Map Problem Research Articles

An Iterative Method with Norm Convergence for a Class of Generalized Equilibrium Problems

Recently, Takahashi and Takahashi proposed an iterative algorithm for solving a problem for finding common solutions of generalized equilibrium problems governed by inverse strongly monotone mappings and of fixed point problems for nonexpansive mappings. In this paper, we provide a result that allows for the removal of one condition ensuring the strong convergence of the algorithm.

Multiple-set split feasibility problems for a finite family of demicontractive mappings in Hilbert spaces

In this paper, we introduce an iterative algorithm for solving the multiple-set split feasibility problems for a finite family demicontractive mappings in Hilbert spaces. The results presented in this paper improve and extend some recent corresponding results in (4), (7), (9), (10), (13), (14).

A note on variational-type inequalities for (η,θ,δ)- pseudomonotone-type set-valued mappings in nonreflexive Banach spaces

In this paper the existence of solutions to variational-type inequalities problems for (η, θ, δ)pseudomonotone-type set-valued mappings in nonreflexive Banach spaces introduced in [4] is considered. Presented theorem does not require a compact set-valued mapping, but requires a weaker condition ’locally bounded’ for the mapping.

On Completely Generalized Co-Quasi-Variational Inequalities

In the present work, we introduce and study completely generalized quasi-variational inequality problem for fuzzy mappings. By using the definitions of strongly accretive and retraction mappings, we propose an iterative algorithm for computing the approximate solutions of this class of variational inequalities. We prove that approximate solutions obtained by the proposed algorithm converge to the exact solutions of completely generalized quasi-variational inequality problem.

Strong Convergence Results for Equilibrium Problems and Fixed Point Problems for Multivalued Mappings

Using viscosity approximation method, we study strong convergence to a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of multivalued mappings satisfying the condition (E) in the setting of Hilbert space. Our results improve and extend some recent results in the literature.

Mapping problems for quasiregular mappings

We study images of the unit ball under certain special classes of quasiregular mappings. For homeomorphic, i.e., quasiconformal mappings problems of this type have been studied extensively in the literature. In this paper we also consider non-homeomorphic quasiregular mappings. In particular, we study (topologically) closed quasiregular mappings originating from the work of J. V\"ais\"al\"a and M. Vuorinen in 1970's. Such mappings need not be one-to-one but they still share many properties of quasiconformal mappings. The global behavior of closed quasiregular mappings is similar to the local behavior of quasiregular mappings restricted to a so-called normal domain.

Uniqueness problem for meromorphic mappings in several complex variables into PN(C) with truncated multiplicities

This paper proves some uniqueness theorems for meromorphic mappings in several complex variables into the complex projective space PN(C) with truncated multiplicities, and our results improve some earlier work.

On a structure of the fixed point set of homogeneous maps

A spectral and inverse spectral problem for homogeneous polynomial maps is discussed.The $m$-independence of vectors based on the symmetric tensor powers performs as a main toolto study the structure of the spectrum. Possible restrictions on this structureare described in terms of syzygies provided by the Euler-Jacobi formula.Applications to projective dynamics are discussed.

Convergence to common solutions of various problems for nonexpansive mappings in Hilbert spaces

In this paper, motivated and inspired by Ceng and Yao (J. Comput. Appl. Math. 214(1):186-201, 2008), Iiduka and Takahashi (Nonlinear Anal. 61(3):341-350, 2005), Jaiboon and Kumam (Nonlinear Anal. 73(5):1180-1202, 2010), Kim (Nonlinear Anal. 73:3413-3419, 2010), Marino and Xu (J. Math. Anal. Appl. 318:43-52, 2006) and Saeidi (Nonlinear Anal. 70:4195-4208, 2009), we introduce a new iterative scheme for finding a common element of the set of solutions of a mixed equilibrium problem for an equilibrium bifunction, the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of some variational inequality problem, and the set of fixed points of a left amenable semigroup of nonexpansive mappings with respect to W-mappings and a left regular sequence of means defined on an appropriate space of bounded real-valued functions of the semigroup S. Furthermore, we prove that the iterative scheme converges strongly to a common element of the above four sets. Our results extend and improve the corresponding results of many others. MSC:43A65, 47H05, 47H09, 47H10, 47J20, 47J25, 74G40.

A general iteration scheme for variational inequality problem and common fixed point problems of nonexpansive mappings in q-uniformly smooth Banach spaces

In this paper, we introduce a general iterative algorithm for finding a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of solutions of systems of variational inequalities for two inverse strongly accretive mappings in a q-uniformly smooth Banach space. Then, we prove a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are novel and different from those in the early and recent literature. Our results can be viewed as improvement, supplementation, development and extension of the corresponding results in some references to a great extent.

Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem

We consider the energy critical Schrodinger map problem with the 2-sphere target for equivariant initial data of homotopy index k=1. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blowup solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy.

