a unifying convergence analysis of second order methods for secular equations existing numerical methods of second order are considered for a so called secular equation we give a brief description of the most important of these methods and show that all of them can be interpreted as improvements of newton s method for an equivalent problem for which newton s method exhibits convergence from any point in a given interval this interpretation unifies the convergence analysis of these methods provides convergence proofs where they were lacking and furnishes ways to construct improved methods in addition we show that some of these methods are in fact equivalent a second secular equation is also briefly considered

the clifford algebra of an infinite dimensional space the clifford algebra of a real infinite dimensional vector space is constructed this algebra contains all finite and infinite dimensional clifford algebras over the real and complex numbers some of the mathematical properties of these algebras are discussed the automorphisms and involutions are specifically addressed the special case with metric g 1 2 follows as a corollary to the more general results

strong convergence theorems for equilibrium problems and relatively nonexpansive mappings in banach spaces the purpose of the paper is to introduce modified halpern and ishikawa iteration for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in banach spaces we also consider two strong convergence theorems for relatively nonexpansive mappings with some proper restriction

fixed point iterations for certain nonlinear mappings abstract in a recent paper schu established the convergence of the mann iterates of a completely continuous asymptotically nonexpansive mapping on a hilbert space in this paper we extend the result of schu to uniformly convex spaces we also obtain the same generalization for the ishikawa iteration process

spectral functions for real symmetric toeplitz matrices we derive separate spectral functions for the even and odd spectra of a real symmetric toeplitz matrix which are given by the roots of those functions these are rational functions also commonly referred to as secular functions two applications are considered spectral evolution as a function of one parameter and the computation of eigenvalues

perturbed proximal point algorithms for generalized quasivariational inclusions abstract in this paper we study a class of generalized quasivariational inclusions by using the properties of the resolvent operator associated with a maximal monotone mapping in hilbert space we have established an existence theorem of solutions for generalized quasivariational inclusions suggesting a new iterative algorithm and a perturbed proximal point algorithm for finding approximate solutions which strongly converge to the exact solution of the generalized quasivariational inclusions as special cases some known results in this field are also discussed

variational conclusions of set valued bifunctions on convex subsets of banach spaces with applications in this paper variational conclusions of set valued bifunctions on convex subsets of banach spaces are investigated and several new results are obtained the results are applied to the fixed point theory and variational inequalities we obtain two fixed point theorems and existence theorems of solutions of variational inequalities

on some algorithms in banach spaces finding fixed points of nonlinear mappings abstract by using a modification of the hybrid method in mathematical programming we give some new algorithm to find out a fixed point of mappings which are in some sense non expansive our results hold in reflexive strictly convex smooth banach spaces with the kadec klee property

optimal solutions to variational inequalities on banach lattices abstract we study the existence of maximum and minimum solutions to generalized variational inequalities on banach lattices the main tools of analysis are the variational characterization of the generalized metric projection operator and order theoretic fixed point theory

a divide and conquer method for unitary and orthogonal eigenproblems leth n xn be a unitary upper hessenberg matrix whose eigenvalues and possibly also eigenvectors are to be determined we describe how this eigenproblem can be solved by a divide and conquer method in which the matrixh is split into two smaller unitary upper hessenberg matricesh 1 andh 2 by a rank one modification ofh the eigenproblems forh 1 andh 2 can be solved independently and the solutions of these smaller eigenproblems define a rational function whose zeros on the unit circle are the eigenvalues ofh the eigenvector ofh can be determined from the eigenvalues ofh and the eigenvectors ofh 1 andh 2 the outlined splitting of unitary upper hessenberg matrices into smaller such matrices is carried out recursively this gives rise to a divide and conquer method that is suitable for implementation on a parallel computer whenh n xn is orthogonal the divide and conquer scheme simplifies and is described separately our interest in the orthogonal eigenproblem stems from applications in signal processing numerical examples for the orthogonal eigenproblem conclude the paper

a variational inequality theory in reflexive smooth banach spaces and applications to p laplacian elliptic inequalities abstract we develop a theory of variational inequality for demicontinuous s contractive maps in reflexive smooth banach spaces variational inequality theorems are proved and applied to study existence of nonzero positive weak solutions for p laplacian elliptic inequalities and a population model of one species arising in mathematical biology

solving variational inequalities in banach spaces in this paper we study the existence of solutions of variational inequalities and complementarity problems in a reflexive strictly convex and smooth banach space by using the generalized projection operator k b k where b is a reflexive strictly convex and smooth banach space with the dual space b and k is a nonempty closed and convex subset of b

