strong convergence for gradient projection method and relatively nonexpansive mappings in banach spaces let c be a nonempty closed convex subset of a 2 uniformly convex and uniformly smooth banach space e a be a single valued monotone and lipschitz continuous mapping of c into e and t be a single valued relatively nonexpansive mapping of c into itself in this paper we consider the composition and the convex combination of t and the gradient projection method for a which goldstein 1964 proposed and proved the strong convergence to a common element of solutions of the variational inequality problem for a and fixed points of t by the hybrid method in mathematical programming haugazeau 1968 and we get several results which improve the well known results in a 2 uniformly convex and uniformly smooth banach space and a hilbert space

approximating fixed points of nonexpansive mappings in a banach space by metric projections abstract in this paper a strong convergence theorem for nonexpansive mappings in a uniformly convex and smooth banach space is proved by using metric projections this theorem is different from the recent strong convergence theorem due to xu h k xu strong convergence of approximating fixed point sequences for nonexpansive mappings bull aust math soc 74 2006 143 151 which was established by generalized projections

strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups strong convergence theorems are obtained for a finite family of asymptotically nonexpansive mappings and semigroups by the modified mann method

strong convergence to common fixed points of families of nonexpansive mappings abstract we introduce iteration schemes for families of nonexpansive mappings in hilbert spaces and prove that the iterates converge strongly to common fixed points of the mappings

on uniformly convex functions abstract in this paper we study uniformly convex functions and uniformly convex functions at a point giving some properties and characterizations of them further we give some examples and applications of these types of functions

strong convergence theorems for solving equilibrium problems and fixed point problems of ξ strict pseudo contraction mappings by two hybrid projection methods in this paper we introduce an iterative scheme by using the hybrid projection methods for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a x strict pseudo contraction mapping in hilbert spaces we obtain two strong convergence theorems under mild assumptions on parameters for the sequences generated by these processes the results presented in the paper extend and improve some recent results of marino and xu g marino h k xu weak and strong convergence theorems for strict pseudo contractions in hilbert spaces j math anal appl 329 2007 336 346 tada and takahashi a tada w takahashi weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem j optim theory appl 133 2007 359 370 and ceng et al l c ceng s al homidan q h ansari j c yao an iterative scheme for equilibrium problems and fixed point problems of strict pseudo contraction mappings j comput appl math 223 2009 967 974 and many others

fast iterative solution of stabilised stokes systems part ii using general block preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems part i of this work described a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems a j wathen and d j silvester siam j numer anal 30 1993 pp 630 649 using simple arguments estimates for the eigenvalue distribution of the discrete stokes operator on which the convergence rate of the iteration depends are easily derived part i discussed the important case of diagonal preconditioning scaling this paper considers the more general class of block preconditioners where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables it is shown that provid

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

convergence theorems of common fixed points for a family of lipschitz quasi pseudocontractions abstract in an infinite dimensional hilbert space the normal mann s iteration has only weak convergence in general even for nonexpansive mappings the purpose of this paper is to modify the normal mann s iteration to have strong convergence for a family of lipschitz quasi pseudocontractions in the framework of hilbert spaces our results improve and extend the corresponding ones announced by many others

a corrected nonlinear uzawa method for solving stabilized saddle point problems in this paper a corrected nonlinear uzawa algorithm for solving stabilized saddle point problems is proposed and the convergence rate is analyzed the results of numerical experiments are presented when we apply the algorithm to stokes equations discretized by mixed finite elements

positive weak solutions of semilinear second order elliptic inequalities via variational inequalities existence uniqueness and convergence of approximants of positive weak solutions for semilinear second order elliptic inequalities are obtained the nonlinearities involved in these inequalities satisfy suitable upper or lower bound conditions or monotonicity conditions the lower bound conditions are allowed to contain the critical sobolev exponents the methodology is to establish variational inequality principles for demicontinuous pseudo contractive maps in hilbert spaces by considering convergence of approximants and apply them to the corresponding variational inequalities arising from the semilinear second order elliptic inequalities examples on the existence uniqueness and convergence of approximants of positive weak solutions of the semilinear second order elliptic inequalities are given

