Zero Of Maximal Monotone Operator Research Articles

An Application of Viscosity Approximation Type Iterative Method in the Generation of Mandelbrot and Julia Fractals

In this paper, we present an application of the viscosity approximation type iterative method introduced by Nandal et al. (Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators, Symmetry, 2019) to visualize and analyse the Julia and Mandelbrot sets for a complex polynomial of the type T(z) = z^{n} + p z + r, where p, rin {mathbb {C}}, and n ge 2. This iterative method has many applications in solving various fixed point problems. We derive an escape criterion to visualize Julia and Mandelbrot sets via the proposed viscosity approximation type method. Moreover, we present several graphical examples of the fractals generated with the proposed iteration method.

A New Approximation Method for Finding Zeros of Maximal Monotone Operators

One of the major problems in the theory of maximal monotone operators is to find a point in the solution set Zer( ), set of zeros of maximal monotone mapping . The problem of finding a zero of a maximal monotone in real Hilbert space has been investigated by many researchers. Rockafellar considered the proximal point algorithm and proved the weak convergence of this algorithm with the maximal monotone operator. Güler gave an example showing that Rockafellar’s proximal point algorithm does not converge strongly in an infinite-dimensional Hilbert space. In this paper, we consider an explicit method that is strong convergence in an infinite-dimensional Hilbert space and a simple variant of the hybrid steepest-descent method, introduced by Yamada. The strong convergence of this method is proved under some mild conditions. Finally, we give an application for the optimization problem and present some numerical experiments to illustrate the effectiveness of the proposed algorithm.

Strong convergence of an inertial algorithm for maximal monotone inclusions with applications

An inertial iterative algorithm is proposed for approximating a solution of a maximal monotone inclusion in a uniformly convex and uniformly smooth real Banach space. The sequence generated by the algorithm is proved to converge strongly to a solution of the inclusion. Moreover, the theorem proved is applied to approximate a solution of a convex optimization problem and a solution of a Hammerstein equation. Furthermore, numerical experiments are given to compare, in terms of CPU time and number of iterations, the performance of the sequence generated by our algorithm with the performance of the sequences generated by three recent inertial type algorithms for approximating zeros of maximal monotone operators. In addition, the performance of the sequence generated by our algorithm is compared with the performance of a sequence generated by another recent algorithm for approximating a solution of a Hammerstein equation. Finally, a numerical example is given to illustrate the implementability of our algorithm for approximating a solution of a convex optimization problem.

Approximation of zeros of m-accretive mappings, with applications to Hammerstein integral equations

"An algorithm for approximating zeros of m-accretive operators is constructed in a uniformly smooth real Banach space. The sequence generated by the algorithm is proved to converge strongly to a zero of an m-accretive operator. In the case of a real Hilbert space, our theorem complements the celebrated proximal point algorithm of Martinet and Rockafellar for approximating zeros of maximal monotone operators. Furthermore, the convergence theorem proved is applied to approximate a solution of a Hammerstein integral equation. Finally, numerical experiments are presented to illustrate the convergence of our algorithm."

The strong convergence of a proximal point algorithm in complete CAT(0) metric spaces

In this paper, we consider a proximal point algorithm for finding zeros of maximal monotone operators in complete CAT(0) spaces. First, a necessary and sufficient condition is presented for the zero set of the operator to be nonempty. Afterwards, we prove that, under suitable conditions, the proposed algorithm converges strongly to the metric projection of some point onto the zero set of the involving maximal monotone operator.

Strong Convergence Theorems for Bregman Demigeneralized Mappings in Banach Spaces with Applications

In this paper, we first introduce a new class of mappings called Bregman -demigeneralized mappings in a Banach space E. Then, using Bregman projection method, we prove a convergence theorem for finding a common fixed point of a family of the new mappings in a Banach space E. Furthermore, we apply our method to prove strong convergence theorems of iterative schemes for finding common fixed points of infinitely many nonlinear mappings satisfying the jointly demiclosedness principle in a Banach space E. Our new technique is based on the strong coerciveness of a Bregman function and the method of proof is completely different from those available in the literature. Some application of our results to the solution of common zeros of maximal monotone operators is presented. Finally, an illustrative numerical example is presented. Our results improve and generalize many known results in the current literature.

