Map Problem Research Articles

Weak and Strong Convergence Theorems for New Demimetric Mappings and the Split Common Fixed Point Problem in Banach Spaces

In this paper, we consider the split common fixed point problem for new demimetric mappings in Banach spaces. Using the idea of Mann’s iteration, we prove a weak convergence theorem for finding a solution of the split common fixed point problem in Banach spaces. Furthermore, using the idea of Halpern’s iteration, we obtain a strong convergence theorem for finding a solution of the problem in Banach spaces. Using these results, we obtain well-known and new weak and strong convergence theorems in Hilbert spaces and Banach spaces.

A regularity theory for intrinsic minimising fractional harmonic maps

We define and develop an interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps are critical points of an energy whose first variation is a Dirichlet to Neumann map for the harmonic map problem on a half-space with a Riemannian metric which can degenerate/become singular along the boundary, depending on the fractional power. Similarly to the approach used to prove regularity for stationary intrinsic semi-harmonic maps, we take advantage of the connection between fractional harmonic maps and free boundary problems for harmonic maps in order to develop a partial regularity theory for the fractional harmonic maps we consider. In particular, we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces with the aforementioned metrics up to the boundary; fractional harmonic maps then inherit this regularity. As a by-product of our methods we shed some new light on the monotonicity of the average energy of solutions of the degenerate linear elliptic equation related to fractional harmonic functions.

Iterative methods for solving the split common fixed point problem of demicontractive mappings in Hilbert spaces

Iterative methods for solving the split common fixed point problem of demicontractive mappings in Hilbert spaces

A strong convergence theorem for monotone inclusion and minimization problems in complete CAT(0) spaces

ABSTRACTIn this paper, a Halpern-type proximal point algorithm for approximating a common solution of monotone inclusion problem, minimization problem (MP) and fixed point problem is introduced. Using our algorithm, we state and prove a strong convergence theorem for approximating an element in the intersection of the set of solutions of a monotone inclusion problem, the set of solutions of an MP and the set of solutions of a fixed point problem for non-expansive mappings in complete CAT(0) spaces.

The modified split generalized equilibrium problem for quasi-nonexpansive mappings and applications

In this paper, we introduce a new problem, the modified split generalized equilibrium problem, which extends the generalized equilibrium problem, the split equilibrium problem and the split variational inequality problem. We introduce a new method of an iterative scheme {x_{n}} for finding a common element of the set of solutions of variational inequality problems and the set of common fixed points of a finite family of quasi-nonexpansive mappings and the set of solutions of the modified split generalized equilibrium problem without assuming a demicloseness condition and T_{omega }:= (1-omega)I+ omega T, where T is a quasi-nonexpansive mapping and omega in ( 0,frac{1}{2} ) ; a difficult proof in the framework of Hilbert space. In addition, we give a numerical example to support our main result.

Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces

The purpose of this article is to propose a splitting algorithm for finding a common zero of a finite family of inclusion problems of accretive operators in Banach space. Under suitable conditions, some strong convergence theorems of the sequence generalized by the algorithm to a common zero of the inclusion problems are proved. Some applications to the convex minimization problem, common fixed point problem of a finite family of pseudocontractive mappings, and accretive variational inequality problem in Banach spaces are presented.

Geodesics on ${\bf SO}(n)$ and a class of spherically symmetric maps as solutions to a nonlinear generalised harmonic map problem

We address questions on existence, multiplicity as well as qualitative features including rotational symmetry for certain classes of geometrically motivated maps serving as solutions to the nonlinear system $$ \begin{cases} -\text{\rm div}[ F'(|x|,|\nabla u|^2) \nabla u] = F'(|x|,|\nabla u|^2) |\nabla u|^2 u &\text{in } \mathbb{X}^n,\\ |u| = 1 &\text{in } \mathbb{X}^n ,\\ u = \varphi &\text{on } \partial \mathbb{X}^n. \end{cases} $$% Here $\varphi \in \mathscr{C}^\infty(\partial {\mathbb{X}}^n, \mathbb S}^{n-1})$ is a suitable boundary map, $F'$ is the derivative of $F$ with respect to the second argument, $u \in W^{1,p}(\mathbb{X}^n, \mathbb S}^{n-1})$ for a fixed $1< p< \infty$ and ${\mathbb{X}}^n=\{x \in \mathbb R^n : a< |x|< b\}$ is a generalised annulus. Of particular interest are spherical twists and whirls, where following \cite{Taheri2012}, a spherical twist refers to a rotationally symmetric map of the form $u\colon x \mapsto \rom{Q}(|x|)x|x|^{-1}$ with $\rom{Q}$ some suitable path in $\mathscr{C}([a, b], {\rm SO}(n))$ and a whirl has a similar but more complex structure with only $2$-plane symmetries. We establish the existence of an infinite family of such solutions and illustrate an interesting discrepancy between odd and even dimensions.

