium problem in a Hilbert space. We show that the iterative sequence converges strongly to a common element of the three sets. The results of this paper extended and improved the results of H. Iiduka and W. Takahashi [Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350] and S. Takahashi and W. Takahashi [Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515]. Therefore, by using the above result, an iterative algorithm for the solution of a optimization problem was obtained.     

A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces

The purpose of this paper is to study the strong convergence of a general iterative scheme to find a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of variational inequality for a relaxed cocoercive mapping and the set of solutions of an equilibrium problem. Our results extend the recent results of Takahashi and Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515], Marino and Xu [G. Marino, H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43–52], Combettes and Hirstoaga [P.L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) 486–491], Iiduka and Takahashi, [H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350] and many others.     

Becker type univalence conditions for harmonic mappings

We obtain Becker type univalence conditions for locally univalent harmonic mappings defined in one of the following domains: the unit disc, a halfplane, the exterior of the unit disc and prove a generalization of John’s univalence condition.     

Inner estimate and quasiconformal harmonic maps between smooth domains

We prove a type of “inner estimate” for quasi-conformal diffeomorphisms, which satisfies a certain estimate concerning their Laplacian. This, in turn, implies that quasiconformal harmonic mappings between smooth domains (with respect to an approximately analytic metric), have bounded partial derivatives; in particular, these mappings are Lipschitz. We discuss harmonic mappings with respect to (a) spherical and Euclidean metrics (which are approximately analytic) (b) the metric induced by a holomorphic quadratic differential.     

Harmonic Properties of Gauss Mappings in H 3

We consider some harmonic mappings related to hyperbolic Gauss mappings and Gauss mappings in the Obata sense.     

On the quasi-isometries of harmonic quasiconformal mappings

We prove versions of the Ahlfors-Schwarz lemma for quasiconformal euclidean harmonic functions and harmonic mappings with respect to the Poincaré metric.     

Landau and Bloch theorems for harmonic mappings

Chen, Gauthier and Hengartner obtained some versions of Landau's theorem for bounded harmonic mappings and Bloch's theorem for harmonic mappings which are quasiregular and for those which are open. Later, Dorff and Nowak improved their estimates concerning Landau's theorem. In this study, we improve these last results by obtaining sharp coefficient estimates for properly normalized harmonic mappings. Furthermore, our estimates allow us to improve Bloch constant for open harmonic mappings.     

Bloch constants for planar harmonic mappings

We give a lower estimate for the Bloch constant for planar harmonic mappings which are quasiregular and for those which are open. The latter includes the classical Bloch theorem for holomorphic functions as a special case. Also, for bounded planar harmonic mappings, we obtain results similar to a theorem of Landau on bounded holomorphic functions.

     

Gelfand–Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff

We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand–Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.