Strong Convergence of a New Generalized Viscosity Implicit Rule and Some Applications in Hilbert Space

In this paper, based on the very recent work by Nandal et al. (Nandal, A.; Chugh, R.; Postolache, M. Iteration process for fixed point problems and zeros of maximal monotone operators. Symmetry 2019, 11, 655.), we propose a new generalized viscosity implicit rule for finding a common element of the fixed point sets of a finite family of nonexpansive mappings and the sets of zeros of maximal monotone operators. Utilizing the main result, we first propose and investigate a new general system of generalized equilibrium problems, which includes several equilibrium and variational inequality problems as special cases, and then we derive an implicit iterative method to solve constrained multiple-set split convex feasibility problem. We further combine forward–backward splitting method and generalized viscosity implicit rule for solving monotone inclusion problem. Moreover, we apply the main result to solve convex minimization problem.

Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators

We introduce an iterative algorithm which converges strongly to a common element of fixed point sets of nonexpansive mappings and sets of zeros of maximal monotone mappings. Our iterative method is quite general and includes a large number of iterative methods considered in recent literature as special cases. In particular, we apply our algorithm to solve a general system of variational inequalities, convex feasibility problem, zero point problem of inverse strongly monotone and maximal monotone mappings, split common null point problem, split feasibility problem, split monotone variational inclusion problem and split variational inequality problem. Under relaxed conditions on the parameters, we derive some algorithms and strong convergence results to solve these problems. Our results improve and generalize several known results in the recent literature.

A strong convergence theorem for Bregman quasi-noexpansive mappings with applications

In this paper we propose a new iterative algorithm for finding a common fixed point of an infinitely countable family of Bregman quasi-nonexpansive mappings in reflexive Banach spaces. We give some applications of our algorithm to solving equilibrium problems, convex feasibility problems, variational inequalities of monotone operators and finding zeros of maximal monotone operators.

A general algorithm for multiple-sets split feasibility problem involving resolvents and Bregman mappings

In this paper, by using Bregman distance, we introduce a new iterative process involving products of resolvents of maximal monotone operators for approximating a common element of the set of common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings and the solution set of the multiple-sets split feasibility problem and common zeros of maximal monotone operators. We derive a strong convergence theorem of the proposed iterative algorithm under appropriate situations. Finally, we mention several corollaries and two applications of our algorithm.

Convergence of over-relaxed contraction-proximal point algorithm in Hilbert spaces

The proximal point algorithm (PPA) is a classical method for finding zeros of maximal monotone operators. It is known that the algorithm only has weak convergence in a general Hilbert space. Recently, Wang, Wang and Xu proposed two modifications of the PPA and established strong convergence theorems on these two algorithms. However, these two convergence theorems exclude an important case, namely, the over-relaxed case. In this paper, we extend the above convergence theorems from under-relaxed case to the over-relaxed case, which in turn improve the performance of these two algorithms. Preliminary numerical experiments show that the algorithm with over-relaxed parameter performs better than that with under-relaxed parameter.

A New Algorithm for Finding Fixed Points of Bregman Quasi-Nonexpansive Mappings and Zeros of Maximal Monotone Operators by Using Products of Resolvents

In this paper, we propose a new iterative algorithm for finding a common fixed point of an infinitely countable family of Bregman quasi-nonexpansive mappings and a common zero of finitely many maximal monotone operators in reflexive Banach spaces. An application of our algorithm for solving equilibrium problems will also be exhibited.

