convergence of a halpern type iteration algorithm for a class of pseudo contractive mappings abstract let e be a real reflexive banach space with uniformly g teaux differentiable norm let k be a nonempty bounded closed and convex subset of e let t k k be a strictly pseudo contractive map and let l 0 denote its lipschitz constant assume f t x k t x x 0 and let z f t fix 0 1 and let be such that l 0 1 define s n x 1 n x n t x x k where n 0 1 and lim n 0 let n be a real sequence in 0 1 which satisfies the following conditions c1 lim n 0 c2 n for arbitrary x 0 u k define a sequence x n k by x n 1 n u 1 n s n x n then x n converges strongly to a fixed point of t

high order numerical approximation of minimal surfaces we present an algorithm for finding high order numerical approximations of minimal surfaces with a fixed boundary the algorithm employs parametrization by high order polynomials and a linearization of the weak formulation of the laplace beltrami operator to arrive at an iterative procedure to evolve from a given initial surface to the final minimal surface for the steady state solution we measure the approximation error in a few cases where the exact solution is known in the framework of parametric interpolation the choice of interpolation points mesh nodes is directly affecting the approximation error and we discuss how to best update the mesh on the evolutionary surface such that the parametrization remains smooth in our test cases we may achieve exponential convergence in the approximation of the minimal surface as the polynomial degree increases but the rate of convergence greatly differs with different choices of mesh update algorithms the present work is also of relevance to high order numerical approximation of fluid flow problems involving free surfaces

a new iterative algorithm for common solutions of a finite family of accretive operators abstract the purpose in this paper is to prove a theorem of strong convergence to a common solution for a finite family of accretive operators in a strictly convex banach space by means of a new iterative algorithm which is a generalization and extension of the results of kim and xu t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 2005 51 60 and zegeye and shahzad h zegeye n shahzad strong convergence theorems for a common zero of a finite family of m accretive mappings nonlinear anal 66 2007 1161 1169 further using the result the theorem of strong convergence to a common fixed point is discussed for a finite family of pseudocontractive mappings under certain conditions

convergence theorems for common fixed points for finite families of nonexpansive mappings in reflexive banach spaces let e be a real reflexive banach space with a uniformly g teaux differentiable norm let k be a nonempty closed convex subset of e suppose that every nonempty closed convex bounded subset of k has the fixed point property for nonexpansive mappings let t1 t2 tn be a family of nonexpansive self mappings of k with f i 1nfix ti 0 f fix tntn 1 t1 fix t1tn t2 fix tn 1tn 2 t1tn let n be a sequence in 0 1 satisfying the following conditions c1 lim n 0 c2 n for a fixed 0 1 define sn k k by snx 1 x tnx x k where tn tnmodn for an arbitrary fixed u x0 k let b x k tntn 1 t1x x 1 u for some 1 be bounded and let the sequence xn be defined iteratively by xn 1 n 1u 1 n 1 sn 1xn for n 0 assume that limn tnxn tn 1xn 0 then xn converges strongly to a common fixed point of the family t1 t2 tn this convergence theorem is also proved for non self maps

a multidimensional gas kinetic bgk scheme for hypersonic viscous flow this paper concerns the development and application of a gas kinetic scheme based on the bhatnagar gross krook bgk model for the navier stokes equations in the study of hypersonic viscous flow firstly we extend the gas kinetic navier stokes solver k xu a gas kinetic bgk scheme for the navier stokes equations and its connection with artificial dissipation and godunov method j comput phys 171 2001 289 335 by implementing a multidimensional particle propagation mechanism in the flux evaluation where the gradients of flow variables in both normal and tangential directions of a cell interface are explicitly included with the construction of a time averaged flux function an implicit bgk scheme with lu sgs method is constructed the main purpose of the current research is to pave the way to extend the current approach directly to the flow computation with unstructured mesh where the flow gradients in both parallel and perpendicular directions around a cell interface can be explicitly taken into account in a viscous flow computation in the numerical parts we concentrate on the computation of heat flux in laminar hypersonic viscous flows where complicated flow phenomena i e shock boundary layer interaction flow separation and viscous inviscid interaction will be encountered the cases studied include the type iv shock shock interaction around a circular cylinder and hypersonic flow passing through a double cone geometry in the hypersonic viscous flow in comparison with the capturing of velocity and pressure fields the accurate computation of stress and heat flux bears large difficulties in all cases studied here the heat fluxes obtained across body surfaces have good agreement with the experimental measurements

