modified numerov s method for inverse sturm liouville problems in this paper we propose a new modified numerov s method for recovering from eigenvalues a symmetric potential of a sturm liouville operator with dirichlet boundary conditions we use interpolation to refine the mesh sufficiently for numerov s method to be effective even without the asymptotic correction technique of andrew and paine accuracy and stability of the method are investigated convergence of the method is established our method is extended to deal with natural boundary conditions numerical experiments confirm its competitiveness

asymptotic correction of numerov s eigenvalue estimates with natural boundary conditions abstract asymptotic correction at negligible extra cost greatly improves the accuracy of higher eigenvalues computed by finite difference or finite element methods and generally increases the accuracy of the lower ones as well this paper gives a brief overview of the technique and describes how its previous use with numerov s method may be extended to problems with natural boundary conditions numerical results indicate that it is just as successful as with dirichlet boundary conditions

iterative approaches to convex feasibility problems in banach spaces abstract the convex feasibility problem cfp of finding a point in the nonempty intersection i 1 n c i is considered where n 1 is an integer and each c i is assumed to be the fixed point set of a nonexpansive mapping t i x x with x a banach space it is shown that the iterative scheme x n 1 n 1 y 1 n 1 t n 1 x n is strongly convergent to a solution of cfp provided the banach space x either is uniformly smooth or is reflexive and has a weakly continuous duality map and provided the sequence n satisfies certain conditions the limit of x n is located as q y where q is the sunny nonexpansive retraction from x onto the common fixed point set of the t i s

correction of numerov s eigenvalue estimates the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by numerov s method with mesh lengthh iso k6h4 we show that a simple correction technique of paine de hoog and anderssen reduces the error to one ofo k3h4 numerical examples demonstrate the usefulness of this correction even for low values ofk

matrix methods for computing eigenvalues of sturm liouville problems of order four abstract this paper examines and develops matrix methods to approximate the eigenvalues of a fourth order sturm liouville problem subjected to a kind of fixed boundary conditions furthermore it extends the matrix methods for a kind of general boundary conditions the idea of the methods comes from finite difference and numerov s methods as well as boundary value methods for second order regular sturm liouville problems moreover the determination of the correction term formulas of the matrix methods is investigated in order to obtain better approximations of the problem with fixed boundary conditions since the exact eigenvalues for q 0 are known in this case finally some numerical examples are illustrated

some problems and results in the study of nonlinear analysis this paper is presented as a survey of the latest and new results on some topics in the study of nonlinear analysis which were obtained by the author and some chinese mathematicians

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

boundary value method for inverse sturm liouville problems in this paper we present a method to recover symmetric and non symmetric potential functions of inverse sturm liouville problems from the knowledge of eigenvalues the linear multistep method coupled with suitable boundary conditions known as boundary value method bvm is the main tool to approximate the eigenvalues in each iteration step of the used newton method the bvm was extended to work for neumann neumann boundary conditions moreover a suitable approximation for the asymptotic correction of the eigenvalues is given numerical results demonstrate that the method is able to give good results for both symmetric and non symmetric potentials

viscosity approximation methods for nonexpansive mappings abstract viscosity approximation methods for nonexpansive mappings are studied consider a nonexpansive self mapping t of a closed convex subset c of a banach space x suppose that the set fix t of fixed points of t is nonempty for a contraction f on c and t 0 1 let x t c be the unique fixed point of the contraction x tf x 1 t tx consider also the iteration process x n where x 0 c is arbitrary and x n 1 n f x n 1 n tx n for n 1 where n 0 1 if x is either hilbert or uniformly smooth then it is shown that x t and under certain appropriate conditions on n x n converge strongly to a fixed point of t which solves some variational inequality

implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings in this paper we propose a new implicit iteration scheme with perturbed mapping for approximation of common fixed points of a finite family of nonexpansive mappings we establish some convergence theorems for this implicit iteration scheme in particular necessary and sufficient conditions for strong convergence of this implicit iteration scheme were obtained

approximation methods for common fixed points of infinite countable family of nonexpansive mappings let e be a real strictly convex and reflexive banach space with a uniformly gateaux differentiable norm and c be a nonempty closed convex subset of e consider the iterative sequence x n email protected n 1f x n email protected n email protected l n 1 w nx n where w n is the w mapping generated by an infinite countable family of nonexpansive mappings t n t n 1 t 1 and a n a n 1 a 1 such that the common fixed point sets email protected email protected n 1 f t n email protected and f c c is a given contractive mapping under very mild conditions on the parameters we prove that x n converges strongly to email protected f where p is the unique solution in f to the following variational inequality 0for all x f