Bifurcation of solutions to Hamiltonian boundary value problems

A bifurcation is a qualitative change in a family of solutions to an equation produced by varying parameters. In contrast to the local bifurcations of dynamical systems that are often related to a change in the number or stability of equilibria, bifurcations of boundary value problems are global in nature and may not be related to any obvious change in dynamical behaviour. Catastrophe theory is a well-developed framework which studies the bifurcations of critical points of functions. In this paper we study the bifurcations of solutions of boundary-value problems for symplectic maps, using the language of (finite-dimensional) singularity theory. We associate certain such problems with a geometric picture involving the intersection of Lagrangian submanifolds, and hence with the critical points of a suitable generating function. Within this framework, we then study the effect of three special cases: (i) some common boundary conditions, such as Dirichlet boundary conditions for second-order systems, restrict the possible types of bifurcations (for example, in generic planar systems only the A-series beginning with folds and cusps can occur); (ii) integrable systems, such as planar Hamiltonian systems, can exhibit a novel periodic pitchfork bifurcation; and (iii) systems with Hamiltonian symmetries or reversing symmetries can exhibit restricted bifurcations associated with the symmetry. This approach offers an alternative to the analysis of critical points in function spaces, typically used in the study of bifurcation of variational problems, and opens the way to the detection of more exotic bifurcations than the simple folds and cusps that are often found in examples.

Asymptotically locally Euclidean/Kaluza–Klein stationary vacuum black holes in five dimensions

We produce new examples, both explicit and analytical, of bi-axisymmetric stationary vacuum black holes in five dimensions. A novel feature of these solutions is that they are asymptotically locally Euclidean, in which spatial cross-sections at infinity have lens space |$L(p,q)$| topology, or asymptotically Kaluza–Klein so that spatial cross-sections at infinity are topologically |$S^1\times S^2$|⁠. These are nondegenerate black holes of cohomogeneity 2, with any number of horizon components, where the horizon cross-section topology is any one of the three admissible types: |$S^3$|⁠, |$S^1\times S^2$|⁠, or |$L(p,q)$|⁠. Uniqueness of these solutions is also established. Our method is to solve the relevant harmonic map problem with prescribed singularities, having target symmetric space |$SL(3,\mathbb{R})/SO(3)$|⁠. In addition, we analyze the possibility of conical singularities and find a large family for which geometric regularity is guaranteed.

Iteration process for solving a fixed point problem of nonexpansive mappings in Banach spaces

In this work, we introduce a new iteration process for solving the fixed point problem of a finite family of nonexpansive mappings. We then prove, in Banach spaces, the strong convergence theorem under some mild conditions. Finally, we give some numerical results to support our main result.

Stability and perturbations of countable Markov maps**Dedicated to the memory of Bernd O Stratmann

Let T and , , be countable Markov maps such that the branches of converge pointwise to the branches of T, as . We study the stability of various quantities measuring the singularity (dimension, Hölder exponent etc) of the topological conjugacy between and T when . This is a well-understood problem for maps with finitely-many branches, and the quantities are stable for small ε, that is, they converge to their expected values if . For the infinite branch case their stability might be expected to fail, but we prove that even in the infinite branch case the quantity is stable under some natural regularity assumptions on and T (under which, for instance, the Hölder exponent of fails to be stable). Our assumptions apply for example in the case of Gauss map, various Lüroth maps and accelerated Manneville–Pomeau maps when varying the parameter α. For the proof we introduce a mass transportation method from the cusp that allows us to exploit thermodynamical ideas from the finite branch case.