a new sufficient condition for the strong convergence of halpern type iterations abstract the aim of this work is to give a new sufficient condition of the strong convergence of the halpern type iteration for a non expansive self mapping defined on a banach space with a uniformly g teaux differentiable norm several examples satisfying our condition are presented our results not only remove the restriction of the space with the fixed point property for non expansive self mappings but also get rid of the dependence on the convergence of the implicit anchor like continuous path z t in the proof

strong convergence theorems for uniformly continuous pseudocontractive maps abstract let k be a nonempty closed convex subset of a real banach space e and let t k k be a uniformly continuous pseudocontraction fix any u k let x n be defined by the iterative process x 0 k x n 1 n n t x n 1 n x n 1 n u let denote the modulus of continuity of t with pseudo inverse if t t 0 t 1 and x n are bounded then under some mild conditions on the sequences n n and n n the strong convergence of x n to a fixed point of t is proved in the special case where t is lipschitz it is shown that the boundedness assumptions on t t 0 t 1 and x n can be dispensed with

compact quasi gas dynamic system and high performance computing abstract new compact hyperbolic quasi gas dynamic system of equations chqgd is derived from kinetic models for the one particle distribution function the system differs from the navier stokes equations by a second order value in the knudsen number the presence of second time derivatives makes the chqgd system hyperbolic in contrast to the navier stokes equations the system of chqgd equations allows one to construct efficient including parallel computational algorithms for modeling viscous flows we consider the properties of the new compact hyperbolic chqgd system and methods for its numerical solution two discretization options are described based on the finite volume method estimates are given of the solution dependence on the small parameter at the second time derivative special attention is paid to the algorithms adaptation to the architecture of computing systems with massive parallelism as an example of the algorithms use the results of numerical simulation of a supersonic separated flow around a compression angle are given a comparison with experimental data and numerical solutions of the navier stokes equations is presented

halpern s iteration for bregman strongly nonexpansive mappings in reflexive banach spaces we investigate strong convergence for bregman strongly nonexpansive mappings by halpern s iteration in the framework of reflexive banach spaces using the obtained result convergence for a family of bregman strongly nonexpansive mappings is also discussed as applications we apply our main result to problems of finding zeros of maximal monotone operators and equilibrium problems in reflexive banach spaces

a hybrid algorithm for pseudo contractive mappings abstract in this paper we suggest and analyze a hybrid algorithm for pseudo contractive mappings in hilbert spaces further we prove the strong convergence of the proposed iterative algorithm for lipschitz pseudo contractive mappings in hilbert spaces

strong convergence of krasnoselskii and mann s type sequences for one parameter nonexpansive semigroups without bochner integrals abstract in this paper we prove krasnoselskii and mann s type convergence theorems for nonexpansive semigroups without using bochner integral and without assuming the strict convexity of banach spaces one of our main results is the following let c be a compact convex subset of a banach space e and let t t t 0 be a one parameter strongly continuous semigroup of nonexpansive mappings on c let t n be a sequence in 0 satisfying lim inf n t n lim sup n t n and lim n t n 1 t n 0 let 0 1 define a sequence x n in c by x 1 c and x n 1 t t n x n 1 x n for n n then x n converges strongly to a common fixed point of t t t 0

the discrete plateau problem convergence results we solve the problem of nding and justifying an optimal fully discrete nite element procedure for approximating minimal including unsta ble surfaces in a previous paper we introduced the general framework and some preliminary estimates developed the algorithm and give the numerical results in this paper we prove the convergence estimate