The uniqueness problem for meromorphic mappings with truncated multiplicities

The purpose of this work is twofold. The first is to solve a uniqueness problem of meromorphic mappings posed by T. Cao and H. Yi in [1]. The second is to generalize several previous uniqueness theorems of meromorphic mappings "partially" sharing a few moving targets, which were given by Z. Chen and M. Ru [2], Z. Chen and Q. Yan [3], D. Thai and S. Quang [13].

A CYCLIC ALGORITHM FOR THE SPLIT COMMON FIXED POINT PROBLEM OF DEMICONTRACTIVE MAPPINGS IN HILBERT SPACES

The split common fixed point problem has been investigated recently, which is a generalization of the split feasibility problem and of the convex feasibility problem. We construct a cyclic algorithm to approximate a solution of the split common fixed point problem for the demicontractive mappings in a Hilbert space. Our results improve and extend previously discussed related problems and algorithms.

Regularity and uniqueness of p-harmonic maps with small range

We prove the uniqueness of solutions to Dirichlet problem for p-harmonic maps with images in a small geodesic ball of the target manifold. As a consequence, we show that such maps have Hölder continuous derivatives. This gives an extension of a result by Hildebrandt et al. (Acta Math 138:1–16, 1977) concerning harmonic maps.

The Complexity of the Empire Colouring Problem

We investigate the computational complexity of the empire colouring problem (as defined by Percy Heawood in 1890) for maps containing empires formed by exactly $r > 1$ countries each. We prove that the problem can be solved in polynomial time using $s$ colours on maps whose underlying adjacency graph has no induced subgraph of average degree larger than $s/r$. However, if $s \geq 3$, the problem is NP-hard even if the graph is a forest of paths of arbitrary lengths (for any $r \geq 2$, provided $s < 2r - \sqrt(2r + 1/4+ 3/2)$. Furthermore we obtain a complete characterization of the problem's complexity for the case when the input graph is a tree, whereas our result for arbitrary planar graphs fall just short of a similar dichotomy. Specifically, we prove that the empire colouring problem is NP-hard for trees, for any $r \geq 2$, if $3 \leq s \leq 2r-1$ (and polynomial time solvable otherwise). For arbitrary planar graphs we prove NP-hardness if $s<7$ for $r=2$, and $s < 6r-3$, for $r \geq 3$. The result for planar graphs also proves the NP-hardness of colouring with less than 7 colours graphs of thickness two and less than $6r-3$ colours graphs of thickness $r \geq 3$.

Nonlinear algorithms approach to split common solution problems

In this paper, we introduce some new iterative algorithms for the split common solution problems for equilibrium problems and fixed point problems of nonlinear mappings. Some examples illustrating our results are also given. MSC:47J25, 47H09, 65K10.

An Iterative Scheme for Generalized Equilibrium Problems and Fixed Point Problems of Strict Pseudo-Contraction Mappings

An Iterative Scheme for Generalized Equilibrium Problems and Fixed Point Problems of Strict Pseudo-Contraction Mappings

Strong convergence theorems for equilibrium problems and fixed point problem of multivalued nonexpansive mappings via hybrid projection method

In this paper, a new iterative process by the hybrid projection method is constructed. Strong convergence of the iterative process to a common element of the set of common fixed points of a finite family of generalized nonexpansive multivalued mappings and the solution set of two equilibrium problems in a Hilbert space is proved. Our results extend some important recent results.

The discrete Markus–Yamabe problem for symmetric planar polynomial maps

We probe deeper into the discrete Markus–Yamabe question for polynomial planar maps and into the normal form for those maps which answer this question in the affirmative. Furthermore, in a symmetric context, we show that the only nonlinear equivariant polynomial maps providing an affirmative answer to the discrete Markus–Yamabe question are those possessing Z2 as their group of symmetries. We use this to establish two new tools which give information about the spectrum of a planar polynomial map.

The two well problem for piecewise affine maps

In the \textit{two-well problem} we look for a map $u$ which satisfies Dirichlet boundary conditions and whose gradient $Du$ assumes values in $SO\left( 2\right) A\cup SO\left( 2\right) B=\mathbb{S}_{A}\cup \mathbb{S}_{B},$ for two given invertible matrices $A,B$ (an element of $SO\left( 2\right) A$ is of the form $RA$ where $R$ is a rotation). In the original approach by Ball and James [1], [2] $A$, $B$ are two matrices such that $\det B>\det A>0$ and $\operatorname*{rank}\left\{ A-B\right\} =1.$ It was proved in the 1990's (see [4], [5], [6], [7], [17]) that a map $u$ satisfying given boundary conditions and such that $Du\in\mathbb{S} _{A}\cup\mathbb{S}_{B}$ exists in the Sobolev class $W^{1,\infty} (\Omega;\mathbb{R}^{2})$ of Lipschitz continuous maps. However, for orthogonal matrices it was also proved (see [3], [8], [9], [10], [11], [12], [16]) that solutions exist in the class of piecewise-$C^{1}$ maps, in particular in the class of piecewise-affine maps. We prove here that this possibility does not exist for other nonsingular matrices $A$, $B$: precisely, the two-well problem can be solved by means of piecewise-affine maps only for orthogonal matrices.