A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping

Many problems to appear in signal processing have been formulated as the variational inequality problem over the fixed point set of a nonexpansive mapping. In particular, convex optimization problems over the fixed point set are discussed, and operators which are considered to the problems satisfy the monotonicity. Hence, the uniqueness of the solution of the problem is not always guaranteed. In this article, we present the variational inequality problem for a monotone, hemicontinuous operator over the fixed point set of a firmly nonexpansive mapping. The main aim of the article is to solve the proposed problem by using an iterative algorithm. To this goal, we present a new iterative algorithm for the proposed problem and its convergence analysis. Numerical examples for the proposed algorithm for convex optimization problems over the fixed point set are provided in the final section.     

Further investigation into split common fixed point problem for demicontractive operators

Our contribution in this paper is to propose an iterative algorithm which does not require prior knowledge of operator norm and prove strong convergence theorem for approximating a solution of split common fixed point problem of demicontractive mappings in a real Hilbert space. So many authors have used algorithms involving the operator norm for solving split common fixed point problem, but as widely known the computation of these algorithms may be difficult and for this reason, authors have recently started constructing iterative algorithms with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm. We introduce a new algorithm for solving the split common fixed point problem for demicontractive mappings with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm and then prove strong convergence of the sequence in real Hilbert spaces. Finally, we give some applications of our result and numerical example at the end of the paper.     

Convergence Theorems for Common Fixed Points of Nonself Asymptotically Nonextensive Mappings

Applying the generalized projection operator, we introduce an iterative algorithm for a common fixed point problem of two asymptotically nonextensive nonself-mappings and hence obtain a related convergence theorem in Banach spaces.     

A new method for split common fixed-point problem without priori knowledge of operator norms

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 linear transformation belongs to another fixed-point set. In this paper, we propose a new algorithm for the split common fixed-point problem that does not need any priori information of the operator norm. Under standard assumptions, we establish a weak convergence theorem of the proposed algorithm.     

Complexity Analysis via Approach Spaces

Complexity of a recursive algorithm typically is related to the solution to a recurrence equation based on its recursive structure. For a broad class of recursive algorithms we model their complexity in what we call the complexity approach space, the space of all functions in X?=? ]0,?∞?] ^Y , where Y can be a more dimensional input space. The set X, which is a dcpo for the pointwise order, moreover carries the complexity approach structure. There is an associated selfmap Φ on the complexity approach space X such that the problem of solving the recurrence equation is reduced to finding a fixed point for Φ. We will prove a general fixed point theorem that relies on the presence of the limit operator of the complexity approach space X and on a given well founded relation on Y. Our fixed point theorem deals with monotone selfmaps Φ that need not be contractive. We formulate conditions describing a class of recursive algorithms that can be treated in this way.     

Weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space

We know that variational inequality problem is very important in the nonlinear analysis. For a variational inequality problem defined over a nonempty fixed point set of a nonexpansive mapping in Hilbert space, the strong convergence theorem has been proposed by I. Yamada. The algorithm in this theorem is named the hybrid steepest descent method. Based on this method, we propose a new weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space. Using this result, we obtain some new weak convergence theorems which are useful in nonlinear analysis and optimization problem.     

Positive solutions for a fractional boundary value problem with p-Laplacian operator

In this paper, we are mainly concerned with positive solutions for a p-Laplacian fractional boundary value problem. By virtue of Jensen’s inequalities and some new properties of the Green function of the problem, we adopt the Krasnoselskii-Zabreiko fixed point theorem to establish the results of existence and multiplicity of the positive solutions. Finally, a uniqueness theorem is established by using a fixed point theorem of concave operator and an example is given to illustrate the result.     

Banach空间中非膨胀映象和α-逆强增生算子的强收敛性

在2-一致光滑的Banach空间中,引入一种新的迭代算法研究非膨胀映象的不动点集与α-逆强增生算子的变分不等式解集的公共元素,并获得了迭代算法的强收敛性定理.而且应用这些结果考虑了非膨胀映象和严格伪压缩映象公共不动点的收敛性问题.     

Viable set computation for hybrid systems

In this paper, we revisit the problem of computing viability sets for hybrid systems with nonlinear continuous dynamics and competing inputs. As usual in the literature, an iterative algorithm, based on the alternating application of a continuous and a discrete operator, is employed. Different cases, depending on whether the continuous evolution and the number of discrete transitions are finite or infinite, are considered. A complete characterization of the reach-avoid computation (involved in the continuous time calculation) is provided based on dynamic programming. Moreover, for a certain class of automata, we show convergence of the iterative process by using a constructive version of Tarski’s fixed point theorem, to determine the maximal fixed point of a monotone operator on a complete lattice of closed sets. The viability algorithm is applied to a benchmark example and to the problem of voltage stability for a single machine-load system in case of a line fault.     

The existence of fixed points of left-continuous monotone operators in spaces with a regular cone

In theorems on the existence of a fixed point of an operator the latter is usually assumed to be continuous. In this paper we prove a theorem with sufficient conditions for the existence of a fixed point of an operator which is not necessarily continuous (possibly it is left-continuous). The obtained theorem with the use of regular cones is applied for proving the existence of a fixed point of a nonlinear integral operator. We give an example illustrating the theorem.     