the effect of the shape of the domain on the existence of solutions of an equation involving the critical sobolev exponent abstract we study the critical sobolev exponent equation by imposing some conditions on the shape of the domain we are able to obtain existence and multiplicity of positive solutions in particular multiplicity results for contractible domains are presented based on an approach that allows us to generalize previous work by ding who had proved an existence result for a contractible domain

degree of convergence of modified averaged iterations for fixed points problems and operator equations abstract let h be a real hilbert space we propose a modification for averaged mappings to approximate the unique fixed point of a mapping t h h such that t is boundedly lipschitzian and t is monotone we not only prove strong convergence theorems but also determine the degree of convergence using this result an iteration process is given for finding the unique solution of the equation a x f where a h h is strongly monotone and boundedly lipschitzian

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

a variational inequality theory for demicontinuous s contractive maps with applications to semilinear elliptic inequalities abstract a variational inequality theory for demicontinuous s contractive maps in hilbert spaces is established by employing the ideas of granas topological transversality such a variational inequality theory has many properties similar to those of fixed point theory for demicontinuous weakly inward s contractive maps and to those of fixed point index for condensing maps the variational inequality theory will be applied to study the existence of positive weak solutions and eigenvalue problems for semilinear second order elliptic inequalities with nonlinearities which satisfy suitable lower bound conditions involving the critical sobolev exponent there has been little discussion for such elliptic inequalities involving the critical sobolev exponent in the literature

strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in this paper we show strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in hilbert spaces by the hybrid method in the mathematical programming

strong convergence theorem of an iterative method for variational inequalities and fixed point problems in hilbert spaces in this paper we introduce a new iteration method based on the hybrid method in mathematical programming and the descent like method for finding a common element of the set of solutions for a variational inequality and the set of fixed points for a nonexpansive mapping in hilbert spaces our method modifies and improves some methods in literature

strong convergence theorems for an infinite family of quasi nonexpansive mappings and generalized equilibrium problems and variational inequality problems in this paper we introduce a new iterative scheme for finding a common element of the set of common fixed points of an infinite family of uniformly continuous quasi nonexpansive mappings and the set of solutions to a generalized equilibrium problem and the set of solutions to a variational inequality problem in a real hilbert space we then prove strong convergence of the scheme to a common element of the three mentioned sets our results extend important recent results

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

implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and lipschitz continuous monotone mappings in this paper we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone lipschitz continuous mapping these implicit iterative algorithms are based on two well known methods extragradient and approximate proximal methods we obtain some weak convergence theorems for these implicit iterative algorithms based on these theorems we also construct some implicit iterative processes for finding a common fixed point of two mappings such that one of these two mappings is taken from the more general class of lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory

a weak to strong convergence principle for feje monotone methods in hilbert spaces we consider a wide class of iterative methods arising in numerical mathematics and optimization that are known to converge only weakly exploiting an idea originally proposed by haugazeau we present a simple modification of these methods that makes them strongly convergent without additional assumptions several applications are discussed

fast iterative solution of stabilised stokes systems part i using simple diagonal preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems the use of stabilisation methods for convenient but unstable mixed elements introduces stabilisation parameters we show how these can be chosen to obtain rapid iterative convergence we propose a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems describe the effects of stabilisation on the algebraic structure of the discrete stokes operator and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends here we discuss the simple case of diagonal preconditioning our results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable we demonstrate that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete laplacian are approachable for the stokes problem

asymptotically strict pseudocontractive mappings in the intermediate sense abstract it is proved that the modified mann iteration process x n 1 1 n x n n t n x n n n where n is a sequence in 0 1 with n 1 for some 0 1 converges weakly to a fixed point of an asymptotically strict pseudocontractive mapping t in the intermediate sense which is not necessarily lipschitzian we also develop cq method for this modified mann iteration process which generates a strongly convergent sequence