existence theorem and algorithm for a general implicit variational inequality in banach space by using the generalized f projection operator the existence theorem of solutions for the general implicit variational inequality givi email protected k is proved without assuming the monotonicity of operators in reflexive and smooth banach space an iterative algorithm for approximating solution of the general implicit variational inequality is suggested also and the convergence for this iterative scheme is shown these theorems extend the corresponding results of wu and huang k q wu n j huang comput math appl 54 2007 399 406 wu and huang k q wu n j huang bull austral math soc 73 2006 307 317 zeng and yao l c zeng j c yao j optimiz theory appl 132 2 2007 321 337 and li j li j math anal appl 306 2005 55 71

deformation and contraction in clifford algebras spin ideals of clifford algebras over quadratic spaces of any rank are constructed through contractive limits of suitably deformed classical spin ideals of nondegenerate clifford algebras the deformation of the algebraic structures including the standard witt basis for the quadratic space results only from a deformation of the underlying quadratic form it is shown that a contractive limit of deformed twistor spaces considered as spin ideals of the dirac clifford algebra provides a decomposable representation space for the galilei clifford algebra the limit spin ideals of degenerate clifford algebras are then decomposed into indecomposable clifford modules

a perturbed algorithm for variational inclusions a perturbed method for solving a new class of variational inclusions is presented and a convergence result which includes as a special case some known results in this field is given

mann and ishikawa type perturbed iterative algorithms for generalized quasivariational inclusions abstract in this paper we introduce a class of generalized quasivariational inclusions and show its equivalence with a class of fixed point problems by making use of the properties of proximal maps using this equivalence we develop the mann and ishikawa type perturbed iterative algorithms for this class of generalized quasivariational inclusions further using fixed point techniques we prove the existence of solutions for the class of generalized quasivariational inclusions and discuss the convergence criteria for the perturbed algorithms our algorithms and results improve and generalize many known corresponding algorithms and results

numerical solution of a secular equation a method is proposed for the solution of a secular equation arising in modified symmetric eigenvalue problems and in several other areas this equation has singularities which make the application of standard root finding methods difficult in order to solve the equation a class of transformations of variables is considered which transform the equation into one for which newton s method converges from any point in a certain given interval in addition the form of the transformed equation suggests a convergence accelerating modification of newton s method the same ideas are applied to the secant method and numerical results are presented

a new method for solving a system of generalized nonlinear variational inequalities in banach spaces abstract the purpose of this paper is by using the generalized projection approach to introduce an iterative scheme for finding a solution to a system of generalized nonlinear variational inequality problem under suitable conditions some existence and strong convergence theorems are established in uniformly smooth and strictly convex banach spaces the results presented in the paper improve and extend some recent results

strong convergence theorems for relatively nonexpansive mappings in banach spaces with applications in this paper some properties of the generalized f projection operator are proved in banach spaces using these results the strong convergence theorems for relatively nonexpansive mappings are studied in banach spaces as applications the strong convergence of general h monotone mappings in banach spaces is also given the results presented in this paper generalize and improve the main results of matsushita and takahashi 2005 9

analysis of third order methods for secular equations third order numerical methods are analyzed for secular equations these equations arise in several matrix problems and numerical linear algebra applications a closer look at an existing method shows that it can be considered as a classical method for an equivalent problem this not only leads to other third order methods it also provides the means for a unifying convergence analysis of these methods and for their comparisons finally we consider approximated versions of the aforementioned methods

convergence of hybrid steepest descent method for variational inequalities in banach spaces abstract let e be a q uniformly smooth real banach space with constant dq q 1 let ti e e i 1 2 r be a finite family of nonexpansive mappings with k i 1 r fix t i and k fix trtr 1 t1 fix t1tr t2 fix tr 1tr 2 tr let g e e be an strongly accretive map which is also lipschitzian a hybrid steepest descent method introduced by yamada 25 and studied by various authors is proved to converge strongly to the unique solution of the variational inequality problem vi g k in q uniformly smooth real banach space in particular in lp spaces 1

the quasi complementarity problem abstract variational inequality techniques are used to suggest a unified and general iterative algorithm for computing the approximate solution of a new class of complementarity problems the convergence properties of this algorithm are also discussed

a clifford algebra quantization of dirac s electron positron field the quantum field theory of free dirac particles four component massive spin 1 2 particles is derived by a segal quantization procedure first details are given on how the spinor space of dirac is actually a minimal left ideal of the clifford algebra associated with a lorentz inner product space and how the homogeneous group actions break the natural two component quaternion structure to give familiar four component complex spinors second wigner s procedure for constructing unitary representations of the poincare group is used to construct the appropriately induced infinite dimensional representation of the inhomogeneous group starting from the above four dimensional nonunitary representation third and finally segal s procedure for quantizing classical fermion fields is adapted to this infinite dimensional hilbert space to obtain the clifford algebra of annihilation creation operators for spin 1 2 particles the familiar fock space appears as a minimal left ideal in this second clifford algebra