凸幂凝聚算子的不动点定理及其对抽象半线性发展方程的应用

从应用问题的需要出发,给出了一类新的算子-凸幂凝聚算子的定义,推广了凝聚算子的概念,并证明了这类新算子的不动点定理,从而推广了著名的Schauder不动点定理和Sadovskii不动点定理.作为应用,获得了Banach空间中一类具有非紧半群的半线性发展方程初值问题整体mild解和正mild解的存在性.     

Block Iterative Methods for a Finite Family of Generalized Nonexpansive Mappings in Banach Spaces

In this paper, we study an operator generated by a finite family of generalized nonexpansive mappings in a Banach space. We first prove that the set of fixed points of this operator is identical to the set of all common fixed points of the mappings. Next, using this operator, we construct an iterative sequence to approximate a common fixed point of the family of generalized nonexpansive mappings. We finally apply our results to solve the feasibility problem in Banach spaces.     

Optimization for Inconsistent Split Feasibility Problems

The split feasibility problem deals with finding a point in a closed convex subset of the domain space of a linear operator such that the image of the point under the linear operator is in a prescribed closed convex subset of the image space. The split feasibility problem and its variants and generalizations have been widely investigated as a means for resolving practical inverse problems in various disciplines. Many iterative algorithms have been proposed for solving the problem. This article discusses a split feasibility problem which does not have a solution, referred to as an inconsistent split feasibility problem. When the closed convex set of the domain space is the absolute set and the closed convex set of the image space is the subsidiary set, it would be reasonable to formulate a compromise solution of the inconsistent split feasibility problem by using a point in the absolute set such that its image of the linear operator is closest to the subsidiary set in terms of the norm. We show that the problem of finding the compromise solution can be expressed as a convex minimization problem over the fixed point set of a nonexpansive mapping and propose an iterative algorithm, with three-term conjugate gradient directions, for solving the minimization problem.     

Fixed point optimization algorithm and its application to power control in CDMA data networks

We discuss the variational inequality problem for a continuous operator over the fixed point set of a nonexpansive mapping. One application of this problem is a power control for a direct-sequence code-division multiple-access data network. For such a power control, each user terminal has to be able to quickly transmit at an ideal power level such that it can get a sufficient signal-to-interference-plus-noise ratio and achieve the required quality of service. Iterative algorithms to solve this problem should not involve auxiliary optimization problems and complicated computations. To ensure this, we devise a fixed point optimization algorithm for the variational inequality problem and perform a convergence analysis on it. We give numerical examples of the algorithm as a power control.     

A new sequential approach for solving the integro-differential equation via Haar wavelet bases

In this work, we present a method for numerical approximation of fixed point operator, particularly for the mixed Volterra–Fredholm integro-differential equations. The main tool for error analysis is the Banach fixed point theorem. The advantage of this method is that it does not use numerical integration, we use the properties of rationalized Haar wavelets for approximate of integral. The cost of our algorithm increases accuracy and reduces the calculation, considerably. Some examples are provided toillustrate its high accuracy and numerical results are compared with other methods in the other papers.     

Random fixed point theorems for Caristi type random operators

We iteratively generate a sequence of measurable mappings and study necessary conditions for its convergence to a random fixed point of random nonexpansive operator. A random fixed point theorem for random nonexpansive operator, relaxing the convexity condition on the underlying space, is also proved. As an application, we obtained random fixed point theorems for Caristi type random operators.     

局部凸空间中集值映射的极小不动点定理及其应用

利用局部凸空间中Fan-Kakutani不动点定理,得到局部凸空间中集值映射的极小不动点定理,应用此定理,证明了半线性不适定的算子方程的最小范数极值解的存在性.此结果可以应用到不适定常微方程的两点边值问题,不适定偏微方程的边值问题.     

General algorithms for split common fixed point problem of demicontractive mappings

In the first part of this paper, we present a new general algorithm for solving the split common fixed point problem for an infinite family of demicontractive mappings. We establish strong convergence of the algorithm in an infinite dimensional Hilbert space. As applications, we consider algorithms for split variational inequality problem and split common null point problem. In the second part of this paper, we present a new algorithm and strong convergence theorem for approximation of solutions of split equality fixed point problems for an infinite family of demicontractive mappings. Our results improve and generalize some recent results in the literature.     

Existence of positive solutions for Sturm-Liouville boundary value problems on the half-line

In this paper, we establish sufficient conditions to guarantee the existence of at least one positive solution, a unique positive solution, and multiple positive solutions for the Sturm-Liouville boundary value problem on the half-line. By using an effective operator, the fixed point theorems in cone, especially Krasnoselskii fixed point theorem, can be applied to such systems and then existence criteria are established. The interesting point of the results is that the nonlinear term f can be sign-changing.     

一类p-Laplacian算子边值问题三个正解的存在性

本文利用一种新的不动点定理得到了一类具有p Laplacian算子的非线性边值问题三个正解的存在性 .