Hierarchical Fixed Point Problem Research Articles

Shrinking projection method for hierarchical fixed point problems on Hadamard manifolds

Shrinking projection method for hierarchical fixed point problems on Hadamard manifolds

A New Inertial-Projection Method for Solving Split Generalized Mixed Equilibrium and Hierarchical Fixed Point Problems

In this paper, we introduce a new iterative algorithm of inertial form for approximating the common solution of Split Generalized Mixed Equilibrium Problem (SGMEP) and Hierarchical Fixed Point Problem (HFPP) in real Hilbert spaces. Motivated by the subgradient extragradient method, we incorporate the inertial technique to accelerate the convergence of the proposed method. Under standard and mild assumption of monotonicity and lower semicontinuity of the SGMEP and HFPP associated mappings, we establish the strong convergence of the iterative algorithm. Some numerical experiments are presented to illustrate the performance and behaviour of our method as well as comparing it with some related methods in the literature.

Novel regularized dynamical systems for solving hierarchical fixed point problems

In this paper, we study some Krasnoselskii-Mann type dynamical systems in solving fixed point problems. The first one can be regarded as a continuous version of the Krasnoselskii-Mann iterations. We prove that the solution of this dynamical system converges weakly to a fixed point of the involving mapping. Next, we focus our attention on a regularized Krasnoselskii-Mann type dynamical system. Besides proving existence and uniqueness of strong global solutions, we show that the generated trajectories converge strongly to a unique solution of a variational inequality over the fixed point set. Also, we provide a convergence rate analysis for the regularized dynamical system.

Strong convergence results of split equality hierarchical fixed point problems

The primary objective of this study is to approximate a nontrivial solution to a significant extension of the hierarchical fixed point problem, known as the split equality hierarchical fixed point problem, in the context of real Hilbert spaces. We present an accelerated simultaneous Halpern‐type iterative algorithm with a step size independent of the operator norms. This method combines accelerated extrapolation and the Halpern‐type iterative algorithm. Moreover, we analyze that the sequence pair induced by the suggested algorithm strongly converges to some solution to the split equality hierarchical fixed point problem under some simple control sequence conditions. Furthermore, we provide numerical experiments to demonstrate the rationality and superiority of our algorithm. Some notable recent findings in this field have been improved, generalized, and extended as an outcome of this paper.

Generalized viscosity approximation method for solving split generalized mixed equilibrium problem with application to compressed sensing

<abstract><p>In this study, we establish a new inertial generalized viscosity approximation method and prove that the resulting sequence strongly converges to a common solution of a split generalized mixed equilibrium problem, fixed point problem for a finite family of nonexpansive mappings and hierarchical fixed point problem in real Hilbert spaces. As an application, we demonstrate the use of our main finding in compressed sensing in signal processing. Additionally, we include numerical examples to evaluate the efficiency of the suggested method and then conduct a comparative analysis of its efficiency with different methods. Our findings can be used in a variety of contexts to improve results.</p></abstract>

Modified Mann-Type Subgradient Extragradient Rules for Variational Inequalities and Common Fixed Points Implicating Countably Many Nonexpansive Operators

In a real Hilbert space, let the CFPP, VIP, and HFPP denote the common fixed-point problem of countable nonexpansive operators and asymptotically nonexpansive operator, variational inequality problem, and hierarchical fixed point problem, respectively. With the help of the Mann iteration method, a subgradient extragradient approach with a linear-search process, and a hybrid deepest-descent technique, we construct two modified Mann-type subgradient extragradient rules with a linear-search process for finding a common solution of the CFPP and VIP. Under suitable assumptions, we demonstrate the strong convergence of the suggested rules to a common solution of the CFPP and VIP, which is only a solution of a certain HFPP.

A simple proof for Kazmi et al.'s iterative scheme

In this paper, a simple proof for the existence iterative scheme using two Hilbert spaces due to Kazmi et al.[K.R. Kazmi, R. Ali, M. Furkan, Hybrid iterative method for split monotone variational inclusion problemand hierarchical fixed point problem for a finite family of nonexpansive mappings, Numer. Algor., 2017] isprovided.

Solving the split equality hierarchical fixed point problem

Solving the split equality hierarchical fixed point problem

Algorithms of common solutions to modified generalized system of variational inclusion problem and hierarchical fixed point problem

This manuscript deals with two problems : the first one is a new problem of the system of variational inclusion that is called modified generalized system of variational inclusion problem(MGSVIP) and the other one is a hierarchical fixed point problem in the framework of real Hilbert space. We establish the important lemma that show the relation between fixed point of nonlinear mapping and solution of MGSVIP for proving the main theorem. To approximate the common solution of these problems, we design an iterative scheme under suitable conditions on parameters. A strong convergence result for the proposed iterative scheme is proved. Applying our main result, we prove strong convergence theorems of the modification system of variational inequalities problem and variational inclusion problem. Moreover, we give the numerical example for supporting our results.

Iterative algorithms of common solutions for a hierarchical fixed point problem, a system of variational inequalities, and a split equilibrium problem in Hilbert spaces

In this paper, we suggest and analyze an iterative algorithm to approximate a common solution of a hierarchical fixed point problem for nonexpansive mappings, a system of variational inequalities, and a split equilibrium problem in Hilbert spaces. Under some suitable conditions imposed on the sequences of parameters, we prove that the sequence generated by the proposed iterative method converges strongly to a common element of the solution set of these three kinds of problems. The results obtained here extend and improve the corresponding results of the relevant literature.

