A hybrid approximation method for equilibrium and fixed point problems for a family of infinitely nonexpansive mappings and a monotone mapping

In this paper, we introduce a new iterative scheme for finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α-inverse-strongly monotone mapping in the framework of a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. Additionally, we utilize our results to study the optimization problem and find a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space. Our results improve and extend the results announced by many others.     

-stability of discrete hybrid systems via variational Lyapunov method

The aim of this paper is to give some results on ₀-stability for discrete hybrid systems via a variational cone-valued Lyapunov method.     

Nonnegative iterations with asymptotically constant coefficients

Let A_k,k=0,1,2,…, be a sequence of real nonsingular n×n matrices which converge to a nonsingular matrix A. Suppose that A has exactly one positive eigenvalue λ and there exists a unique nonnegative vector u with properties Au=λu and u=1. Under further additional conditions on the spectrum of A, it is shown that if x₀≠0 and the iterates

are nonnegative, then converges to u and converges to λ as k→∞.     

Solutions to operator equations on Hilbert -modules

Let be a C^*-algebra, E,F and G be Hilbert -modules, , and . We generalize the Douglas theorem about the operator equation TX=T^′ from Hilbert space to Hilbert C^*-module. To the equation TX=T^′ and to the system of two equations TX=T^′ and XS=S^′, we get the forms of general solutions (in the case that there exists a solution), and give some sufficient and necessary conditions for the existence of solutions, and the existence of hermitian solutions and positive solutions (in the case G=E). In addition, the forms of general hermitian solution and general positive solution (in the case that there exists a solution and G=E) to the equation TX=T^′ are given too.     

On the convergence of modified Noor iterations with errors for asymptotically nonexpansive mappings

In this paper, several weak and strong convergence theorems are established for a modified Noor iterative scheme with errors for three asymptotically nonexpansive mappings in Banach spaces. Mann-type, Ishikawa-type, and Noor-type iterations are covered by the new iteration scheme. Our results extend and improve the recently announced ones [B.L. Xu, M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002) 444-453; Y.J. Cho, H. Zhou, G. Guo, Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl. 47 (2004) 707-717] and many others.     

Rates of -statistical convergence of positive linear operators

In this paper we study the rates of A-statistical convergence of sequences of positive linear operators mapping the weighted space C_ρ1 into the weighted space B_ρ2.     

The convergence of three spline methods for scattered data interpolation and fitting

The convergences of three L₁ spline methods for scattered data interpolation and fitting using bivariate spline spaces are studied in this paper. That is, L₁ interpolatory splines, splines of least absolute deviation, and L₁ smoothing splines are shown to converge to the given data function under some conditions and hence, the surfaces from these three methods will resemble the given data values.     

The behavior of best -approximations as . A counterexample of convergence

In the space of summable sequences we give an example of a one-dimensional affine subspace C such that the best L_p-approximations of 0 from C fail to converge as p↓1. We thus give an answer to this problem of convergence in infinite measure spaces.     

A note on preconditioned AOR method for -matrices

In this note, some errors in a recent article by Li et al. (Improvements of preconditioned AOR iterative methods for L-matrices, J. Comput. Appl. Math. 206 (2007) 656–665) are pointed out and some correct results are presented.     

Strong Haagerup inequalities with operator coefficients

We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, denotes the subspace of the von Neumann algebra of a free group F_I spanned by the words of length d in the generators (but not their inverses), then we provide in this paper an explicit upper bound on the norm on , which improves and generalizes previous results by Kemp–Speicher (in the scalar case) and Buchholz and Parcet–Pisier (in the non-holomorphic setting). Namely the norm of an element of the form ∑_{i=(i1,…,id)}a_iλ(g_i1g_id) is less than , where M₀,…,M_d are d+1 different block-matrices naturally constructed from the family (a_i)_iI^d for each decomposition of I^dI^l×I^d−l with l=0,…,d. It is also proved that the same inequality holds for the norms in the associated non-commutative L_p spaces when p is an even integer, pd and when the generators of the free group are more generally replaced by *-free -diagonal operators. In particular it applies to the case of free circular operators. We also get inequalities for the non-holomorphic case, with a rate of growth of order d+1 as for the classical Haagerup inequality. The proof is of combinatorial nature and is based on the definition and study of a symmetrization process for partitions.     

