Map Problem Research Articles

Auxiliary Principle for a System of Generalized Set-Valued Nonlinear Mixed Variational-Like Inequalities in Banach Spaces

In this paper, the auxiliary principle technique is extended to study a system of generalized nonlinear mixed variational-like inequalities problem for set-valued mappings without compact values in Banach spaces with p-uniformly convex bidual spaces. First, the existence of the solutions of the related auxiliary problem is proved. Then, a new iterative algorithm based on the system of auxiliary variational inequalities is constructed. Finally, both the existence of the solutions of the original problem and the convergence of the iterative sequences generated by the algorithm are proved. And we also present a numerical example to demonstrate the result. Our results improve and extend some known results.

A strong convergence theorem for the split common fixed-point problem of demicontractive mappings

A strong convergence theorem for the split common fixed-point problem of demicontractive mappings

Formal versus analytic CR mappings

We discuss the convergence/divergence problem for formal holomorphic mappings sending real-analytic CR submanifolds into real-analytic sets both lying in complex Euclidean spaces of arbitrary dimension. In particular, we survey the recent developments on

Simultaneous extragradient iterative method to a split equality variational inequality problem and a multiple-sets split equality fixed point problem for multi-valued demicontractive mappings

This paper deals with a strong convergence theorem for a simultaneous extragradient iterative method to approximate a common solution to a split equality variational inequality problem and a multiple-sets split equality fixed point problem for two countable families of multi-valued demicontractive mappings in real Hilbert spaces. Further, we give a numerical example to justify the main result. The method and results presented in this paper extend and unify some recent known results in the literature.

Self-Adaptive Algorithms for the Split Common Fixed Point Problem of the Demimetric Mappings

The split common fixed point problem is an inverse problem that consists in finding an element in a fixed point set such that its image under a bounded linear operator belongs to another fixed-point set. In this paper, we present new iterative algorithms for solving the split common fixed point problem of demimetric mappings in Hilbert spaces. Moreover, our algorithm does not need any prior information of the operator norm. Weak and strong convergence theorems are given under some mild assumptions. The results in this paper are the extension and improvement of the recent results in the literature.

Algorithms for Common Solutions to Generalized Mixed Equilibrium Problems and Fixed Point Problems under Nonlinear Transformations in Banach Spaces

The purpose of this paper is to present a new iterative scheme for finding a common solution of the generalized mixed equilibrium problems with an infinite family of inverse strongly monotone mappings and the fixed point problems of demimetric mappings under nonlinear transformations in Banach spaces. Applications are also included. The results in this paper are the extension and improvement of the recent results in the literature.

An iterative algorithm for solving equilibrium problems, variational inequalities and fixed point problems of multivalued quasi-nonexpansive mappings

An iterative algorithm for solving equilibrium problems, variational inequalities and fixed point problems of multivalued quasi-nonexpansive mappings

Hamiltonian Boundary Value Problems, Conformal Symplectic Symmetries, and Conjugate Loci

In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we present a convenient, coordinate free framework to analyse separated Lagrangian boundary value problems which include classical Dirichlet, Neumann and Robin boundary value problems. The framework is then used to prove the existence of obstructions arising from conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems. Under non-degeneracy conditions, a group action by conformal symplectic symmetries has the effect that the flow map cannot degenerate in a direction which is tangential to the action. This imposes restrictions on which singularities can occur in boundary value problems. Our results generalise classical results about conjugate loci on Riemannian manifolds to a large class of Hamiltonian boundary value problems with, for example, scaling symmetries.

A Flexible Stochastic Method for Solving the MAP Problem in Topic Models

The estimation of the posterior distribution is the core problem in topic models, unfortunately it is intractable. There are approximation and sampling methods proposed to solve it. However, most of them do not have any clear theoretical guarantee of neither quality nor rate of convergence. Online Maximum a Posteriori Estimation (OPE) is another approach with concise guarantee on quality and convergence rate, in which we cast the estimation of the posterior distribution into a non-convex optimization problem. In this paper, we propose a more general and flexible version of OPE, namely Generalized Online Maximum a Posteriori Estimation (G-OPE), which not only enhances the flexibility of OPE in different real-world situations but also preserves key advantage theoretical characteristics of OPE when comparing to the state-of-the-art methods. We employ G-OPE as inference a document within large text corpora. The experimental and theoretical results show that our new approach performs better than OPE and other state-of-the-art methods.

Convex feasibility problems on uniformly convex metric spaces

ABSTRACTIn this paper, we extend two important notions, weighted average method and Mann's iterative method, in the study of convex feasibility problem for general maps defined on p-uniformly convex metric spaces. Then we prove the Δ-convergence theorems for the weighted average sequence and the Mann's alternating sequence in p-uniformly convex metric spaces.

A modified successive projection method for Mann’s iteration process

Mann’s iteration process, which is often used to solve the fixed point problems of nonexpansive mappings, has only weak convergence in infinite-dimensional Hilbert spaces. One way to overcome this weakness is the hybrid method proposed by Nakajo and Takahashi in 2003. But the hybrid method is difficult to implement, since the projection operator on an intersection of the domain and two half-spaces in each step has no closed-form formulae in general. A modified successive projection method for Mann’s iteration process proposed in this paper not only has strong convergence, but also is easy to implement. Therefore, our method effectively improves the hybrid method. We also extend our method to solve the nonexpansive semigroup problems. The numerical results show the superiority of the method in this paper.

