An efficient iterative method for finding common fixed point and variational inequalities in Hilbert spaces

ABSTRACT

In this paper, we investigate the problem of finding a common solution to a fixed point problem involving demi-contractive operator and a variational inequality with monotone and Lipschitz continuous mapping in real Hilbert spaces. Inspired by the projection and contraction method and the hybrid descent approximation method, a new and efficient iterative method for solving the problem is introduced. Strong convergence theorem of the proposed method is established under standard and mild conditions. Our scheme generalizes and extends some of the existing results in the literature, and moreover, its computational effort is less per each iteration compared with related works.     

Convergence results for a self-dual regularization of convex problems

We study a one-parameter regularization technique for convex optimization problems whose main feature is self-duality with respect to the Legendre–Fenchel conjugation. The self-dual technique, introduced by Goebel, can be defined for both convex and saddle functions. When applied to the latter, we show that if a saddle function has at least one saddle point, then the sequence of saddle points of the regularized saddle functions converges to the saddle point of minimal norm of the original one. For convex problems with inequality and state constraints, we apply the regularization directly on the objective and constraint functions, and show that, under suitable conditions, the associated Lagrangians of the regularized problem hypo/epi-converge to the original Lagrangian, and that the associated value functions also epi-converge to the original one. Finally, we find explicit conditions ensuring that the regularized sequence satisfies Slater's condition.     

Clouding and Hydrodynamic Behavior of Pluronic-Surfactant Interaction in Aqueous Salt Solutions

Clouding behavior of PEO-PPO-PEO and PPO-PEO-PPO block copolymers were studied in presence of sodium dodecyl sulfate (SDS) and NaCl. Extensive study of Pluronic P84 (EO₁₉PO₄₃EO₁₉) with different salts and ionic surfactants, were carried out using cloud point, viscosity and dynamic light scattering (DLS) measurements. The change in cloud point, as well as the size of P84 micelles in aqueous salt solution obeys the Hofmeister lyotropic series. Results on P84-ionic surfactant mixture indicate stronger interaction in case of SDS compared to those in presence of dodecyl trimethylammonium chloride (DTAC); here interaction seems to diminish in the presence of salts.     

Chebyshev centers and fixed point theorems

Brodskii and Milman proved that there is a point in C(K)$C(K)$, the set of all Chebyshev centers of K, which is fixed by every surjective isometry from K into K whenever K   is a nonempty weakly compact convex subset having normal structure in a Banach space. Motivated by this result, Lim et al. raised the following question namely “does there exist a point in C(K)$C(K)$ which is fixed by every isometry from K into K?”. In fact, Lim et al. proved that “if K is a nonempty weakly compact convex subset of a uniformly convex Banach space, then the Chebyshev center of K is fixed by every isometry T from K into K”. In this paper, we prove that if K   is a nonempty weakly compact convex set having normal structure in a strictly convex Banach space and F$\mathfrak{F}$ is a commuting family of isometry mappings on K   then there exists a point in C(K)$C(K)$ which is fixed by every mapping in F$\mathfrak{F}$.     

Electron and positron pair emission by low energy positron impact on surfaces

The emission of electron pairs from surfaces has the power to reveal details about the electron–electron interaction in condensed matter. This process, stimulated by a primary electron or photon beam, has been studied both in experiment and theory over the last two decades. An additional pathway, namely positron–electron pair emission, holds the promise to provide additional information. It is based on the notion that the Pauli exclusion principle does not need to be considered for this process.We have commissioned a laboratory based positron source and performed a systematic study on a variety of solid surfaces. In a symmetric emission geometry we can explore the fact that positron and electron are distinguishable particles. Following fundamental symmetry arguments we have to expect that the available energy is shared unequally among positron and electron. Experimentally we observe such a behavior for all materials studied. We find an universal feature for all materials in the sense that on average the positron carries a larger fraction of the available energy. This is qualitatively accounted for by a simplified scattering model. Numerical results, which we obtained by a microscopic theory of positron–electron emission from surfaces, reveal however that there are also cases in which the electron carries more energy. Whether the positron or the electron is more energetic depends on details of the bound electron state and of the emission geometry. The coincidence intensity is strongly material dependent and there exists an almost monotonic relation between the singles and coincidence intensity. These results resemble the findings obtained in electron and photon stimulated electron pair emission. An additional reaction channel is the emission of an electron pair upon positron impact. We will discuss the energy distributions and the material dependence of the coincidence signal which shows similar features as those for positron–electron pairs.     

Grain boundary,triple junction and quadruple point mobility controlled normal grain growth

Reduction in stored free energy provides the thermodynamic driving force for grain and bubble growth in polycrystals and foams. Evolution of polycrystalline networks exhibit the additional complication that grain growth may be controlled by several kinetic mechanisms through which the decrease in network energy occurs. Polyhedral boundaries, triple junctions (TJs), and quadruple points (QPs) are the geometrically distinct elements of three dimensional networks that follow Plateau’s rules, provided that grain growth is limited by diffusion through, and motion of, cell boundaries. Shvindlerman and co-workers have long recognized the kinetic influences on polycrystalline grain growth of network TJs and QPs. Moreover, the emergence of interesting polycrystalline nanomaterials underscored that TJs can indeed influence grain growth kinetics. Currently there exist few detailed studies concerned either with network distributions of grain size, number of faces per grain, or with ‘grain trajectories’, when grain growth is limited by the motion of its TJs or QPs. By contrast there exist abundant studies of classical grain growth limited by boundary mobility. This study is focused on a topological/geometrical representation of polycrystals to obtain statistical predictions of the grain size and face number distributions, as well as growth ‘trajectories’ during steady-state grain growth. Three limits to grain growth are considered, with grain growth kinetics controlled by boundary, TJ, and QP mobilities.     

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.     

一类二阶多点边值问题正解的存在性

利用锥上不动点定理讨论了二阶常微分方程组多点边值问题正解的存在性，所得结论推广了最近的一些结果．