Banach operator pairs,common fixed-points,invariant approximations,and ∗ -nonexpansive multimaps

Common fixed-point results are proved for I$I$-nonexpansive maps, Ciric type maps, and ∗-nonexpansive multimaps. Invariant approximation results are obtained for these types of maps as applications. Our results extend or generalize several known results including the recent results of Chen and Li [J. Chen, Z. Li, Common fixed-points for Banach operators pairs in best approximation, J. Math. Anal. Appl. 336 (2007) 1466–1475]. In particular, we show that the results of Chen and Li on Banach operator pairs are particular cases of the results of Al-Thagafi and Shahzad [M.A. Al-Thagafi, N. Shahzad, Noncommuting selfmaps and invariant approximations, Nonlinear Anal. 64 (2006) 2778–2786].     

Subweakly Biased Pairs and Jungck Contractions with Applications

The existence of common fixed point results for a weakly biased pair under Jungck contraction is established. We define a new class of subweakly biased pair which properly contains the classes of C _q -commuting, pointwise R-subweakly commuting and R-subweakly commuting maps and establish common fixed point results for this class of maps. As applications several invariant best approximation results are proved which unify, extend, and complement the well-known results.     

&rho|混合随机变量加权和的收敛性

该文研究了ρ 混合随机变量加权和的强大数律及完全收敛性, 获得了一些新的结果. 该文的结果推广和改进了Bai 等^[1]及Baum 等^[18] 在 i.i.d. 情形时相应的结果, 也推广和改进了Volodin 等^[4]在实值独立时相应的结果. 该文还得到了一关于任意随机变量阵列加权和的完全收敛性定理.     

Jacobi Spectral Methods for Volterra-Urysohn Integral Equations of Second Kind with Weakly Singular Kernels

Abstract

In this article, we discuss Jacobi spectral Galerkin and iterated Jacobi spectral Galerkin methods for Volterra-Urysohn integral equations with weakly singular kernels and obtain the convergence results in both the infinity and weighted L²-norm. We show that the order of convergence in iterated Jacobi spectral Galerkin method improves over Jacobi spectral Galerkin method. We obtain the convergence results in two cases when the exact solution is sufficiently smooth and non-smooth. For finding the improved convergence results, we also discuss Jacobi spectral multi-Galerkin and iterated Jacobi spectral multi-Galerkin method and obtain the convergence results in weighted L²-norm. In fact, we prove that the iterated Jacobi spectral multi-Galerkin method improves over iterated Jacobi spectral Galerkin method. We provide numerical results to verify the theoretical results.     

Ekeland’s variational principle for vectorial multivalued mappings in a uniform space

In this paper, we establish Ekeland’s variational principle and an equilibrium version of Ekeland’s variational principle for vectorial multivalued mappings in the setting of separated, sequentially complete uniform spaces. Our approaches and results are different from those in Chen et al. (2008), Hamel (2005), and Lin and Chuang (2010) [13], [14] and [15]. As applications of our results, we study vectorial Caristi’s fixed point theorems and Takahashi’s nonconvex minimization theorems for multivalued mappings and their equivalent forms in a separated, sequentially complete uniform space. We also apply our results to study maximal element theorems, which are unified methods of several variational inclusion problems. Our results contain many known results in the literature Fang (1996) [21], and will have many applications in nonlinear analysis.     

Approximation by rational operators in Lp spaces

The present paper introduces a kind of Kantorovich type Shepard operators. Complete results including direct and converse results, equivalence results are established. As Della Vecchia and Mastroianni ([4], [7]) did, our results involve a weighted modulus of smoothness related to step‐functions φ (x) vanishing algebraically at the endpoints ±1 (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized I-nonexpansive selfmaps and invariant approximations

Common fixed point results for new classes of noncommuting selfmaps satisfying generalized I-contraction or I-nonexpansive type conditions are established. We apply them to obtain several invariant approximation results which unify, extend, and complement the well-known results.     

t-Schreier Domains

We study, under the name t-Schreier, the class of those integral domains whose group of t-invertible t-ideals satisfies the Riesz interpolation property. The so-called Prüfer v-multiplication domains (PVMDs) and Prüfer domains are special cases of t-Schreier domains. We show that, while a number of results known for Prüfer domains and PVMDs hold for these domains, the t-Schreier domains have a remarkable capability of unifying various results in that the results proved for t-Schreier domains can also be translated to results on pre-Schreier domains and hence on GCD and Bezout domains.     

Applications of time scale symplectic systems without normality

Recently, the authors obtained new characterizations of the positivity and nonnegativity of a time scale quadratic functional F with separable endpoints related to a time scale symplectic system (S). In these results, the assumption of normality is absent. In this paper we present applications of such results. Namely, without assuming normality we derive Sturmian comparison theorems, results for general jointly varying endpoints, and characterizations of the positivity of F via the corresponding time scale Riccati equation, a certain perturbed quadratic functional, and a time scale Riccati inequality. These results generalize and unify many recent as well as classical ones.     

