Common Best Proximity Points: Global Optimal Solutions

Let S:A→B and T:A→B be given non-self mappings, where A and B are non-empty subsets of a metric space. As S and T are non-self mappings, the equations Sx=x and Tx=x do not necessarily have a common solution, called a common fixed point of the mappings S and T. Therefore, in such cases of non-existence of a common solution, it is attempted to find an element x that is closest to both Sx and Tx in some sense. Indeed, common best proximity point theorems explore the existence of such optimal solutions, known as common best proximity points, to the equations Sx=x and Tx=x when there is no common solution. It is remarked that the functions x→d(x,Sx) and x→d(x,Tx) gauge the error involved for an approximate solution of the equations Sx=x and Tx=x. In view of the fact that, for any element x in A, the distance between x and Sx, and the distance between x and Tx are at least the distance between the sets A and B, a common best proximity point theorem achieves global minimum of both functions x→d(x,Sx) and x→d(x,Tx) by stipulating a common approximate solution of the equations Sx=x and Tx=x to fulfill the condition that d(x,Sx)=d(x,Tx)=d(A,B). The purpose of this article is to elicit common best proximity point theorems for pairs of contractive non-self mappings and for pairs of contraction non-self mappings, yielding common optimal approximate solutions of certain fixed point equations. Besides establishing the existence of common best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.     

A system of generalized variational inclusions involving generalized H(·, ·)-accretive mapping in real q-uniformly smooth Banach spaces

In this paper, we consider a class of accretive mappings called generalized H(·, ·)-accretive mappings in Banach spaces. We prove that the proximal-point mapping of the generalized H(·, ·)-accretive mapping is single-valued and Lipschitz continuous. Further, we consider a system of generalized variational inclusions involving generalized H(·, ·)-accretive mappings in real q-uniformly smooth Banach spaces. Using proximal-point mapping method, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions. Furthermore, we discuss the convergence criteria of the iterative algorithm under some suitable conditions. Our results can be viewed as a refinement and improvement of some known results in the literature.     

Convergence of Ishikawa’s iteration method for pseudocontractive mappings

Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let T_i:C→C,i=1,2,…,N, be a finite family of Lipschitz pseudocontractive mappings. It is our purpose, in this paper, to prove strong convergence of Ishikawa’s method to a common fixed point of a finite family of Lipschitz pseudocontractive mappings provided that the interior of the common fixed points is nonempty. No compactness assumption is imposed either on T or on C. Moreover, computation of the closed convex set C_n for each n≥1 is not required. The results obtained in this paper improve on most of the results that have been proved for this class of nonlinear mappings.     

Search for a lighter neutral custodial fiveplet scalar in the Georgi-Machacek model

Many researches from both theoretical and experimental perspectives have been performed to search for a new Higgs Boson that is lighter than the 125 \begin{document}$ {\rm GeV} $\end{document} Higgs boson, which was discovered at the LHC in 2012. In this study, we explore the possibility of constraining a lighter neutral custodial fiveplet scalar \begin{document}$ H_{5}^{0} $\end{document} in the Georgi-Machacek (GM) model using the latest results of the search for a lighter Higgs boson decaying into two photons from LHC data. The custodial-singlet mass eigenstate h or H is considered to be the LHC observed 125 \begin{document}$ {\rm GeV} $\end{document} Higgs boson. A new set of constrained parameters that is favoured by low-mass \begin{document}$ H_{5}^{0} $\end{document} is proposed to generate events efficiently. The production of \begin{document}$ H_{5}^{0} $\end{document} from a scan based on the constrained parameters is compared to the latest results of the search for a lighter Higgs boson decaying into two photons by the CMS Collaboration after applying theoretical constraints from the GM model and constraints from all existing relevant experimental measurements, including the recent results of the Higgs boson searches by the LHC. Numerical analyses of the surviving GM parameter space are performed. The tendencies and correlations of the GM input parameters from phenomenological studies are summarized. In addition, the discovery potential of the other interesting decay channels of this low-mass neutral custodial fiveplet scalar are discussed.     

Redox switching of polyoxometalate-methylene blue-based layer-by-layer films

Iron-substituted crown-type polyoxometalate (POM) [P(8)W(48)O(184)Fe(16)(OH)(28)(H(2)O)(4)](20-) has been successfully immobilized onto glassy carbon electrode surfaces by means of the layer-by-layer (LBL) technique employing the cationic redox active dye, methylene blue (MB). The constructed multilayers exhibit pH-dependent redox activity for both the anionic POM and the cationic dye moieties, which is in good agreement with their solution behavior. The films have been characterized by alternating current impedance, atomic force microscopy, and X-ray photoelectron spectroscopy, whereby the nature of the outer layer within the assemblies was found to have an effect upon the film's behavior. Preliminary investigations show that the POM dye-based films show electrocatalytic ability toward the reduction of hydrogen peroxide, however, only when there is an outer anionic POM layer.     

