Strong Convergence Theorems for Nonexpansive Mappings by Viscosity Approximation Methods in Banach Spaces

In this paper, we introduce a modified Ishikawa iterative process for a pair of nonexpansive mappings and obtain a strong convergence theorem in the framework of uniformly Banach spaces. Our results improve and extend the recent ones announced by Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51-60], Xu [H.K. Xu, Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298 (2004) 279-291] and some others.</p

Strong Convergence Theorems for a Finite Family of Nonexpansive Mappings

We modified the classic Mann iterative process to have strong convergence theorem for a finite family of nonexpansive mappings in the framework of Hilbert spaces. Our results improve and extend the results announced by many others

Hybrid Algorithm for Common Fixed Points of Uniformly Closed Countable Families of Hemirelatively Nonexpansive Mappings and Applications

The authors have obtained the following results: (1) the definition of uniformly closed countable family of nonlinear mappings, (2) strong convergence theorem by the monotone hybrid algorithm for two countable families of hemirelatively nonexpansive mappings in a Banach space with new method of proof, (3) two examples of uniformly closed countable families of nonlinear mappings and applications, (4) an example which is hemirelatively nonexpansive mapping but not weak relatively nonexpansive mapping, and (5) an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping. Therefore, the results of this paper improve and extend the results of Plubtieng and Ungchittrakool (2010) and many others

Composite Implicit General Iterative Process for a Nonexpansive Semigroup in Hilbert Space

Let C be nonempty closed convex subset of real Hilbert space H. Consider C a nonexpansive semigroup ℑ={T(s):s≥0} with a common fixed point, a contraction f with coefficient 00. Let 0<γ<γ¯/α. It is proved that the sequence {xn} generated iteratively by xn=(I−αnA)(1/tn)∫0tnT(s)ynds+αnγf(xn),yn=(I−βnA)xn+βnγf(xn) converges strongly to a common fixed point x∗∈F(ℑ) which solves the variational inequality 〈(γf−A)x∗,z−x∗〉≤0 for all z∈F(ℑ)

Wiener-Hopf Equations Technique for General Variational Inequalities Involving Relaxed Monotone Mappings and Nonexpansive Mappings

We show that the general variational inequalities are equivalent to the general Wiener-Hopf equations and use this alterative equivalence to suggest and analyze a new iterative method for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality involving multivalued relaxed monotone operators. Our results improve and extend recent ones announced by many others

Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings

The purpose of this article is to prove strong convergence theorems for fixed points of closed hemi-relatively nonexpansive mappings. In order to get these convergence theorems, the monotone hybrid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S. Matsushita and W. Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping. The results of this paper modify and improve the results of S. Matsushita and W. Takahashi (2005), and some others

Cloud-Magnetic Resonance Imaging System: In the Era of 6G and Artificial Intelligence

Magnetic Resonance Imaging (MRI) plays an important role in medical diagnosis, generating petabytes of image data annually in large hospitals. This voluminous data stream requires a significant amount of network bandwidth and extensive storage infrastructure. Additionally, local data processing demands substantial manpower and hardware investments. Data isolation across different healthcare institutions hinders cross-institutional collaboration in clinics and research. In this work, we anticipate an innovative MRI system and its four generations that integrate emerging distributed cloud computing, 6G bandwidth, edge computing, federated learning, and blockchain technology. This system is called Cloud-MRI, aiming at solving the problems of MRI data storage security, transmission speed, AI algorithm maintenance, hardware upgrading, and collaborative work. The workflow commences with the transformation of k-space raw data into the standardized Imaging Society for Magnetic Resonance in Medicine Raw Data (ISMRMRD) format. Then, the data are uploaded to the cloud or edge nodes for fast image reconstruction, neural network training, and automatic analysis. Then, the outcomes are seamlessly transmitted to clinics or research institutes for diagnosis and other services. The Cloud-MRI system will save the raw imaging data, reduce the risk of data loss, facilitate inter-institutional medical collaboration, and finally improve diagnostic accuracy and work efficiency.Comment: 4pages, 5figures, letter