Coupled common fixed point theorems for w-compatible mappings in ordered cone metric spaces

We establish coupled coincidence point results for mixed g-monotone mappings under general contractive conditions in partially ordered cone metric spaces over solid cones. We also present results on existence and uniqueness of coupled common fixed points. Our results generalize, extend and unify several well known comparable results in the literature. To illustrate our results and to distinguish them from the earlier ones, we equip the paper with examples.     

On the convergence of the midpoint method

The midpoint method is an iterative method for the solution of nonlinear equations in a Banach space. Convergence results for this method have been studied in [3, 4, 9, 12]. Here we show how to improve and extend these results. In particular, we use hypotheses on the second Fréchet derivative of the nonlinear operator instead of the third-derivative hypotheses employed in the previous results and we obtain Banach space versions of some results that were derived in [9, 12] only in the real or complex space. We also provide various examples that validate our results.      

Monochromatic and Heterochromatic Subgraphs in Edge-Colored Graphs - A Survey

Nowadays the term monochromatic and heterochromatic (or rainbow, multicolored) subgraphs of an edge-colored graph appeared frequently in literature, and many results on this topic have been obtained. In this paper, we survey results on this subject. We classify the results into the following categories: vertex-partitions by monochromatic subgraphs, such as cycles, paths, trees; vertex partition by some kinds of heterochromatic subgraphs; the computational complexity of these partition problems; some kinds of large monochromatic and heterochromatic subgraphs. We have to point out that there are a lot of results on Ramsey type problem of monochromatic and heterochromatic subgraphs. However, it is not our purpose to include them in this survey because this is slightly different from our topics and also contains too large amount of results to deal with together. There are also some interesting results on vertex-colored graphs, but we do not include them, either. Supported by NSFC, PCSIRT and the “973” program.     

A new stability number of the Bresse‐Cattaneo system

In this paper, we consider the Bresse‐Cattaneo system with a frictional damping term and prove some optimal decay results for the L²‐norm of the solution and its higher order derivatives. In fact, we show that there is a completely new stability number δ that controls the decay rate of the solution. To prove our results, we use the energy method in the Fourier space to build some very delicate Lyapunov functionals that give the desired results. We also prove the optimality of the results by using the eigenvalues expansion method. In addition, we show that for the absence of the frictional damping term, the solution of our problem does not decay at all. This result improves some early results     

The Fermat–Torricelli Problem in Normed Planes and Spaces

We investigate the Fermat–Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat–Torricelli locus in a geometric way. We present many new results, as well as give an exposition of known results that are scattered in various sources, with proofs for some of them. Together, these results can be considered to be a minitheory of the Fermat–Torricelli problem in Minkowski spaces and especially in Minkowski planes. This demonstrates that substantial results about locational problems valid for all norms can be found using a geometric approach.     

Abundant semigroups with a quasi-ideal quasi-adequate transversal

Characterization theorems for abundant sernigroups having a quasi-ideal quasi-adequate transversal are obtained. Our results generalize and amplify the related results of Satio on regular semigroup obtained in 1985 and Kong obtained in 2007 respectively. Some recent results on this topic given by Guo-Shum are strengthened. In particular, the structure of such an abundant semigroup is described.     

Geometric Heat Comparison Criteria for Riemannian Manifolds

The main results of this paper are small-time heat comparison results for two points in two manifolds with characteristic functions as initial temperature distributions (Theorems 1 and 2). These results are based on the geometric concepts of (essential) distance from the complement and spherical area function. We also discuss some other geometric results about the heat development and illustrate them by examples. Mathematics subject classifications (2000): 58J35, 35K05     

On vector variational inequalities

In this paper, we study vector variational inequalities. The concept of weaklyC-pseudomonotone operator is introduced. By employing the Fan lemma, we establish several existence results. The new results extend and unify existence results of vector variational inequalities for monotone operators under a Banach space setting. In particular, existence results for the generalized vector complementarity problem with weaklyC-pseudomonotone operators in Banach space are obtained.This research was partially supported by the National Science Council of the Republic of China under Contract NSC 84-2121-M-110-008.     

一类A-调和方程的障碍问题的很弱解的全局正则性

应用Hodge分解定理,得到了非齐次A-调和方程-div(A(x,Du(x)))=f(x,u(x))对应的障碍问题很弱解的局部和全局的W~(1,q)(Ω)-正则性,其中,A(x,Du(x)),f(x,u(x))满足文中所给的条件,从而推广了相关文献中的有关结果.该结果在优化控制问题中有着广泛的应用.     

Asymptotic behaviour of the containment of certain mesh patterns

We present some results on the proportion of permutations of length n containing certain mesh patterns as n grows large, and give exact enumeration results in some cases. In particular, we focus on mesh patterns where entire rows and columns are shaded. We prove some general results which apply to mesh patterns of any length, and then consider mesh patterns of length four. An important consequence of these results is to show that the proportion of permutations containing a mesh pattern can take a wide range of values between 0 and 1.     

Periodic solutions for a kind of Liénard equation

Using inequality techniques and coincidence degree theory, new results are provided concerning the existence and uniqueness of T-periodic solutions for a Liénard equations with delay. An illustrative example is provided to demonstrate that the results in this paper hold under weaker conditions than existing results, and are more effective.