on the existence of solutions of variational inequalities in banach spaces abstract in this paper we study the existence of the solution of the variational inequality tx y x 0 by applying the generalized projection operator k b b where b is a banach space with dual space b and by using the well known fankkm theorem

the generalized projection operator on reflexive banach spaces and its applications abstract in this paper we extend the definition of the generalized projection operator k b k where b is a reflexive banach space with dual space b and k is a nonempty closed and convex subset of b and we study its properties and applications to solving variational inequality

iterative schemes for approximating solutions of generalized variational inequalities in banach spaces abstract in this paper we introduce two iterative schemes for approximating solutions of generalized variational inequalities in the setting of banach spaces the existence of solutions of this general problem and the convergence of the proposed iterative schemes to a solution are established

on chidume s open questions and approximate solutions of multivalued strongly accretive mapping equations in banach spaces abstract some new convergence theorems of the ishikawa iteration process for multi valued pseudo contractive mappings and multi valued accretive mappings are obtained which affirmatively answer the open questions put forth by 4

on systems of generalized nonlinear variational inequalities in banach spaces abstract in this paper we introduce a new algorithm for a system for generalized nonlinear variational inequalities in banach spaces by using convergence of the generalized projection methods our results extend and improve the results of fan j h fan a mann type iterative scheme for variational inequalities in noncompact subsets of banach spaces j math anal appl 337 2007 1041 1047 li j li on the existence of solutions of variational inequalities in banach spaces j math anal appl 295 2004 115 126 verma r u verma projection methods algorithms and a new system of nonlinear variational inequalities comput math appl 41 2001 1025 1031 and many others

proximal projection methods for variational inequalities and cesaro averaged approximations abstract in this paper we prove convergence and stability of cesaro averages generated by the proximal projection method applied to nonlinear equations and variational inequalities in uniformly convex and uniformly smooth banach spaces we first consider the stability of the approximations with respect to perturbations of the operator and constraint sets weak convergence of cesaro averages is shown to hold with only a monotonicity condition for the operator involved if in addition proximal iterations are also satisfying some boundedness requirements then we show that the weak convergence of cesaro averages is stable

generalised prox regularity in reflexive smooth banach spaces with smooth dual norm abstract in this paper we continue the study of the properties of the generalised projection on nonconvex sets in reflexive smooth banach spaces with smooth dual norm started by the first author in 9 we define a new class of nonconvex sets in banach spaces containing the convex closed sets and having various important properties whenever the space is assumed to be uniformly convex with smooth norm

a uniqueness result concerning a robust regularized least squares solution in solving a robust version of regularized least squares with weighting a certain scalar valued optimization problem is required in order to determine the regularized robust solution and the corresponding robustified weighting parameters this letter establishes that the required optimization problem does not have local non global minima over the interval of interest this property is proved by resorting to a useful schur complementation argument the result is reassuring in that it demonstrates that the robust design procedure is well defined and that its optimal global solution can be determined without concerns about local minima

on the nonlinear complementarity problem abstract we consider and study an algorithm for a new class of complementarity problems of finding u such that u 0 tu a u 0 u tu a u 0 where t is a continuous mapping and a is a diagonally nonlinear mapping from r n into itself furthermore it is proved that the approximate solution obtained by the iterative scheme converges to the exact solution special cases are also discussed

a constrained eigenvalue problem in this paper we consider the following mathematical and computational problem given the quantities a n m by n m matrix symmetric n 0 n n m by m matrix with full rank t vector of dimension m with nt t 1 determine an x such that operatorname x t ax min subject to the constraints n t x t i x t x 1 ii variants of this problem occur in many applications 1 5 7 8 11 the problem has been studied previously when t 0 the null vector cf 4 6 when t 0 then the problem becomes more complicated

a numerical comparison of methods for solving secular equations numerical comparisons are carried out by the most prominent methods both second and higher order for solving secular equations these equations arise in several matrix problems and numerical linear algebra applications in addition to the comparisons we propose a new starting point our results show a clear advantage to higher order methods

generalized projection method for a variational inequality system with different mapping in banach spaces the approximate solution of a system for variational inequality with different mapping in hilbert spaces has been studied based on the convergence of projection methods however little research has been done in banach space the primary reason is that projection mapping lacks preferable properties in banach space in this paper we introduce generalized projection methods by using these methods the results presented in this paper have extended the main results of chang et al s and huang and noor s recent works s s chang h w joseph lee c k chan generalized system for relaxed cocoercive variational inequalities in hilbert spaces appl math lett 20 2007 329 334 zhenyu huang muhammad aslam noor an explicit projection method for a system of nonlinear variational inequalities with different c r cocoercive mappings appl math comput 190 2007 356 361 from hilbert spaces to banach space