strong convergence of an iterative method for nonexpansive mappings with new control conditions abstract the purpose of this paper is to study the convergence problem of an iterative method for nonexpansive mappings in banach spaces under some new control conditions on parameters

lattice boltzmann method for the compressible euler equations the lattice boltzmann model for the compressible euler equations is proposed together with its rigorous theoretical background the proposed model has completely overcome the defects of the previous model that the specific heat ratio cannot be chosen freely the macroscopic variables obtained from the solution are shown to satisfy in the limit of the small knudsen number the compressible euler equations if the variation of the solution is moderate this is the case where no shock waves or contact discontinuities appear in contrast when the solution makes steep variation at several localized regions due to the appearance of shock waves and contact discontinuities the corresponding macroscopic variables satisfy the weak form of the euler equations their derivation is carried out rigorously by taking into account the scale of variation of the solution correctly this is the first study that has laid the theoretical foundation of the lattice boltzmann model for the simulation of flows with shock waves and contact discontinuities numerical examples and the error estimates are also given which are consistent with the above theoretical arguments

iterative approximation of fixed points of nonexpansive mappings let k be a nonempty closed convex subset of a real banach space e which has a uniformly g teaux differentiable norm and t k k be a nonexpansive mapping with f t x k tx x for a fixed 0 1 define s k k by sx 1 x tx x k assume that zt converges strongly to a fixed point z of t as t 0 where zt is the unique element of k which satisfies zt tu 1 t tzt for arbitrary u k let n be a real sequence in 0 1 which satisfies the following conditions c1 lim n 0 c2 n for arbitrary x0 k let the sequence xn be defined iteratively by xn 1 nu 1 n sxn then xn converges strongly to a fixed point of t

the viscosity approximation process for quasi nonexpansive mappings in hilbert spaces in this paper we discuss the strong convergence of the viscosity approximation method in hilbert spaces relatively to the computation of fixed points of operators in the wide class of quasi nonexpansive mappings our convergence results improve previously known ones obtained for the class of nonexpansive mappings

an iterative method for elliptic problems on regions partitioned into substructures some new preconditioners for discretizations of elliptic boundary problems are studied with these preconditioners the domain under consideration is broken into subdomains and preconditioners are defined which only require the solution of matrix problems on the subdomains analytic estimates are given which guarantee that under appropriate hypotheses the preconditioned iterative procedure converges to the solution of the discrete equations with a rate per iteration that is independent of the number of unknowns numerical examples are presented which illustrate the theoretically predicted iterative convergence rates

numerical navier stokes solutions from gas kinetic theory abstract in order to test the gas kinetic based hydrodynamic scheme given in paper i k h prendergast and k xu j comput phys 109 53 1993 as a navier stokes solver we extend the scheme to two dimensions and exhibit some navier stokes solutions the scheme is a high resolution gas kinetic scheme in both space and time the advective and diffusive fluxes are coupled and solved at the same time by following the time dependent velocity distribution function for particles numerical results for some well defined euler and navier stokes test cases are presented the kolmogorov and the laminar boundary layer problems exhibit the dominance of the real viscosity effects the strong shock interaction test cases show the applicability of the scheme to supersonic gas flow

on two methods for approximating minimal surfaces in parametric form etude de deux methodes d approximation des surfaces minimales sous forme parametrique minimisation de l aire et de l energie convergence transformation conforme approchee exemples

an algorithm for prescribed mean curvature using isogeometric methods we present a newton type algorithm to find parametric surfaces of prescribed mean curvature with a fixed given boundary in particular it applies to the problem of minimal surfaces the algorithm relies on some global regularity of the spaces where it is posed which is naturally fitted for discretization with isogeometric type of spaces we introduce a discretization of the continuous algorithm and present a simple implementation using the recently released isogeometric software library igatools finally we show several numerical experiments which highlight the convergence properties of the scheme

