Fixed Points Of Mappings Research Articles

Fixed Point Results for Hardy Roger Type Contraction in Ordered Complete Dislocated G d Metric Space

In this paper we discuss the fixed points of mappings satisfying a contractive condition on a closed ball in an ordered complete dislocated quasi Gd-metric space. The notion of dominated mappings is applied to approximate the unique solution of non linear functional equations. An example is given to show the validity of our work. Our results improve/generalize several well known recent and classical results.

ON THE QUASI-STATIONARY DISTRIBUTION OF SIS MODELS

In this paper, we propose a novel method for constructing upper bounds of the quasi-stationary distribution of SIS processes. Using this method, we obtain an upper bound that is better than the state-of-the-art upper bound. Moreover, we prove that the fixed point map Φ [7] actually preserves the equilibrium reversed hazard rate order under a certain condition. This allows us to further improve the upper bound. Some numerical results are presented to illustrate the results.

Linesearch algorithms for split equilibrium problems and nonexpansive mappings

AbstractIn this paper, we first propose a weak convergence algorithm, called the linesearch algorithm, for solving a split equilibrium problem and nonexpansive mapping (SEPNM) in real Hilbert spaces, in which the first bifunction is pseudomonotone with respect to its solution set, the second bifunction is monotone, and fixed point mappings are nonexpansive. In this algorithm, we combine the extragradient method incorporated with the Armijo linesearch rule for solving equilibrium problems and the Mann method for finding a fixed point of an nonexpansive mapping. We then combine the proposed algorithm with hybrid cutting technique to get a strong convergence algorithm for SEPNM. Special cases of these algorithms are also given.

Extragradient algorithms for equilibrium problems and symmetric generalized hybrid mappings

In this paper, we propose new algorithms for finding a common point of the solution set of a pseudomonotone equilibrium problem and the set of fixed points of a symmetric generalized hybrid mapping in a real Hilbert space. The convergence of the iterates generated by each method is obtained under assumptions that the fixed point mapping is quasi-nonexpansive and demiclosed at 0, and the bifunction associated with the equilibrium problem is weakly continuous. The bifunction is assumed to be satisfying a Lipschitz-type condition when the basic iteration comes from the extragradient method. It becomes unnecessary when an Armijo back tracking linesearch is incorporated in the extragradient method.

Local Linear Convergence of the ADMM/Douglas--Rachford Algorithms without Strong Convexity and Application to Statistical Imaging

We consider the problem of minimizing the sum of a convex function and a convex function composed with an injective linear mapping. For such problems, subject to a coercivity condition at fixed points of the corresponding Picard iteration, iterates of the alternating directions method of multipliers converge locally linearly to points from which the solution to the original problem can be computed. Our proof strategy uses duality and strong metric subregularity of the Douglas--Rachford fixed point mapping. Our analysis does not require strong convexity and yields error bounds to the set of model solutions. We show in particular that convex piecewise linear-quadratic functions naturally satisfy the requirements of the theory, guaranteeing eventual linear convergence of both the Douglas--Rachford algorithm and the alternating directions method of multipliers for this class of objectives under mild assumptions on the set of fixed points. We demonstrate this result on quantitative image deconvolution and deno...

Two algorithms for solving single-valued variational inequalities and fixed point problems

In this paper, we suggest two new iterative methods for finding a common element of the solution set of a variational inequality problem and the set of fixed points of a contraction mapping in Hilbert space. We also present weak and strong convergence theorems for these new methods, provided that the fixed point mapping is a θ-strict pseudocontraction and the mapping associated with the variational inequality problem is monotone. The results presented in this paper improve and unify important recent results announced by many authors.

On the existence and uniqueness of solutions to a new class of fractional boundary value problems

In this article, we verify the existence and uniqueness of solution for a class of nonlinear boundary value problem for fractional differential equation of order, with the standard fractional derivative in the sense of Riemann–Liouville. By using a variety of tools including the Schauder fixed-point theorem, the Krasnoseĺskii fixed-point theorem and contraction mapping principle, existence and uniqueness results are obtained. Some examples are given to expression the results.

Phase structure of higher spin black holes

We revisit the study of the phase structure of higher spin black holes carried out in arXiv: 1210.0284 using the “canonical formalism”. In particular we study the low as well as the high temperature regimes. We show that the Hawking-Page transition takes place in the low temperature regime. The thermodynamically favoured phase changes from conical surplus to black holes and then again to conical surplus as we increase temperature. We then show that in the high temperature regime the diagonal embedding gives the appropriate description. We also give a map between the parameters of the theory near the IR and UV fixed points with the effect that the “good” solutions near IR fixed point map to the “bad” solutions near the UV fixed point and vice versa.

