Article
Mathematics, Applied
Nuttapol Pakkaranang, Poom Kumam, Ching-Feng Wen, Jen-Chih Yao, Yeol Je Cho
Summary: This paper shows the existence of solutions for convex minimization problems and common fixed point problems in CAT(1) spaces under certain conditions, proposing a modified proximal point algorithm. Additionally, applications for these problems in CAT(kappa) spaces with bounded positive real number kappa are given. The results improve and generalize many recent important findings in the literature.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Chainarong Khunpanuk, Chanchal Garodia, Izhar Uddin, Nuttapol Pakkaranang
Summary: This article presents a new modified proximal point algorithm in the framework of CAT(1) spaces, which is used to solve common fixed point problem and minimization problems. Convergence results of the obtained process under some mild conditions are proved. Our results extend and improve several corresponding results of the existing literature.
Article
Operations Research & Management Science
Shih-sen Chang, J. C. Yao, Ching-Feng Wen, Li Juan Qin
Summary: This paper aims to find a common element in the intersection of the set of zeros of the inclusion problem of sum of two monotone mappings and the set of fixed points of a Bregman quasi nonexpansive mapping in a reflexive Banach space using Bregman distance and shrinking projection method. Some strong convergence theorems are proved under suitable conditions, and the results are applied to study the convex minimization problem and the variational inequality problem.
Article
Mathematics, Applied
Patrick L. Combettes, Lilian E. Glaudin
Summary: Various strategies are available to construct a common fixed point of nonexpansive operators iteratively by activating only a block of operators at each iteration. The proposed method achieves this goal and maintains convergence while updating only blocks of operators. Weak, strong, and linear convergence results are established by exploiting a connection with the theory of concentrating arrays.
ADVANCES IN NONLINEAR ANALYSIS
(2021)
Article
Mathematics
Mujahid Abbas, Hira Iqbal, Manuel De la Sen, Khushdil Ahmad
Summary: This article introduces the concept of monotone multivalued generalized (alpha, beta)-nonexpansive mappings and explores the iterative approximation of fixed points in an ordered CAT(0) space. The S-iteration algorithm is used to prove convergence results, and examples and useful results related to the proposed mapping are provided. Numerical experiments are included to illustrate and compare the convergence of the iteration scheme, with an application presented in solving integral differential equations.
Article
Mathematics, Applied
Yan Tang, Honghua Lin, Aviv Gibali, Yeol Je Cho
Summary: Nonlinear operator theory is a significant field in nonlinear functional analysis, covering various nonlinear problems in mathematics, physical sciences, and engineering. This work focuses on the problem of finding common solutions to a monotone operator equation and fixed points of a nonexpansive mapping in real Hilbert spaces. A simple inertial forward-backward splitting method derived from dynamical systems is proposed and analyzed under mild and standard assumptions. Numerical examples and comparisons with related works demonstrate the theoretical advantages and potential applicability of the proposed scheme.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Olaniyi S. Iyiola, Yekini Shehu
Summary: This paper proposes a two-point inertial proximal point algorithm for finding zero of maximal monotone operators in Hilbert spaces. The weak convergence results and non-asymptotic O(1/n) convergence rate of the proposed algorithm in a non-ergodic sense are obtained. Applications of the results to various well-known convex optimization methods, such as the proximal method of multipliers and the alternating direction method of multipliers, are given. Numerical results are provided to demonstrate the accelerating behaviors of the proposed method compared to other related methods in the literature.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics
Li-Jun Zhu, Yonghong Yao
Summary: In this paper, the split fixed point and variational inclusion problem are studied. An iterative algorithm is proposed with the help of fixed point technique, Tseng-type splitting method, and self-adaptive rule. The involved operators S and T are considered to be demicontractive operators and g is assumed to be plain monotone. Under some mild conditions, a strong convergence theorem is proved.
Article
Mathematics
Konrawut Khammahawong, Parin Chaipunya, Kamonrat Sombut
Summary: The aim of this research is to propose a new iterative procedure for approximating common fixed points of nonexpansive mappings in Hadamard manifolds, and discuss the convergence theorem of the proposed method under certain conditions. Numerical examples are provided to support the results for clarity. Furthermore, the suggested approach is applied to solve inclusion problems and convex feasibility problems.
Article
Mathematics, Applied
Sani Salisu, Poom Kumam, Songpon Sriwongsa
Summary: This paper proposes a one step convex combination of proximal point algorithms for countable collection of monotone vector fields in CAT(0) spaces, establishing convergence theorems and applying the methods to solve minimization problems, compute mean and median, and solve kinematic problems in robotic motion control. A numerical example is provided to demonstrate the efficiency and robustness of the proposed scheme compared to existing literature.
Article
Mathematics, Applied
Rahul Shukla
Summary: This paper focuses on finding solutions to constrained minimization problems and zeros of monotone operators in geodesic spaces using nonexpansive mappings. By employing the general Picard-Mann iterative method, fixed points of nonexpansive mappings are approximated under various conditions, leading to theorems regarding Delta and strong convergence.
Article
Multidisciplinary Sciences
Jinhua Zhu, Jinfang Tang, Shih-sen Chang, Min Liu, Liangcai Zhao
Summary: This paper introduces an iterative algorithm for finding a common solution to a finite family of problems on Hadamard manifolds, proving some strong convergence theorems. The results extend recent findings in literature.
Article
Operations Research & Management Science
Shin-ya Matsushita
Summary: This paper addresses the problem of finding the resolvent of the sum of two maximal monotone operators and introduces a new mapping for this purpose. It proves the existence of a solution using the fixed point property of the mapping, and proposes a splitting method for solving the problem in a real Hilbert space, demonstrating strong convergence of the generated sequences to the solution under certain assumptions. The efficiency of the method is illustrated through convergence rate analysis and application to optimization problems.
Article
Operations Research & Management Science
Getahun B. Wega, Habtu Zegeye, Oganeditse A. Boikanyo
Summary: This paper studies the method of approximation for zeros of the sum of a finite family of maximal monotone mappings in Banach spaces. It establishes strong convergence results of the proposed approximation method under certain conditions. It also provides applications to minimization problems and a numerical example to support the main result.
Article
Operations Research & Management Science
Feng Xue
Summary: Based on degenerate proximal point analysis, this study demonstrates that the Douglas-Rachford splitting can be reduced to a well-defined resolvent, but it generally fails to be a proximal mapping. This extends the recent findings of [Bauschke, Schaad and Wang. Math. Program. 2018;168:55-61] to a more general setting. The concepts and consequences related to maximal and cyclic monotonicity are also examined, which proves to be crucial for analyzing various operator splitting algorithms.
