An efficient iterative method for finding common fixed point and variational inequalities in Hilbert spaces

ABSTRACT

In this paper, we investigate the problem of finding a common solution to a fixed point problem involving demi-contractive operator and a variational inequality with monotone and Lipschitz continuous mapping in real Hilbert spaces. Inspired by the projection and contraction method and the hybrid descent approximation method, a new and efficient iterative method for solving the problem is introduced. Strong convergence theorem of the proposed method is established under standard and mild conditions. Our scheme generalizes and extends some of the existing results in the literature, and moreover, its computational effort is less per each iteration compared with related works.     

Can the LHC rule out the MSSM?

If supersymmetry (SUSY) exists in nature and is a solution to the hierarchy problem then it should be detectable at the TeV energy scale which the large hadron collider (LHC) is now exploring. One of the main goals of the LHC is the discovery or exclusion of the R-parity conserving minimal supersymmetric standard model (MSSM). So far, the SUSY search results are presented in the context of the constrained MSSM and other specific simplified SUSY models. A model-independent analysis necessarily relies on the trigger-system of the LHC detectors. By using the posterior samples of a 20-parameter MSSM, the phenomenological MSSM, from a fit to indirect collider and cosmological data we find that there is a significant volume in the MSSM parameter space that would escape the standard trigger-systems of the detectors. As such, in the absence of discovery in the current and future LHC runs, it would be difficult if not impossible to exclude the MSSM unless some dedicated and special triggers are commissioned or a Higgs boson with mass as predicted by the supersymmetric models is not found.     

Convergence analysis for system of equilibrium problems and left bregman strongly relatively nonexpansive mapping

In this paper, we prove strong convergence theorems for approximation of a fixed point of a left Bregman strongly relatively nonexpansive mapping which is also a solution to a finite system of equilibrium problems in the framework of reflexive real Banach spaces. We also discuss the approximation of a common fixed point of a family of left Bregman strongly nonexpansive mappings which is also solution to a finite system of equilibrium problems in reflexive real Banach spaces. Our results complement many known recent results in the literature.     

Strong convergence result for monotone variational inequalities

Our aim in this paper is to study strong convergence results for L-Lipschitz continuous monotone variational inequality but L is unknown using a combination of subgradient extra-gradient method and viscosity approximation method with adoption of Armijo-like step size rule in infinite dimensional real Hilbert spaces. Our results are obtained under mild conditions on the iterative parameters. We apply our result to nonlinear Hammerstein integral equations and finally provide some numerical experiments to illustrate our proposed algorithm.     

Beyond sumudu transform and natural transform: J-transform properties and applications

In this paper, we introduce an efficient integral transform called the ${\mathbb J}$-transform which is a modification of the well-known Sumudu transform and the Natural transform for solving differential equations with real applications in applied physical sciences and engineering. The ${\mathbb J}$-transform is more advantageous than both the Sumudu transform and the Natural transform. Interestingly, our proposed ${\mathbb J}$-transform can be applied successfully to solve complex problems that are ordinarily beyond the scope of either Sumudu transform or Natural transform. As a proof of concept, we consider some classic examples and highlight the limitations of the previously reported integral transforms and lastly demonstrate the superiority of the proposed ${\mathbb J}$-transform in addressing those limitations.     

Iterative methods for solving proximal split minimization problems

In this paper, we propose two iterative algorithms for finding the minimum-norm solution of a split minimization problem. We prove strong convergence of the sequences generated by the proposed algorithms. The iterative schemes are proposed in such a way that the selection of the step-sizes does not need any prior information about the operator norm. We further give some examples to numerically verify the efficiency and implementation of our new methods and compare the two algorithms presented. Our results act as supplements to several recent important results in this area.     

Solving nonlinear ordinary differential equations using the NDM

In this research paper, we examine a novel method called the Natural Decomposition Method (NDM). We use the NDM to obtain exact solutions for three different types of nonlinear ordinary differential equations (NLODEs). The NDM is based on the Natural transform method (NTM) and the Adomian decomposition method (ADM). By using the new method, we successfully handle some class of nonlinear ordinary differential equations in a simple and elegant way. The proposed method gives exact solutions in the form of a rapid convergence series. Hence, the Natural Decomposition Method (NDM) is an excellent mathematical tool for solving linear and nonlinear differential equation. One can conclude that the NDM is efficient and easy to use.     

Fixed point solutions of variational inequality and generalized equilibrium problems with applications

In this paper, we introduce a new iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solution of generalized equilibrium problem and the set of solutions of the variational inequality problem for a co-coercive mapping in a real Hilbert space. Then strong convergence of the scheme to a common element of the three sets is proved. Furthermore, new convergence results are deduced and finally we apply our results to solving optimization problems and obtaining zeroes of maximal monotone operators and co-coercive mappings.     

An efficient projection-type method for monotone variational inequalities in Hilbert spaces

Numerical Algorithms - We consider the monotone variational inequality problem in a Hilbert space and describe a projection-type method with inertial terms under the following properties: (a) The...