Convergence Analysis for Anderson Acceleration

Anderson($m$) is a method for acceleration of fixed point iteration which stores m+1 prior evaluations of the fixed point map and computes the new iteration as a linear combination of those evaluations. Anderson(0) is fixed point iteration. In this paper we show that Anderson($m$) is locally r-linearly convergent if the fixed point map is a contraction and the coefficients in the linear combination remain bounded. Without assumptions on the coefficients, we prove q-linear convergence of Anderson(1) and, in the case of linear problems, Anderson($m$). We observe that the optimization problem for the coefficients can be formulated and solved in nonstandard ways and report on numerical experiments which illustrate the ideas.

On the Analysis of the Discretized Kohn--Sham Density Functional Theory

In this paper, we study a few theoretical issues in the discretized Kohn--Sham (KS) density functional theory. The equivalence between either a local or global minimizer of the KS total energy minimization problem and the solution to the KS equation is established under certain assumptions. The nonzero charge densities of a strong local minimizer are shown to be bounded from below by a positive constant uniformly. We analyze the self-consistent field (SCF) iteration by formulating the KS equation as a fixed point map with respect to the potential. The Jacobian of these fixed point maps is derived explicitly. Both global and local convergence of the simple mixing scheme can be established if the gap between the occupied states and unoccupied states is sufficiently large. This assumption can be relaxed in certain cases. Numerical experiments based on the MATLAB toolbox KSSOLV show that it holds on a few simple examples. Although our assumption on the gap is very stringent, our analysis is still valuable for a better understanding of the KS minimization problem, the KS equation, and the SCF iteration.

Periodic nonnegative solutions in neutral nonlinear integro-differential equations with functional delay

We use a variation of Krasnoselskii’s fixed point theorem due to Burton to investigate the existence of solutions of the nonlinear integro-differential neutral equation $$\begin{aligned} x^{\prime }\left( t\right) =-\int _{t-\tau \left( t\right) }^{t}a\left( t,s\right) g(x\left( s\right) )ds+c\left( t\right) x^{\prime }\left( t-\tau \left( t\right) \right) +G\left( t,x(t),x(t-\tau (t)\right) ). \end{aligned}$$ We transform and then invert this equation to obtain a fixed point mapping. We, carely, express this mapping as sum of a compact map and a large contraction and choose suitable hypotheses to show that this fits very nicely into the framework of the modification of Krasnoselskii’s theorem so that existence of nonnegative T-periodic solutions is concluded.

Recent Development in Fixed-Point Theory, Optimization, and Their Applications

This special volume was originally conceived to provide authors and readers a publication to present the most recent advances in the study of applications of various fixed point theorems. Since 1909, when Luitzen Brouwer proved the first fixed point theorem named after him, fixed point theory has played very important roles in many different fields. We can find a lot of demonstrations in optimization theory, approximation theory, differential equations, variational inequalities, complementary problems, equilibrium theory, game theory, economics theory, and so forth. Fixed point theorems are developed for single-valued or set-valued mappings of metric spaces, topological vector spaces, posets and lattices, Banach lattices, . . .. Among the themes of fixed point theory, the topic of approximation of fixed points of mappings is particularly important because it is useful for proving the existence of fixed points ofmappings. It can be applied to prove the solvability of optimization problems, differential equations, variational inequalities, and equilibrium problems. Due to the importance of and the high volume of active research in the nonlinear analysis and optimization and, in particular, many new tools in studying them involving fixed point approximations, it is worthwhile to publish a special issue on this topic to highlight the recent advances in this field. The selection of the papers included in this volume, which has been based on a strict international peer review procedure, contains a representative list of papers with newfangled results which covers the different topics considered in our original proposal. In the papers in this special issue, the following topics are discussed.

On Stability of Fixed Points for Multi-Valued Mappings with an Application

This paper studies the stability of fixed points for multi-valued mappings in relation to selections. For multi-valued mappings admitting Michael selections, some examples are given to show that the fixed point mapping of these mappings are neither upper semi-continuous nor almost lower semi-continuous. Though the set of fixed points may be not compact for multi-valued mappings admitting Lipschitz selections, by finding sub-mappings of such mappings, the existence of minimal essential sets of fixed points is proved, and we show that there exists at least an essentially stable fixed point for almost all these mappings. As an application, we deduce an essentially stable result for differential inclusion problems.