Random fixed point theorems for multivalued acyclic random maps

Some random fixed point theorems for acyclic valued random operators are proved. The class of acyclic – valued random operators includes convex – valued and star – shaped – valued random operators. This leads to the discovery of some new results     

Implicit and explicit algorithms for solving the split feasibility problem

Very recently, Dang and Gao (Inverse Probl 27:015007, 2011) introduced a KM-CQ algorithm with strong convergence for the split feasibility problem. In this paper, we will continue to consider the split feasibility problem. We present two algorithms. First, we introduce an implicit algorithm. Consequently, by discretizing the continuous implicit algorithm, we obtain an explicit algorithm. Under some weaker conditions, we show the strong convergence of presented algorithms to some solution of the split feasibility problem which solves some special variational inequality. As special cases, we obtain two algorithms which converge strongly to the minimum norm solution of the split feasibility problem. Results obtained in this paper include the corresponding results of Dang and Gao (2011) and extend a recent result of Wang and Xu (J Inequalities Appl 2010, doi:10.1155/2010/102085).     

Reduction of Breakdown Pressure by Filter Cake Removal Using Thermochemical Fluids and Solvents: Experimental and Numerical Studies

The process of well cleanup involves the removal of an impermeable layer of filter cake from the face of the formation. The inefficient removal of the filter cake imposes difficulty on fracturing operations. Filter cake’s impermeable features increase the required pressure to fracture the formation. In this study, a novel method is introduced to reduce the required breakdown pressure to fracture the formation containing the water-based drilling fluid filter cake. The breakdown pressure was tested for five samples of similar properties using different solutions. A simulated borehole was drilled in the core samples. An impermeable filter cake using barite-weighted drilling fluid was built on the face of the drilled hole of each sample. The breakdown pressure for the virgin sample without damage (filter cake) was 6.9 MPa. The breakdown pressure increased to 26.7 MPa after the formation of an impermeable filter cake. Partial removal of filter cake by chelating agent reduced the breakdown pressure to 17.9 MPa. Complete dissolution of the filter cake with chelating agents resulted in the breakdown pressure approximately equivalent to the virgin rock breakdown pressure, i.e., 6.8 MPa. The combined thermochemical and chelating agent solution removed the filter cake and reduced the breakdown pressure to 3.8 MPa. Post-treatment analysis was carried out using nuclear magnetic resonance (NMR) and scratch test. NMR showed the pore size redistributions with good communication between different pores after the thermochemical removal of filter cake. At the same time, there was no communication between the different pores due to permeability impairment after filter cake formation. The diffusion coupling through NMR scans confirmed the higher interconnectivity between different pores systems after the combined thermochemical and chelating agent treatment. Compressive strength was measured from the scratch test, confirming that filter cake formation caused added strength to the rock that impacts the rock breakdown pressure. The average compressive strength of the original specimen was 44.5 MPa that increased to 73.5 MPa after the formation of filter cake. When the filter cake was partially removed, the strength was reduced to 61.7 MPa. Complete removal with chelating agents removed the extra strength that was added due to the filter cake presence. Thermochemical and chelating agents resulted in a significantly lower compressive strength of 25.3 MPa. A numerical model was created to observe the reduction in breakdown pressure due to the thermochemical treatment of the filter cake. The result presented in this study showed the engineering applications of thermochemical treatment for filter cake removal.     

Hydrodynamics of viscous fluid through porous slit with linear absorption

The exact solutions for the viscous fluid through a porous slit with linear ab-sorption are obtained. The Stokes equation with non-homogeneous boundary conditions is solved to get the expressions for the velocity components, pressure distribution, wall shear stress, fractional absorption, and leakage flux. The volume flow rate and mean flow rate are found to be useful in obtaining a convenient form of the longitudinal velocity component and pressure difference. The points of the maximum velocity components for a fixed axial distance are identified. The value of the linear absorption parameter is ran-domly chosen, and the rest available data of the rat kidney to the tabulate pressure drop and fractional absorption are incorporated. The effects of the linear absorption, uniform absorption, and flow rate parameters on the flow properties are discussed by graphs. It is found that forward flow occurs only if the volume flux per unit width is greater than the absorption velocity throughout the length of the slit, otherwise back flow may occur. The leakage flux increases with the increase in the linear absorption parameter. Streamlines are drawn to help the analysis of the flow behaviors during the absorption of the fluid flow through the renal tubule and purification of blood through an artificial kidney.