A self-adaptive iterative algorithm for the split common fixed point problems

In this paper, we use the dual variable to propose a self-adaptive iterative algorithm for solving the split common fixed point problems of averaged mappings in real Hilbert spaces. Under suitable conditions, we get the weak convergence of the proposed algorithm and give applications in the split feasibility problem and the split equality problem. Some numerical experiments are given to illustrate the efficiency of the proposed iterative algorithm. Our results improve and extend the corresponding results announced by many others.

Multiple values and unicity problem of meromorphic mappings sharing different families of moving hyperplanes

Abstract The purpose of this paper is to deal with the multiple values and unicity problem of meromorphic mappings of ℂ m {\mathbb{C}^{m}} into ℙ n ⁢ ( ℂ ) {\mathbb{P}^{n}(\mathbb{C})} sharing different families of moving hyperplanes with a small identity set. We prove several uniqueness theorems for meromorphic mappings, which extend and improve some recent results.

Some comments on the paper of Khuangsatung and Kangtunyakarn

In this paper, we discuss the validity of the result of Khuangsatung and Kangtunyakarn [Existence and convergence theorem for fixed point problem of various nonlinear mappings and variational inequality problems without some assumptions, Filomat 32(1) (2018) 305–309].

Solution of the Ulam stability problem for Euler–Lagrange k-quintic mappings

Abstract The “oldest quartic” functional equation f ⁢ ( x + 2 ⁢ y ) + f ⁢ ( x - 2 ⁢ y ) = 4 ⁢ [ f ⁢ ( x + y ) + f ⁢ ( x - y ) ] - 6 ⁢ f ⁢ ( x ) + 24 ⁢ f ⁢ ( y ) f(x+2y)+f(x-2y)=4[f(x+y)+f(x-y)]-6f(x)+24f(y) was introduced and solved by the second author of this paper (see J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) 1999, 2, 243–252). Similarly, an interesting “quintic” functional equation was introduced and investigated by I. G. Cho, D. Kang and H. Koh, Stability problems of quintic mappings in quasi-β-normed spaces, J. Inequal. Appl. 2010 2010, Article ID 368981, in the following form: 2 ⁢ f ⁢ ( 2 ⁢ x + y ) + 2 ⁢ f ⁢ ( 2 ⁢ x - y ) + f ⁢ ( x + 2 ⁢ y ) + f ⁢ ( x - 2 ⁢ y ) = 20 ⁢ [ f ⁢ ( x + y ) + f ⁢ ( x - y ) ] + 90 ⁢ f ⁢ ( x ) . 2f(2x+y)+2f(2x-y)+f(x+2y)+f(x-2y)=20[f(x+y)+f(x-y)]+90f(x). In this paper, we generalize this “Cho–Kang–Koh equation” by introducing pertinent Euler–Lagrange k-quintic functional equations, and investigate the “Ulam stability” of these new k-quintic functional mappings.

Split Null Point Problems and Fixed Point Problems for Demicontractive Multivalued Mappings

In this paper, we consider the split null point problem and the fixed point problem for multivalued mappings in Hilbert spaces. We introduce a Halpern-type algorithm for solving the problem for maximal monotone operators and demicontractive multivalued mappings, and establish a strong convergence result under some suitable conditions. Also, we apply our problem of main result to other split problems, that is, the split feasibility problem, the split equilibrium problem, and the split minimization problem. Finally, a numerical result for supporting our main result is also supplied.

Strong convergence theorem for monotone inclusion problem in CAT(0) spaces

In this paper, we introduce Halpern-type proximal point algorithm for approximating a common solution of monotone inclusion problem and fixed point problem. We obtain a strong convergence of the proposed algorithm to a common solution of finite family of monotone inclusion problem and fixed point problem for nonexpansive mappings in complete CAT(0) spaces. Nontrivial application and numerical example were given. Our results complement and extend some recent results in literature.

The split common fixed point problem for generalized demimetric mappings in two Banach spaces

ABSTRACTIn this paper, we consider the split common fixed point problem for new demimetric mappings in two Banach spaces. Using the hybrid method, we prove a strong convergence theorem for finding a solution of the split common fixed point problem in two Banach spaces. Furthermore, using the shrinking projection method, we obtain another strong convergence theorem for finding a solution of the problem in two Banach spaces. Using these results, we obtain well-known and new strong convergence theorems in Hilbert spaces and Banach spaces.

A Unified Algorithm for Solving Split Generalized Mixed Equilibrium Problem, and for Finding Fixed Point of Nonspreading Mapping in Hilbert Spaces

AbstractThe purpose of this paper is to study a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces.We introduce a new iterative algorithm and prove its strong convergence for approximating a common solution of a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces. Our algorithm is developed by combining a modified accelerated Mann algorithm and a viscosity approximation method to obtain a new faster iterative algorithm for finding a common solution of these problems in real Hilbert spaces. Also, our algorithm does not require any prior knowledge of the bounded linear operator norm. We further give a numerical example to show the efficiency and consistency of our algorithm. Our result improves and compliments many recent results previously obtained in this direction in the literature.

A viscosity iterative algorithm for split common fixed-point problems of demicontractive mappings

A viscosity iterative algorithm for split common fixed-point problems of demicontractive mappings