One Contractive Inequality onQuasi-normed Space

We analyze the existence of fixed points for mappings defined on quasi normed Banach spaces satisfying a general contractive inequality of integral type. We are affected from the similar results achieved by A. Branciari and B. E. Rhoades. This condition is extension to Banach-Caccioppoli’s one. We study mappings for which there exists a real number such that for each we have where is a strong measurable mapping which is summable [7],[8, p.1]as well as ||f|| on each compact subset of and such that for each , . We give a general condition which enables one to easily establish fixed point theorems for a pair of maps satisfying a contractive inequality of integral type.

FIXED POINT AND WEAK COMMUTING MAPPING

In this paper we prove some new fixed point theorems for weak commuting mapping on complete metric space. Our results generalize several corresponding relations in metric space of weak commuting mapping.

Elliptic Preconditioner for Accelerating the Self-Consistent Field Iteration in Kohn--Sham Density Functional Theory

We discuss techniques for accelerating the self-consistent field iteration for solving the Kohn--Sham equations. These techniques are all based on constructing approximations to the inverse of the Jacobian associated with a fixed point map satisfied by the total potential. They can be viewed as preconditioners for a fixed point iteration. We point out different requirements for constructing preconditioners for insulating and metallic systems, respectively, and discuss how to construct preconditioners to keep the convergence rate of the fixed point iteration independent of the size of the atomistic system. We propose a new preconditioner that can treat insulating and metallic systems in a unified way. The new preconditioner, which we call an elliptic preconditioner, is constructed by solving an elliptic partial differential equation. The elliptic preconditioner is shown to be more effective in accelerating the convergence of a fixed point iteration than the existing approaches for large inhomogeneous systems at low temperature.

Existence and uniqueness of solution for a class of nonlinear fractional differential equations

AbstractIn this paper, we present some new existence and uniqueness results for nonlinear fractional differential equations with a kind of general irregular boundary condition in Banach space by using a fixed-point theorem and contraction mapping principle. Moreover, the boundary condition is extended, therefore, some conclusions from other references are special cases of our results.MSC:26A33, 34A08.

Extragradient Methods and Linesearch Algorithms for Solving Ky Fan Inequalities and Fixed Point Problems

In this paper, we introduce some new iterative methods for finding a common element of the set of points satisfying a Ky Fan inequality, and the set of fixed points of a contraction mapping in a Hilbert space. The strong convergence of the iterates generated by each method is obtained thanks to a hybrid projection method, under the assumptions that the fixed-point mapping is a ξ-strict pseudocontraction, and the function associated with the Ky Fan inequality is pseudomonotone and weakly continuous. A Lipschitz-type condition is assumed to hold on this function when the basic iteration comes from the extragradient method. This assumption is unnecessary when an Armijo backtracking linesearch is incorporated in the extragradient method. The particular case of variational inequality problems is examined in a last section.

Resolvents for weakly singular kernels and fractional differential equations

The solution R(t,s) of the resolvent equation (1)R(t,s)=B(t,s)+∫stB(t,u)R(u,s)du, where B(t,s) denotes a given weakly singular matrix, is obtained by means of fixed point mappings. The result is a series that begins with some singular terms after which the remainder of the terms defines a continuous function. In particular, the resolvent is calculated for the kernel B(t,s)=λ(t−s)q−1 of the scalar Abel integral equation of the second kind (2)x(t)=a(t)+λ∫0t1(t−s)1−qx(s)ds where 0<q<1. It is then used to derive closed-form formulas for the resolvents corresponding to q=1/2 and 1/3. Furthermore, a closed-form formula for the resolvent for B(t,s)=λs(t2−s2)q−1 is derived thereby demonstrating that the results of this paper apply not only to kernels of convolution type but also to those of non-convolution type as well. Finally, the resolvent for the kernel of (2) is used to find a general solution of a linear fractional differential equation of Caputo type.

Sequential Estimation of Structural Models With a Fixed Point Constraint

This paper considers the estimation problem of structural models for which empirical restrictions are characterized by a fixed point constraint, such as structural dynamic discrete choice models or models of dynamic games. We analyze a local condition under which the nested pseudo likelihood (NPL) algorithm converges to a consistent estimator, and derive its convergence rate. We find that the NPL algorithm may not necessarily converge to a consistent estimator when the fixed point mapping does not have a local contraction property. To address the issue of divergence, we propose alternative sequential estimation procedures that can converge to a consistent estimator even when the NPL algorithm does not.