Nonlinear Ergodic Theorem for Positively Homogeneous Nonexpansive Mappings in Banach Spaces

Recently, two retractions (projections), which are different from the metric projection and the sunny nonexpansive retraction in a Banach space, were found. In this article, using nonlinear analytic methods and new retractions, we prove a nonlinear ergodic theorem for positively homogeneous and nonexpansive mappings in a uniformly convex Banach space. The limit points are characterized by using new retractions.     

Shrinking Projection Methods for a Pair of Asymptotically Quasi-φ-Nonexpansive Mappings

Shrinking projection methods are investigated for a pair of asymptotically quasi-φ-nonexpansive mappings. Conditions ensuring strong convergence to common fixed points of the pair of mappings are established.     

Minimax Problems for Set-Valued Mappings

In this article, by virtue of the Fan-Browder fixed point theorem, we first obtain a minimax theorem and establish an equivalent relationship between the minimax theorem and a cone saddle point theorem for scalar set-valued mappings. Then, by using scalarization functions, we investigate an existence theorem of the cone saddle point for set-valued mappings. Simultaneously, we also establish a minimax theorem for set-valued mappings.     

Error estimates for the finite element solution of quasilinear obstacle problems *

This paper focusses on quasilinear second order differential operators. Concerning their strong monotonicity and Lipschitz continuity sufficient conditions are stated which are both explicit and easy to show. Applying these functional analytic properties to nonlinear obstacle problems we prove error estimates in energy norm for the finite element method with

linear trial functions.     

Approximation in eigenvalue problems for holomorphic fredholm operator functions I

The approximation of a holomorphic eigenvalue problem is considered. The main purpose is to present a construction by which the derivation of the asymptotic error estimations for the approximate eigenvalues of Fredholm operator functions can be reduced to the derivation of these estimations for the case of matrix functions. (Some estimations for the latter problem can be derived, in one's turn, from the error estimations for the zeros of the corresponding determinants.) The asymptotic error estimations are considered in part II of this paper, in [10]. By the presented construction also the stability of the algebraic multiplicity of eigenvalues by regular approximation is proved in Section 3

The presented construction, in essence, reproduces the constructions in [7] for the case of the compact approximation in subspaces and in [9] for the case of projection—like methods. It is simpler to use than similiar construction in [8], and allows unified consideration of the general case and the case of projection—like methods, what in [8, 9] was not achieved     

Parallel Algorithms for Solving a Class of Variational Inequalities over the Common Fixed Points Set of a Finite Family of Demicontractive Mappings

Abstract

We propose parallel algorithms for solving a class of variational inequalities over the set of common fixed points for a finite family of demicontractive mappings in real Hilbert spaces. Under some suitable conditions, we prove that the sequence generated by the proposed algorithms converges strongly to a solution of the problem. We apply the proposed algorithms to strongly monotone variational inequality problems with pseudomonotone equilibrium constraints by defining a quasi-nonexpansive and demi-closed mapping whose fixed point set coincides with the solution set of the equilibrium problem.     

Improved B0 field map estimation for high field EPI

Echo planar imaging (EPI) is an ultrafast magnetic resonance imaging (MRI) technique that allows one to acquire a 2D image in about 100 ms. Unfortunately, the standard EPI images suffer from substantial geometric distortions, mainly originating from susceptibility differences in adjacent tissues. To reduce EPI distortions, correction methods based on a field map, which is a map of the off-resonance frequencies, have been developed. In this work, a nonlinear least squares estimator is used to optimize the estimation of the field map of the B₀ field. The model of the EPI and reference data includes parameters for the phase evolution, the complex magnitude, the relaxation of the MRI signal and the EPI-specific phase difference between odd and even echoes, and from these parameters, additional corrections might be computed. The reference data required to estimate the field map can be acquired with a modified EPI-sequence. The proposed method is tested on simulated as well as experimental data and proves to be significantly more robust against noise, compared to the previously suggested method.     

Ubiquitylation of Fe65 adaptor protein by neuronal precursor cell expressed developmentally down regulated 4-2 (Nedd4-2) via the WW domain interaction with Fe65

Fe65 has been characterized as an adaptor protein, originally identified as an expressed sequence tag (EST) corresponding to an mRNA expressed at high levels in the rat brain. It contains one WW domain and two phosphotyrosine interaction/phosphotyrosine binding domains (PID1/PID2). As the neuronal precursor cell expressed developmentally down regulated 4-2 (Nedd4-2) has a putative WW domain binding motif (⁷²PPLP⁷⁵) in the N-terminal domain, we hypothesized that Fe65 associates with Nedd4-2 through a WW domain interaction, which has the characteristics of E3 ubiquitin-protein ligase. In this paper, we present evidence for the interaction between Fe65 WW domain and Nedd4-2 through its specific motif, using a pull down approach and co-immunoprecipitation. Additionally, the co-localization of Fe65 and Nedd4-2 were observed via confocal microscopy. Co-localization of Fe65 and Nedd4-2 was disrupted by either the mutation of Fe65 WW domain or its putative binding motif of Nedd4-2. When the ubiquitin assay was performed, the interaction of Nedd4-2 (wt) with Fe65 is required for the cell apoptosis and the ubiquitylation of Fe65. We also observed that the ubiquitylation of Fe65 (wt) was augmented depending on Nedd4-2 expression levels, whereas the Fe65 WW domain mutant (W243KP245K) or the Nedd4-2 AL mutant (⁷²PPLP⁷⁵ was changed to ⁷²APLA⁷⁵) was under-ubiquitinated significantly. Thus, our observations implicated that the protein-protein interaction between the WW domain of Fe65 and the putative binding motif of Nedd4-2 down-regulates Fe65 protein stability and subcellular localization through its ubiquitylation, to contribute cell apoptosis.     

Remarks on the structure of the fixed-point sets of uniformly lipschitzian mappings in uniformly convex Banach spaces

It is shown, by asymptotic center techniques, that the set of fixed points of any uniformly k-lipschitzian mapping in a uniformly convex Banach space is a retract of the domain when k is less than a constant bigger than the constant from the paper [K. Goebel, W.A. Kirk, A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math. 47 (1973) 135-140]. Our result improves a recently result presented in [E. S?d?ak, A. Wi?nicki, On the structure of fixed-point sets of uniformly lipschitzian mappings, Topol. Methods Nonlinear Anal. 30 (2007) 345-350].