Krasnoselski\u2013Mann-type inertial method for solving split generalized mixed equilibrium and hierarchical fixed point problems

In this paper, we present Krasnoselski–Mann-type inertial method for solving split generalized mixed equilibrium and hierarchical fixed point problems for k-strictly pseudocontractive nonself-mappings. We establish that the weak convergence of the proposed accelerated iterative method with inertial terms involves a step size which does not require any prior knowledge of the operator norm under several suitable conditions in Hilbert spaces. Finally, the application to a Nash–Cournot oligopolistic market equilibrium model is discussed, and numerical examples are provided to demonstrate the effectiveness of our iterative method.

An iterative scheme for split equality equilibrium problems and split equality hierarchical fixed point problem

This paper deals with a split equality equilibrium problem for pseudomonotone bifunctions and a split equality hierarchical fixed point problem for nonexpansive and quasinonexpansive mappings. We suggest and analyze an iterative scheme where the stepsizes do not depend on the operator norms, the so-called simultaneous projected subgradient-proximal iterative scheme for approximating a common solution of the split equality equilibrium problem and the split equality hierarchical fixed point problem. Further, we prove a weak convergence theorem for the sequences generated by this scheme. Furthermore, we discuss some consequences of the weak convergence theorem. We present a numerical example to justify the main result.

Modified Krasnoselski–Mann type iterative algorithm with strong convergence for hierarchical fixed point problem and split monotone variational inclusions

In this paper, we introduce a modified Krasnoselski–Mann type method for solving the hierarchical fixed point problem and split monotone variational inclusions in real Hilbert spaces. We prove that the sequence generated by the modified algorithm converges strongly to a common element of the set of hierarchical fixed point problem and split monotone variational inclusions only basing on the coefficients. Our results extend and improve the weak convergence results of Kazmi et al. (Kazmi et al., 2018) and others. Moreover, an application and its numerical examples illustrate the feasibility and strong convergence of the proposed method and results.

Inertial KM-type extragradient scheme for solving a variational inequality and a hierarchical fixed point problems

We propose an inertial KM-type extragradient scheme to approximate a common solution of a variational inequality problem and a hierarchical fixed point problem for nonexpansive mappings. This scheme generalizes and unifies a number of known iterative schemes. Furthermore, we discuss the weak convergence for the proposed scheme. We also discuss an example to illustrate the main theorem.

Modified Krasnoselski\u2013Mann iterative method for hierarchical fixed point problem and split mixed equilibrium problem

In this paper, we introduce a modified Krasnoselski–Mann type iterative method for capturing a common solution of a split mixed equilibrium problem and a hierarchical fixed point problem of a finite collection of k-strictly pseudocontractive nonself-mappings. Many of the algorithms for solving the split mixed equilibrium problem involve a step size which depends on the norm of a bounded linear operator. Since the computation of the operator norm is very difficult, we formulate our iterative algorithm in such a way that the implementation of the proposed algorithm does not require any prior knowledge of operator norm. Weak convergence results are established under mild conditions. We also establish strong convergence results for a certain class of hierarchical fixed point and split equilibrium problem. Our results generalize some important results in the recent literature.

The Split Various Variational Inequalities Problems for Three Hilbert Spaces

There are many methods for finding a common solution of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem in the setting of real Hilbert spaces. They proved the strong convergence theorem. Many split feasibility problems are generated in real Hillbert spaces. The open problem is proving a strong convergence theorem of three Hilbert spaces with different methods from the lasted method. In this research, a new split variational inequality in three Hilbert spaces is proposed. Important tools which are used to solve classical problems will be developed. The convergence theorem for finding a common element of the set of solution of such problems and the sets of fixed-points of discontinuous mappings has been proved.

Systems of unrelated generalized mixed equilibrium problems and unrelated hierarchical fixed point problems in Hilbert space

Systems of unrelated generalized mixed equilibrium problems and unrelated hierarchical fixed point problems in Hilbert space

Hybrid iterative scheme for solving split equilibrium and hierarchical fixed point problems

We suggest and analyze a hybrid projected subgradient–proximal iterative scheme to approximate a common solution of a split equilibrium problem for pseudomonotone and monotone bifunctions and a hierarchical fixed point problem for nonexpansive and quasi-nonexpansive mappings. We prove that sequences generated by the proposed scheme converge weakly to a common solution of these problems. Further, we discuss some consequences of the main result and a numerical example for the proposed scheme.

Hybrid viscosity approximation methods for systems of variational inequalities and hierarchical fixed point problems

In this paper, we propose an implicit iterative method and an explicit iterative method for solving a general system of variational inequalities with a hierarchical fixed point problem constraint for an infinite family of nonexpansive mappings. We show that the proposed algorithms converge strongly to a solution of the general system of variational inequalities, which is a unique solution of the hierarchical fixed point problem.

Strong convergence of inertial Mann algorithms for solving hierarchical fixed point problems

Strong convergence of inertial Mann algorithms for solving hierarchical fixed point problems