Convergence theorems of fixed points for asymptotically hemi-pseudocontractive mappings

In this paper, the concept of asymptotically hemi-pseudocontractive mapping in Banach space, which is a proper generalization of asymptotically pseudocontractive mapping with nonempty fixed point set as shown by an example, is introduced. Some sufficient and necessary conditions on the strong convergence of the modified Ishikawa and the modified Mann iteration sequences with mean errors for uniformly L-Lipshitzian asymptotically hemi-pseudocontractive mappings are presented.     

Best approximation and moduli of smoothness: Computation and equivalence theorems

In this paper we investigate the trigonometric series with the β-general monotone coefficients. First, we study the uniform convergence criterion. The estimates of best approximations and moduli of smoothness of the series in uniform metrics are obtained in terms of coefficients. These results imply several important relations between moduli of smoothness of different orders (in particular, Marchaud-type inequality) and best approximations.     

Accelerating gradient projection methods for -constrained signal recovery by steplength selection rules

We propose a new gradient projection algorithm that compares favorably with the fastest algorithms available to date for ℓ₁-constrained sparse recovery from noisy data, both in the compressed sensing and inverse problem frameworks. The method exploits a line-search along the feasible direction and an adaptive steplength selection based on recent strategies for the alternation of the well-known Barzilai–Borwein rules. The convergence of the proposed approach is discussed and a computational study on both well conditioned and ill-conditioned problems is carried out for performance evaluations in comparison with five other algorithms proposed in the literature.     

Strong convergence of approximate solutions for nonlinear hyperbolic equation without convexity

Schonbek [M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations 7 (1982) 959-1000] obtained the strong convergence of uniform bounded approximate solutions to hyperbolic scalar equation under the assumption that the flux function is strictly convex. While in this paper, by constructing four families of Lax entropies, we succeed in dealing with the non-convexity with the aid of the well-known Bernstein-Weierstrass theorem, and obtaining the strong convergence of uniform L^∞ or bounded viscosity solutions for scalar conservation law without convexity.     

Coincidence and common fixed points for hybrid strict contractions without the weakly commuting condition

In this work, we establish new coincidence and common fixed point theorems for hybrid strict contraction maps by dropping the assumption “f is T-weakly commuting” for a hybrid pair (f,T) of multivalued maps in Theorem 3.10 of T. Kamran (2004) [8]. As an application, an invariant approximation result is derived.     

Dunford–Pettis properties, Hilbert spaces and projective tensor products

We study complete continuity properties of operators onto ℓ₂ and prove several results in the Dunford–Pettis theory of JB^*-triples and their projective tensor products, culminating in characterisations of the alternative Dunford–Pettis property for where E and F are JB^*-triples.     

Algorithm for solving a new class of general mixed variational inequalities in Banach spaces

In this paper, a new concept of η-proximal mapping for a proper subdifferentiable functional (which may not be convex) on a Banach space is introduced. An existence and Lipschitz continuity of the η-proximal mapping are proved. By using properties of the η-proximal mapping, a new class of general mixed variational inequalities is introduced and studied in Banach spaces. An existence theorem of solutions is established and a new iterative algorithm for solving the general mixed variational inequality is suggested. A convergence criteria of the iterative sequence generated by the new algorithm is also given.     

Strength of convergence in the orbit space of a groupoid

Let G be a second-countable locally-compact Hausdorff groupoid with a Haar system, and let {xn} be a sequence in the unit space G(0) of G. We show that the notions of strength of convergence of {xn} in the orbit space G(0)/G and measure-theoretic accumulation along the orbits are equivalent ways of realising multiplicity numbers associated to a sequence of induced representation of the groupoid C^?-algebra.