The split common fixed point problem for multivalued demicontractive mappings and its applications

In this article, we consider the split common fixed point problem for two infinite families of multivalued mappings in real Hilbert spaces. We introduce an algorithm based on the viscosity method for solving the split common fixed point problem for two infinite families of multivalued demicontractive mappings. We establish a strong convergence result under some suitable conditions. As applications, we also apply our main result to the split variational inequality problem and the split common null point problem. Finally, we give the numerical example for supporting our main theorem.

Hybrid projected subgradient-proximal algorithms for solving split equilibrium problems and split common fixed point problems of nonexpansive mappings in Hilbert spaces

In this paper, we propose two strongly convergent algorithms which combines diagonal subgradient method, projection method and proximal method to solve split equilibrium problems and split common fixed point problems of nonexpansive mappings in a real Hilbert space: fixed point set constrained split equilibrium problems (FPSCSEPs) in real Hilbert spaces. The computations of first algorthim requires prior knowledge of operator norm. To estimate the norm of an operator is not always easy, and if it is not easy to estimate the norm of an operator, we purpose another iterative algorithm with a way of selecting the step-sizes such that the implementation of the algorithm does not need any prior information as regards the operator norm. The strong convergence properties of the algorithms are established under mild assumptions on equilibrium bifunctions. We also report some applications and numerical results to compare and illustrate the convergence of the proposed algorithms.

A strong convergence theorem for split mixed equilibrium and fixed point problems for nonexpansive mappings

The aim of this work is twofold; first, to give an extension to split mixed equilibrium problem. Secondly, we suggest and analyze an iterative method based on Mann iterative method and Halpern iterative method to find a common solution of split mixed equilibrium problem and fixed point problem for a nonexpansive mapping in the framework of real Hilbert spaces. Further, under some mild conditions, strong convergence theorem is obtained by the sequences generated by the proposed iterative method, which solves the variational inequality problem. Furthermore, we derived some consequences from our main result and provide some numerical experiments. Our results can be viewed as significant extension and generalization of the previously known results for solving equilibrium problems.

The general split equality problem for Bregman quasi-nonexpansive mappings in Banach spaces

The purpose of this paper is to propose and study an algorithm for solving the general split equality problem governed by Bregman quasi-nonexpansive mappings in Banach spaces. Under some mild conditions, we established the norm convergence of the proposed algorithm. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

Weak Convergence Theorems on the Split Common Fixed Point Problem for Demicontractive Continuous Mappings

We are concerned with the split common fixed point problem in Hilbert spaces. We propose a new method for solving this problem and establish a weak convergence theorem whenever the involved mappings are demicontractive and Lipschitz continuous. As an application, we also obtain a new method for solving the split equality problem in Hilbert spaces.

Polyhedral Complementarity on a Simplex. Potentiality of Regular Mappings

We consider a special class of the fixed point problems for piecewise constant mappings of a simplex into itself. These are polyhedral complementarity problems arising in studying the classical exchange model and its variations. We study the problems that stem from the consideration of models with fixed budgets and possessing a certain property of monotonicity (logarithmic monotonicity). Our considerations are purely mathematical and not associated with the economic models that gave rise to these mathematical objects. The class of regular mappings is investigated, and their potentiality is proved.

Existence and convergence theorem for fixed point problem of various nonlinear mappings and variational inequality problems without some assumptions

The purpose of this article, we give a necessary and sufficient condition for the modified Mann iterative process in order to obtain a strong convergence theorem for finding a common element of the set of fixed point of a finite family of nonexpansive mappings and variational inequality problem in Hilbert space without the conditions ?Ni=1 Fix(Ti)? VI(C,A)??. Moreover, we utilize our main result to fixed point problems of strictly pseudocontractive mappings and the set of solutions of variational inequality problem.

On system of vector quasi-equilibrium problems for multivalued mappings

On system of vector quasi-equilibrium problems for multivalued mappings

Strong convergence of a cyclic iterative algorithm for split common fixed-point problems of demicontractive mappings

Strong convergence of a cyclic iterative algorithm for split common fixed-point problems of demicontractive mappings