A Geometrical Look at Iterative Methods for Operators with Fixed Points

ABSTRACT

We distinguish classes of operators T with fixed points on a real Hilbert space by comparing the distances of a point x and its image Tx to the (set of) fixed points of T; this leads to a ranking of those classes, based on a nonnegative parameter. That same parameter also lets us conclude about the sign of and an upper bound for a characteristic inner product result that arises in iterative processes to obtain a common fixed point of a set of operators. We use that parameter as the starting point for a geometrically-inclined study of specific iterative algorithms intended to find a common fixed point of operators belonging to such class.     

General local convergence theory for a class of iterative processes and its applications to Newton’s method

General local convergence theorems with order of convergence r≥1$r\ge 1$ are provided for iterative processes of the type _xn+1=T_xn$x_{n+1}=T{x}_n$, where T:D⊂X→X$T:D\subset X\to X$ is an iteration function in a metric space X$X$. The new local convergence theory is applied to Newton iteration for simple zeros of nonlinear operators in Banach spaces as well as to Schröder iteration for multiple zeros of polynomials and analytic functions. The theory is also applied to establish a general theorem for the uniqueness ball of nonlinear equations in Banach spaces. The new results extend and improve some results of [K. Do?ev, Über Newtonsche Iterationen, C. R. Acad. Bulg. Sci. 36 (1962) 695–701; J.F. Traub, H. Wo?niakowski, Convergence and complexity of Newton iteration for operator equations, J. Assoc. Comput. Mach. 26 (1979) 250–258; S. Smale, Newton’s method estimates from data at one point, in: R.E. Ewing, K.E. Gross, C.F. Martin (Eds.), The Merging of Disciplines: New Direction in Pure, Applied, and Computational Mathematics, Springer, New York, 1986, pp. 185–196; P. Tilli, Convergence conditions of some methods for the simultaneous computation of polynomial zeros, Calcolo 35 (1998) 3–15; X.H. Wang, Convergence of Newton’s method and uniqueness of the solution of equations in Banach space, IMA J. Numer. Anal. 20 (2000) 123–134; I.K. Argyros, J.M. Gutiérrez, A unified approach for enlarging the radius of convergence for Newton’s method and applications, Nonlinear Funct. Anal. Appl. 10 (2005) 555–563; M. Giusti, G. Lecerf, B. Salvy, J.-C. Yakoubsohn, Location and approximation of clusters of zeros of analytic functions, Found. Comput. Math. 5 (3) (2005) 257–311], and others.     

An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings

In this paper, we propose an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudo-contraction mapping in the setting of real Hilbert spaces. We establish some weak and strong convergence theorems of the sequences generated by our proposed scheme. Our results combine the ideas of Marino and Xu’s result [G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–346], and Takahashi and Takahashi’s result [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515]. In particular, necessary and sufficient conditions for strong convergence of our iterative scheme are obtained.     

一个新的网格多翅膀混沌系统及其电路实现

提出了一个新的三维二次自治混沌系统,与大多数广义Lorenz系统一样,该系统只能产生双翅膀吸引子.依据该双翅膀混沌系统平衡点和吸引子的拓扑结构,设计合适的非线性函数可以将其改进为一个产生网格多翅膀吸引子的混沌系统.对该网格多翅膀混沌系统的基本动力学特性进行了分析,证实了多翅膀吸引子的混沌特性.最后设计了混沌电路,给出了多翅膀混沌吸引子的电路仿真结果,证实了理论设计与电路实现的一致性.     

A regularized trace formula and oscillation of eigenfunctions of a Sturm‐Liouville operator with retarded argument at 2 points of discontinuity

In this study, first, a formula for regularized sums of eigenvalues of a Sturm‐Liouville problem with retarded argument at 2 points of discontinuity which contains a spectral parameter in the boundary conditions is obtained. After that, oscillation properties of the related problem is investigated. Finally, under the condition that a subset of nodal points is dense in definition set, the potential function is determined.     

Necessary and Sufficient Conditions for Weak Quasi Efficiency in Nonsmooth Multiobjective Problems

In this article, a multiobjective problem with a feasible set defined by inequality, equality and set constraints is considered, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given and the relations between them are analyzed. We establish Kuhn-Tucker and strong Kuhn-Tucker necessary optimality conditions for (weak) quasi e?ciency in terms of the Clarke subdifferential. By using two new classes of generalized convex functions, su?cient conditions for local (weak) quasi e?cient are also provided. Furthermore, we study the Mond-Weir type dual problem and establish weak, strong and converse duality results.     

用拓扑指数S_i研究碳氢化合物的沸点(Ⅲ)

定义了拓扑指数Ｓｉ（ｉ＝１，２，３，４），并选取包含双、三键、环及苯环的碳氢化合物为研究对象，计算了这些化合物的拓扑指数Ｓｉ，同时用Ｓｉ相关了它们的沸点．结果较文献值有所改善．并从Ｓｉ中筛选出了Ｓ     

衍生气相色谱-质谱法测定水中氯代酸性除草剂

采用H2SO4加入法改变了酸性除草剂在水中的存在状态,对碱性活化点的存在与消除进行了分析,同时对不同试剂的萃取效率进行了对比,建立了衍生气相色谱-质谱法测定水中10种氯代酸性除草剂的方法。化合物检出限均低于10.0 ng/L,回收率范围为60%～120%。     

Stacking Control by Molecular Symmetry of Sterically Protected Phthalocyanines

The synthesis and characterization of two phthalocyanine (Pc) structural isomers, 1 and 2, in which four 2,6-di(hexyloxy)phenyl units were attached directly to the 1,8,15,22- or 1,4,15,18-positions of the Pc rings, are described. Both Pcs 1 and 2 exhibited low melting points, i.e., 120 and 130 °C respectively, due to the reduction in intermolecular π-π interaction among the Pc rings caused by the steric hindrance of 2,6-dihexyloxybenzene units. The thermal behaviors were investigated with temperature-controlled polarizing optical microscopy, differential scanning calorimetry, powder X-ray diffraction, and absorption spectral analyses. Pc 1, having C_4h molecular symmetry, organized into a lamellar structure containing lateral assemblies of Pc rings. In contrast, the other Pc 2 revealed the formation of metastable crystalline phases, including disordered stacks of Pcs due to rapid cooling from a melted liquid.