properties of the generalized f projection operator and its applications in banach spaces in this paper we prove some properties of the generalized f projection operator in banach spaces and show an interesting relation between the generalized f projection operator and the resolvent operator for the subdifferential of a proper convex and lower semi continuous functional in reflexive and smooth banach spaces moreover we prove that the generalized f projection operator is maximal monotone as applications we propose an iterative method of approximating solutions for a class of generalized variational inequalities and give a convergence result for the iterative method in uniformly convex and uniformly smooth banach spaces

some aspects of variational inequalities abstract in this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence uniqueness computational properties various generalizations sensitivity analysis and their applications we also propose some open problems with sufficient information and references so that someone may attempt solution s in his her area of special interest we also include some new results which we have recently obtained

perturbed algorithms and sensitivity analysis for a general class of variational inclusions in the present paper we study a perturbed iterative method for solving a general class of variational inclusions an existence result which generalizes some known results in this field a convergence result and a new iterative method are given we also prove the continuity of the perturbed solution to a parametric variational inclusion problem several special cases are discussed

strong convergence of a proximal type algorithm in a banach space in this paper we study strong convergence of the proximal point algorithm it is known that the proximal point algorithm converges weakly to a solution of a maximal monotone operator but it fails to converge strongly then in math program 87 2000 pp 189 202 solodov and svaiter introduced the new proximal type algorithm to generate a strongly convergent sequence and established a convergence property for it in hilbert spaces our purpose is to extend solodov and svaiter s result to more general banach spaces using this we consider the problem of finding a minimizer of a convex function

projection algorithms for the system of generalized mixed variational inequalities in banach spaces some projection algorithms are suggested for solving the system of generalized mixed variational inequalities and the convergence of the proposed iterative methods are proved without any monotonicity assumption for the mappings in banach spaces our theorems generalize some known results

mann and ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasi variational inclusions abstract in this paper we introduce a new class of generalized nonlinear implicit quasi variational inclusions and prove its equivalence with a class of fixed point problems by making use of the properties of maximal monotone we also prove the existence of solutions for this generalized nonlinear implicit quasi variational inclusions and the convergence of iterative sequences generated by the perturbed algorithms

a mann type iterative scheme for variational inequalities in noncompact subsets of banach spaces abstract in this paper we study the existence of solutions and approximation of the solutions by using a mann type iterative scheme for variational inequalities in noncompact subsets of banach spaces the results presented in this paper generalize the corresponding results of j li j li on the existence of solutions of variational inequalities in banach spaces j math anal appl 295 2004 115 126

existence of solutions for generalized variational inequality problems in banach spaces abstract in this paper we prove the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over compact convex subsets in a reflexive banach space with a frechet differentiable norm moreover we give some conditions that guarantee the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over unbounded closed convex subsets the result obtained in this paper improves and extends the recent ones announced by yu and yang j yu h yang existence of solutions for generalized variational inequality problems nonlinear anal 71 2009 e2327 e2330 and many others

the generalized f projection operator and set valued variational inequalities in banach spaces abstract in this paper we further study some properties of the generalized f projection operator introduced by wu and huang by employing the properties of the generalized f projection operator and the well known kkm and kakutani fan glicksberg theorems we establish some new existence theorems of solutions for two classes of generalized set valued variational and quasi variational inequalities in reflexive and smooth banach spaces under some suitable conditions

structure of spin groups associated with degenerate clifford algebras clifford algebras over finite dimensional vector spaces endowed with degenerate quadratic form contain a nontrivial two sided nilpotent ideal the jacobson radical generated by the orthogonal complement of such spaces thus they cannot be faithfully represented by matrix algebras following the theory of spin representations of classical clifford algebras the left regular spin representations of these degenerate algebras can be studied in suitably constructed left ideals first structure of the group of units of such algebras is examined for a quadratic form of arbitrary rank it is shown to be a semidirect product of a group generated by the radical and the group of units of a maximal nondegenerate clifford subalgebra next in the special case of corank 1 clifford pin and spin groups are defined an their structures are described as an example a galilei clifford algebra over the galilei space time is considered a covering theorem is then proved analogous to the one well known in the theory of

on the generalized implicit quasivariational inequalities abstract in the present paper we introduce and study a new class of generalized nonlinear implicit quasivariational inequalities for set valued mappings and construct some new iterative algorithms we prove the existence of solutions for this class of generalized nonlinear implicit quasivariational inequalities for set valued mappings without compactness and the convergence of iterative sequences generated by the algorithms we also study a new perturbed iterative algorithm for solving a class of general strongly nonlinear quasivariational inequalities