strong convergence theorems for finitely many nonexpansive mappings abstract under the assumption that e is a reflexive banach space whose norm is uniformly geteaux differentiable and which has a weakly continuous duality mapping j with gauge function ceng cubiotti yao strong convergence theorems for finitely many nonexpansive mappings and applications nonlinear analysis 67 2007 1464 1473 introduced a new iterative scheme for a finite commuting family of nonexpansive mappings and proved strong convergence theorems about this iteration in this paper only under the hypothesis that e is a reflexive banach space which has a weakly continuous duality mapping j with gauge function and several control conditions about the iterative coefficient are removed we present a short and simple proof of the above theorem

effective metastability of halpern iterates in cat 0 spaces abstract this paper provides an effective uniform rate of metastability in the sense of tao on the strong convergence of halpern iterations of nonexpansive mappings in cat 0 spaces the extraction of this rate from an ineffective proof due to saejung is an instance of the general proof mining program which uses tools from mathematical logic to uncover hidden computational content from proofs this methodology is applied here for the first time to a proof that uses banach limits and hence makes a substantial reference to the axiom of choice

halpern s iteration in banach spaces abstract we prove strong convergence theorems for a sequence which is generated by halpern s iteration we also apply our result for finding zeros of an accretive operator our result improves the recent result of aoyama et al k aoyama y kimura w takahashi m toyoda approximation of common fixed points of a countable family of nonexpansive mappings in a banach space nonlinear anal 67 2007 2350 2360 by removing some assumptions on the parameters finally we discuss the new sufficient condition studied by song y song a new sufficient condition for the strong convergence of halpern type iterations appl math comput 198 2 2008 721 728 y song new strong convergence theorems for nonexpansive nonself mappings without boundary conditions comput math appl 56 6 2008 1473 1478 and correct the main result of song and chai y song x chai halpern iteration for firmly type nonexpansive mappings nonlinear anal 71 10 2009 4500 4506

variational convergence of discrete minimal surfaces this work is concerned with the convergence behavior of the solutions to parametric variational problems an emphasis is put on sequences of variational problems that arise as discretizations of either infinite dimensional optimization problems or infinite dimensional operator problems finally the results are applied to discretizations of the douglas plateau problem and of a boundary value problem in nonlinear elasticity

on convergence criteria of generalized proximal point algorithms we analyze some generalized proximal point algorithms which include the previously known proximal point algorithms as special cases weak and strong convergence of the proposed proximal point algorithms are proved under some mild conditions

finite element approximations to surfaces of prescribed variable mean curvature we give an algorithm for finding finite element approximations to surfaces of prescribed variable mean curvature which span a given boundary curve we work in the parametric setting and prove optimal estimates in the h 1 norm the estimates are verified computationally

strong convergence for a countable family of strict pseudocontractions in q uniformly smooth banach spaces we introduce a new iterative algorithm for finding a common fixed point of a countable family of strict pseudocontractions in q uniformly smooth and uniformly convex banach spaces we then prove that the sequence generated by the proposed algorithm converges strongly to a common fixed point of an infinite family of strict pseudocontractions our results mainly improve and extend the results announced by yao et al y yao y c liou g marino strong convergence of two iterative algorithms for nonexpansive mappings in hilbert spaces fixed point theory appl 2009 2009 7 pages doi 10 1155 2009 279058 art id 279058

four parameter proximal point algorithms several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm thus our results address the two important problems related to the proximal point algorithm one being that of strong convergence instead of weak convergence and the other one being that of acceptable errors one of the algorithms discussed was introduced by yao and noor 2008 7 while the other one is new and it is a generalization of the regularization method initiated by lehdili and moudafi 1996 9 and later developed by xu 2006 8 the new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals although these algorithms are distinct it turns out that for a particular case they are equivalent the results of this paper extend and generalize several existing ones in the literature

iterative construction for common fixed points of finitely many nonexpansive mappings abstract many problems encountered in applied mathematics can be recast as the problem of selecting a particular common fixed point of a family of nonexpansive mappings in a banach space in this paper under the hypotheses that e is a reflexive and strictly convex banach spaces with a uniformly g eaux differentiable norm the strong convergence of a iteration algorithm is proved for finding a common fixed point of a finite family of nonexpansive self mappings defined on a closed convex subset k of e