Contraction of the proximal mapping and applications to the equilibrium problem

In this paper, we introduce a new definition of Lipschitz-type continuity of a bifunction. Using this definition, we prove the contraction of the proximal mapping and apply it to the equilibrium problem over the fixed-point set of a nonexpansive mapping. We present a new algorithm for this problem. Under classical conditions, the convergence of the algorithm is proved. Finally, we present some numerical results for the proposed algorithm.     

Quantitative asymptotic regularity results for the composition of two mappings

In this paper, we use techniques which originate from proof mining to give rates of asymptotic regularity and metastability for a sequence associated to the composition of two firmly nonexpansive mappings.     

正交约束优化问题的一阶算法

带有正交约束的矩阵优化问题在材料计算、统计及数据分析等领域中有着广泛的应用.由于正交约束的可行域是Stiefel流形,一直以来流形上的优化方法是求解这一问题的主要方法.近年来,随着实际应用问题所要求的变量规模的扩大,传统的流形优化方法在计算上的劣势显现出来,而一些迭代简单、收敛快的新算法逐渐被提出.通过收缩方法、非收缩可行方法、不可行方法三个类别分别来介绍求解带有正交约束的矩阵优化问题的最新算法.通过分析这些方法的主要特性,以及应用问题的要求,对这类问题算法设计的研究进行了展望.     

The free retraction of natural rubber: A momentum-based model

The free retraction of vulcanised strips of natural rubber released from simple uniaxial deformation is studied using high speed cinematography in the context of a simple momentum theory. Good agreement between the theory and experiment is observed when vulcanisates are released from stresses below 1 MPa, which corresponds to tensile strains rates below 1 × 10³ s⁻¹. Above this critical stress and corresponding strain rate value, an increasing dispersion is observed in the form of slowing down of the characteristic retraction pulse, and also by a relaxation of strain ahead of the pulse front (a dispersion of the pulse). Holding samples at high strains for an extended period of time prior to releasing results in a further, significant retardation of the retraction pulse velocity. These effects are related to the increasing non-linearity of high strain rate retraction stress–strain behaviour. Energy balance arguments show that the dispersion of the retraction pulse is a prerequisite for pulse propagation, and that its magnitude underpins the deviation from the momentum model outlined in this paper.     

On the Problem of Dissipative Perturbations of Nonexpansive Mappings

ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｅｌｉｍｉｎａｒｉｅｓＴｈｒｏｕｇｈｏｕｔｔｈｉｓｐａｐｅｒ,ｗｅａｓｓｕｍｅｔｈａｔＨｉｓａｒｅａｌＨｉｌｂｅｒｔｓｐａｃｅ ,〈· ,·〉ｉｓｔｈｅｉｎｎｅｒｐｒｏｄｕｃｔｏｎＨ ,ＰｉｓａｃｏｎｅｉｎＨ .ＢｙｖｉｒｔｕｅｏｆｔｈｅｃｏｍｅＰ ,ａｎｏｒｄｅｒ“≤”ｉｓｉｎｄｕｃｅｄｉｎＨ ,ｉ.ｅ .,ｆｏｒａｎｙｇｉｖｅｎｘ,ｙ∈Ｈ ,ｘ≤ｙｉｆａｎｄｏｎｌｙｉｆｙ -ｘ∈Ｐ .Ａｍｕｌｔｉ＿ｖａｌｕｅｄｍａｐｐｉｎｇＡ :Ｄ(Ａ) Ｈ → 2 Ｈｉｓｓａｉｄｔｏｂｅａｃｃｒｅｔｉｖ…     

Strong Convergence Theorems for Mixed Typ e Asymptotically Nonexpansive Mappings

The purpose of this paper is to study a new two-step iterative scheme with mean errors of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappin...     

Fixed points of multivalued quasi-nonexpansive mappings using a faster iterative process

In this article, we prove some strong and weak convergence theorems for quasi-nonexpansive multivalued mappings in Banach spaces. The iterative process used is independent of Ishikawa iterative process and converges faster. Some examples are provided to validate our results. Our results extend and unify some results in the contemporary literature.     

Spreading and retraction dynamics of smectic liquid crystal domains at interfaces

The spreading of a liquid drop over liquid subphase can be driven by change in interfacial tension mediated through a surfactant, volatile solvent or photoinduced reaction. In contrast to the spreading dynamics of a liquid drop, a liquid crystal drop with anisotropic structure can lead to interesting behaviour due to its viscoelasticity and anchoring at the interfaces. Recently, we have reported studies on unusual spreading and retraction dynamics of a smectic domain doped with a fluorescent dye in the collapsed state of a Langmuir monolayer. Under epifluorescence microscope, during excitation, a stack of layers of the dye-doped smectic domain gets sheared causing the domain to spread asymmetrically. Further, due to line tension, the domain transforms into a circular shape. We also find the domain size to be about twice that of the initial size. Interestingly, in the absence of excitation, the domain retracts to a smaller area. During retraction of the domain, successive generation of edge dislocation loops arising from a nucleus results in an increase in the domain thickness. The dynamics of spreading and retraction of the domain can be understood by invoking changes in the spreading coefficient due to photoinduced modification of the interfacial tension.     

Hybrid projection method for generalized mixed equilibrium problem and fixed point problem of infinite family of asymptotically quasi-?-nonexpansive mappings in Banach spaces

In this paper, we consider a hybrid projection method for finding a common element in the set of fixed points of a infinite family of asymptotically quasi-?-nonexpansive mappings and in the set of solutions of a generalized mixed equilibrium problem. Some strong convergence theorems of common elements are established in a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property. The results presented in the paper improve and extend some recent results.     

Strong Convergence Theorems for the Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in the Intermediate Sense in Hilbert Spaces

Abstract

The aim of this paper is to introduce the generalized viscosity implicit rules of one asymptotically nonexpansive mapping in the intermediate sense in Hilbert spaces. We obtain some strong convergence theorems under certain assumptions imposed on the parameters. We also give a numerical example to support our main results. The results obtained in this paper improve and extend many recent ones in this culture.