Error Sensitivity for Strongly Convergent Modifications of the Proximal Point Algorithm

The proximal point algorithm plays an important role in finding zeros of maximal monotone operators. It has however only weak convergence in the infinite-dimensional setting. In the present paper, we provide two contraction-proximal point algorithms. The strong convergence of the two algorithms is proved under two different accuracy criteria on the errors. A new technique of argument is used, and this makes sure that our conditions, which are sufficient for the strong convergence of the algorithms, are weaker than those used by several other authors.

Convergence theorems for finding zero points of maximal monotone operators and equilibrium problems in Banach spaces

In this paper, we construct a new hybrid projection method for approximating a common element of the set of zeroes of a finite family of maximal monotone operators and the set of common solutions to a system of generalized equilibrium problems in a uniformly smooth and strictly convex Banach space. We prove strong convergence theorems of the algorithm to a common element of these two sets. As application, we also apply our results to find common solutions of variational inequalities and zeroes of maximal monotone operators.

Halpern-Mann’s iterations for Bregman strongly nonexpansive mappings in reflexive Banach spaces with applications

We investigate strong convergence for Bregman strongly nonexpansive mappings by modifying Halpern and Mann’s iterations in the framework of a reflexive Banach space. As applications, we apply our main result to problems of finding zeros of maximal monotone operators and equilibrium problems in reflexive Banach spaces.MSC:47H05, 47H09, 47J25.

A new iteration process for equilibrium, variational inequality, fixed point problems, and zeros of maximal monotone operators in a Banach space

In this article, a new iterative process is introduced to approximate a common element of a fixed point set, the solutions of equilibrium problems, the solution set of variational inequality problems, and the set of zeros of maximal monotone operators in a uniformly smooth and strictly convex Banach space by using a hybrid projection method. Also, we prove new strong convergence theorems for this proposed iterative precess in a Banach space.MSC:47H05, 47H09, 47H10.

Hybrid Projection Algorithm for Two Countable Families of Hemirelatively Nonexpansive Mappings and Applications

Two countable families of hemirelatively nonexpansive mappings are considered based on a hybrid projection algorithm. Strong convergence theorems of iterative sequences are obtained in an uniformly convex and uniformly smooth Banach space. As applications, convex feasibility problems, equilibrium problems, variational inequality problems, and zeros of maximal monotone operators are studied.

The hybrid projection algorithm for finding the common fixed points of nonexpansive mappings and the zeroes of maximal monotone operators in Banach spaces

The proposal of this article is to construct a new modified block by using the hybrid projection method and prove the strong convergence theorem for this method, which include the fixed point set of an infinite family of weak relatively nonexpansive mappings and zeroes of a finite family of maximal monotone operators in a uniformly smooth and strictly convex Banach space with the Kadec–Klee property. The results presented in this article improve and generalize some well-known results in the literature.

Halpern’s iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces

We investigate strong convergence for Bregman strongly nonexpansive mappings by Halpern’s iteration in the framework of reflexive Banach spaces. Using the obtained result, convergence for a family of Bregman strongly nonexpansive mappings is also discussed. As applications, we apply our main result to problems of finding zeros of maximal monotone operators and equilibrium problems in reflexive Banach spaces.

Hölder Metric Subregularity with Applications to Proximal Point Method

This paper is mainly devoted to the study and applications of Hölder metric subregularity (or metric $q$-subregularity of order $q\in(0,1]$) for general set-valued mappings between infinite-dimensional spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive neighborhood and point-based sufficient conditions as well as necessary conditions for $q$-metric subregularity with evaluating the exact subregularity bound, which are new even for the conventional (first-order) metric subregularity in both finite and infinite dimensions. In this way we also obtain new fractional error bound results for composite polynomial systems with explicit calculating fractional exponents. Finally, metric $q$-subregularity is applied to conduct a quantitative convergence analysis of the classical proximal point method (PPM) for finding zeros of maximal monotone operators on Hilbert spaces.