fast divergence conforming reduced basis methods for steady navier stokes flow abstract reduced basis methods rb methods or rbms form one of the most promising techniques to deliver numerical solutions of parametrized pdes in real time with reasonable accuracy for incompressible flow problems rbms based on lbb stable velocity pressure spaces do not generally inherit the stability of the underlying high fidelity model and instead additional stabilization techniques must be introduced one way of bypassing the loss of lbb stability in the rbm is to inflate the velocity space with supremizer modes this however deteriorates the performance of the rbm in the performance critical online stage as additional dofs must be introduced to retain stability while these dofs do not effectively contribute to accuracy of the rb approximation in this work we consider a velocity only rb approximation exploiting a solenoidal velocity basis the solenoidal reduced basis emerges directly from the high fidelity velocity solutions in the offline stage by means of piola transforms the solenoidality of the velocity space is retained under geometric transformations making the proposed rb method suitable also for the investigation of geometric parameters to ensure exact solenoidality of the high fidelity velocity solutions that constitute the rb we consider approximations based on divergence conforming compatible b splines we show that the velocity only rb method leads to a significant improvement in computational efficiency in the online stage and that the pressure solution can be recovered a posteriori at negligible extra cost we illustrate the solenoidal rb approach by modeling steady two dimensional navier stokes flow around a naca0015 airfoil at various angles of attack

a note on approximating fixed points of nonexpansive mappings by the ishikawa iteration process abstract letxbe a uniformly convex banach space that satisfies opial s condition or whose norm is frechet differentiable letcbe a bounded closed convex subset ofx and lett c cbe a nonexpansive mapping it is shown that for any initial datax0 c the ishikawa iterates xn defined byxn 1 tnt sntxn 1 sn xn 1 tn xn n 0 with the restrictions that lim n snis less than 1 and for any subsequence nk k 0of n n 0 k 0 tnk 1 tnk diverges converge to a fixed point oftweakly thus such a result complements theorem 1 of tan and xu j math anal appl 178 1993 301 308 and generalizes to a certain extent theorem 2 of reich j math anal appl 67 1979 274 276

relaxed viscosity approximation methods for fixed point problems and variational inequality problems abstract let x be a real strictly convex and reflexive banach space with a uniformly g teaux differentiable norm and c be a nonempty closed convex subset of x let t n n 1 be a sequence of nonexpansive self mappings on c such that the common fixed point set f n 1 f t n 0 and f c c be a given contractive mapping and n be a sequence of nonnegative numbers in 0 1 consider the following relaxed viscosity approximation method x n 1 1 n n x n n f y n n w n y n y n 1 n x n n w n x n n 1 where w n is the w mapping generated by t n t n 1 t 1 and n n 1 1 for each n 1 it is proven that under very mild conditions on the parameters the sequence x n of approximate solutions generated by the proposed method converges strongly to some p f where p is the unique solution in f to the following variational inequality i f p j p x 0 x f

viscosity approximation methods for a finite family of nonexpansive mappings in banach spaces abstract by using viscosity approximation methods for a finite family of nonexpansive mappings in banach spaces some sufficient and necessary conditions for the iterative sequence to converging to a common fixed point are obtained the results presented in the paper extend and improve some recent results in h k xu viscosity approximation methods for nonexpansive mappings j math anal appl 298 2004 279 291 h k xu remark on an iterative method for nonexpansive mappings comm appl nonlinear anal 10 2003 67 75 h h bauschke the approximation of fixed points of compositions of nonexpansive mappings in banach spaces j math anal appl 202 1996 150 159 b halpern fixed points of nonexpansive maps bull amer math soc 73 1967 957 961 j s jung iterative approaches to common fixed points of nonexpansive mappings in banach spaces j math anal appl 302 2005 509 520 p l lions approximation de points fixes de contractions c r acad sci paris ser a 284 1977 1357 1359 a moudafi viscosity approximation methods for fixed point problems j math anal appl 241 2000 46 55 s reich strong convergence theorems for resolvents of accretive operators in banach spaces j math anal appl 75 1980 128 292 r wittmann approximation of fixed points of nonexpansive mappings arch math 58 1992 486 491

stability of iterative procedures with errors for approximating common fixed points of a couple of q contractive like mappings in banach spaces abstract recently agarwal cho li and huang r p agarwal y j cho j li n j huang stability of iterative procedures with errors approximating common fixed points for a couple of quasi contractive mappings in q uniformly smooth banach spaces j math anal appl 272 2002 435 447 introduced the new iterative procedures with errors for approximating the common fixed point of a couple of quasi contractive mappings and showed the stability of these iterative procedures with errors in banach spaces in this paper we introduce a new concept of a couple of q contractive like mappings q 1 in a banach space and apply these iterative procedures with errors for approximating the common fixed point of the couple of q contractive like mappings the results established in this paper improve extend and unify the corresponding ones of agarwal cho li and huang r p agarwal y j cho j li n j huang stability of iterative procedures with errors approximating common fixed points for a couple of quasi contractive mappings in q uniformly smooth banach spaces j math anal appl 272 2002 435 447 chidume c e chidume approximation of fixed points of quasi contractive mappings in l p spaces indian j pure appl math 22 1991 273 386 chidume and osilike c e chidume m o osilike fixed points iterations for quasi contractive maps in uniformly smooth banach spaces bull korean math soc 30 1993 201 212 liu q h liu on naimpally and singh s open questions j math anal appl 124 1987 157 164 q h liu a convergence theorem of the sequence of ishikawa iterates for quasi contractive mappings j math anal appl 146 1990 301 305 osilike m o osilike a stable iteration procedure for quasi contractive maps indian j pure appl math 27 1996 25 34 m o osilike stability of the ishikawa iteration method for quasi contractive maps indian j pure appl math 28 1997 1251 1265 and many others in the literature

