An iterative algorithm based on -proximal mappings for a system of generalized implicit variational inclusions in Banach spaces

In this paper, we give the notion of M-proximal mapping, an extension of P-proximal mapping given in [X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369–383], for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space and prove its existence and Lipschitz continuity. Further, we consider a system of generalized implicit variational inclusions in Banach spaces and show its equivalence with a system of implicit Wiener–Hopf equations using the concept of M-proximal mappings. Using this equivalence, we propose a new iterative algorithm for the system of generalized implicit variational inclusions. Furthermore, we prove the existence of solution of the system of generalized implicit variational inclusions and discuss the convergence and stability analysis of the iterative algorithm.     

A projected subgradient method for solving generalized mixed variational inequalities

We consider the projected subgradient method for solving generalized mixed variational inequalities. In each step, we choose an ε_k-subgradient u^k of the function f and w^k in a set-valued mapping T, followed by an orthogonal projection onto the feasible set. We prove that the sequence is weakly convergent.     

System of generalized variational inclusions with -accretive operators in uniformly smooth Banach spaces

In this paper, we consider and study a system of generalized variational inclusions with H-accretive operators in uniformly smooth Banach spaces. We prove the convergence of iterative algorithm for this system of generalized variational inclusions. A new definition of H-resolvent operator as a retraction is introduced, and in support of the definition, we have constructed an example and a numerical example through Matlab programming. Some special cases are also discussed.     

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.     

King type modification of Meyer-König and Zeller operators based on the -integers

In the present paper, introducing a King type modification of the Meyer-König and Zeller (MKZ) operators, we prove that the error estimation of these operators is better than the classical MKZ operators. Furthermore, a King type modification of the q-MKZ is also introduced and the rate of convergence of this modification is examined.     

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.     

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.     

A regularization semismooth Newton method based on the generalized Fischer–Burmeister function for -NCPs

We consider a regularization method for nonlinear complementarity problems with F being a P₀-function which replaces the original problem with a sequence of the regularized complementarity problems. In this paper, this sequence of regularized complementarity problems are solved approximately by applying the generalized Newton method for an equivalent augmented system of equations, constructed by the generalized Fischer–Burmeister (FB) NCP-functions φ_p with p>1. We test the performance of the regularization semismooth Newton method based on the family of NCP-functions through solving all test problems from MCPLIB. Numerical experiments indicate that the method associated with a smaller p, for example p[1.1,2], usually has better numerical performance, and the generalized FB functions φ_p with p[1.1,2) can be used as the substitutions for the FB function φ₂.     

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.     

A completeness result for the simply typed -calculus

In this paper, we define a realizability semantics for the simply typed λμ-calculus. We show that, if a term is typable, then it inhabits the interpretation of its type. This result serves to give characterizations of the computational behavior of some closed typed terms. We also prove a completeness result of our realizability semantics using a particular term model.     

Bayesian factor analysis with fat-tailed factors and its exact marginal likelihood

The traditional Bayesian factor analysis method is extended. In contrast to the case for previous studies, the matrix variate t-distribution is utilized to provide a prior density on the latent factors. This is a natural extension of the traditional model and yields many advantages. The crucial issue is the selection of the number of factors. The marginal likelihood, constructed by asymptotic and computational approaches, is generally utilized for this problem. However, both theoretical and computational problems have arisen.In this paper, the exact marginal likelihood is derived. It enables us to evaluate posterior model probabilities without inducing the above problems. Monte Carlo experiments were conducted to examine the performance of the proposed Bayesian factor analysis modelling methodology. The simulation results show that the proposed methodology performs well.     

The p-step iterative algorithm for a system of generalized mixed quasi-variational inclusions with -accretive operators in q-uniformly smooth banach spaces

In this paper, we introduce and study a new system of generalized mixed quasi-variational inclusions with (A,η)-accretive operators in q-uniformly smooth Banach spaces. By using the resolvent operator technique associated with (A,η)-accretive operators, we construct a new p-step iterative algorithm for solving this system of generalized mixed quasi-variational inclusions in real q-uniformly smooth Banach spaces. We also prove the existence of solutions for the generalized mixed quasi-variational inclusions and the convergence of iterative sequences generated by algorithm. Our results improve and generalize many known corresponding results.     

Dynamical analysis of the transmission of seasonal diseases using the differential transformation method

The aim of this paper is to apply the differential transformation method (DTM) to solve systems of nonautonomous nonlinear differential equations that describe several epidemic models where the solutions exhibit periodic behavior due to the seasonal transmission rate. These models describe the dynamics of the different classes of the populations. Here the concept of DTM is introduced and then it is employed to derive a set of difference equations for this kind of epidemic models. The DTM is used here as an algorithm for approximating the solutions of the epidemic models in a sequence of time intervals. In order to show the efficiency of the method, the obtained numerical results are compared with the fourth-order Runge–Kutta method solutions. Numerical comparisons show that the DTM is accurate, easy to apply and the calculated solutions preserve the properties of the continuous models, such as the periodic behavior. Furthermore, it is showed that the DTM avoids large computational work and symbolic computation.     

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.     

A new iterative method for equilibrium problems and fixed point problems of infinitely nonexpansive mappings and monotone mappings

In this paper, we introduce a new iterative scheme for finding the common element of the set of common fixed points of infinitely many 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 Hilbert spaces. We prove that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. This main result improve and extend Plubtieng and Punpaeng’s corresponding result [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Applied Mathematics and Computation 197 (2008), 548–558]. Using this theorem, we obtain three corollaries.     

Solutions for an operator equation under the conditions of pairs of paralleled lower and upper solutions

In this paper, under the condition of two pairs of strict lower and upper solutions and using the concept of (e₁,B)-limit increasing operator, some multiplicity results for an operator equation are obtained by the method of the fixed point index.     

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.     

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.     

-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.     

Central limit theorem and the bootstrap for -statistics of strongly mixing data

The asymptotic normality of U-statistics has so far been proved for iid data and under various mixing conditions such as absolute regularity, but not for strong mixing. We use a coupling technique introduced in 1983 by Bradley [R.C. Bradley, Approximation theorems for strongly mixing random variables, Michigan Math. J. 30 (1983),69–81] to prove a new generalized covariance inequality similar to Yoshihara’s [K. Yoshihara, Limiting behavior of U-statistics for stationary, absolutely regular processes, Z. Wahrsch. Verw. Gebiete 35 (1976), 237–252]. It follows from the Hoeffding-decomposition and this inequality that U-statistics of strongly mixing observations converge to a normal limit if the kernel of the U-statistic fulfills some moment and continuity conditions.The validity of the bootstrap for U-statistics has until now only been established in the case of iid data (see [P.J. Bickel, D.A. Freedman, Some asymptotic theory for the bootstrap, Ann. Statist. 9 (1981), 1196–1217]. For mixing data, Politis and Romano [D.N. Politis, J.P. Romano, A circular block resampling procedure for stationary data, in: R. Lepage, L. Billard (Eds.), Exploring the Limits of Bootstrap, Wiley, New York, 1992, pp. 263–270] proposed the circular block bootstrap, which leads to a consistent estimation of the sample mean’s distribution. We extend these results to U-statistics of weakly dependent data and prove a CLT for the circular block bootstrap version of U-statistics under absolute regularity and strong mixing. We also calculate a rate of convergence for the bootstrap variance estimator of a U-statistic and give some simulation results.