viscosity approximation methods for fixed points problems the aim of this work is to propose viscosity approximation methods which amounts to selecting a particular fixed point of a given nonexpansive self mapping

iterative approaches to common fixed points of nonexpansive mappings in banach spaces abstract the iteration scheme for families of nonexpansive mappings essentially due to halpern bull amer math soc 73 1967 957 961 is established in a banach space the main theorem extends a recent result of o hara et al nonlinear anal 54 2003 1417 1426 to a banach space setting for the same iteration scheme with finitely many mappings a complementary result to a result of jung and kim bull korean math soc 34 1997 93 102 also bauschke j math anal appl 202 1996 150 159 is obtained by imposing other condition on the sequence of parameters our results also improve results in c r acad sci ser a b paris 284 1977 1357 1359 j math anal appl 211 1997 71 83 arch math 59 1992 486 491 in framework of a hilbert space

sampling for the fourth order sturm liouville differential operator we extend the sampling method to compute the eigenvalues of a fourth order differential operator we show that the shannon sampling theorem is not applicable due to the growth of the solutions on the real line the inverse spectral theory as developed by mclaughlin and kramer s theorem allow us to express the characteristic function explicitly by a new sampling formula

accurate solutions of fourth order sturm liouville problems recently we introduced a new method which we call the extended sampling method to compute the eigenvalues of second order sturm liouville problems with eigenvalue dependent potential we shall see in this paper how we use this method to compute the eigenvalues of fourth order sturm liouville problems and present its practical use on a few examples

positive solutions of fourth order nonlinear singular sturm liouville eigenvalue problems abstract we consider the existence of positive solutions for the following fourth order singular sturm liouville eigenvalue problems 1 p t p t u t g t f t u u 0 0 t 1 1 u 0 1 u 0 0 1 u 1 1 u 1 0 2 u 0 2 lim t 0 p t u t 0 2 u 1 2 lim t 1 p t u t 0 where 0 g p may be singular at t 0 and or 1 moreover f t x y may also have singularity at x 0 and or y 0 by using fixed point theory in cones an explicit interval for is derived such that for any in this interval the existence of at least one positive solution to the boundary value problem is guaranteed our results extend and improve many known results including singular and nonsingular cases

spline functions the biharmonic operator and approximate eigenvalues the biharmonic operator plays a central role in a wide array of physical models such as elasticity theory and the streamfunction formulation of the navier stokes equations its spectral theory has been extensively studied in particular the one dimensional case over an interval constitutes the basic model of a high order sturm liouville problem the need for corresponding numerical simulations has led to numerous works the present paper relies on a discrete biharmonic calculus the primary object of this calculus is a high order compact discrete biharmonic operator dbo the dbo is constructed in terms of the discrete hermitian derivative however the underlying reason for its accuracy remained unclear this paper is a contribution in this direction expounding the strong connection between cubic spline functions on an interval and the dbo the first observation is that the scaled fourth order distributional derivative of the cubic spline is identical to the action of the dbo on grid functions it is shown that the kernel of the inverse of the discrete operator is up to scaling equal to the grid evaluation of the kernel of left left frac d dx right 4 right 1 and explicit expressions are presented for both kernels as an important application the relation between the infinite set of eigenvalues of the fourth order sturm liouville problem and the finite set of eigenvalues of the discrete biharmonic operator is studied the discrete eigenvalues are proved to converge at an optimal o h 4 rate to the continuous ones another consequence is the validity of a comparison principle it is well known that there is no maximum principle for the fourth order equation however a positivity result is derived both for the continuous and the discrete biharmonic equation showing that in both cases the kernels are order preserving

exponentially convergent symbolic algorithm of the functional discrete method for the fourth order sturm liouville problems with polynomial coefficients abstract a new symbolic algorithmic implementation of the functional discrete fd method is developed and justified for the solution of fourth order sturm liouville problem on a finite interval in the hilbert space the eigenvalue problem for the fourth order ordinary differential equation with polynomial coefficients is investigated the sufficient conditions of an exponential convergence rate of the proposed approach are received the obtained estimates of the absolute errors of fd method significantly improve the accuracy of the estimates obtained earlier by i p gavrilyuk v l makarov and a m popov in 2010 our algorithm is symbolic and operates with the decomposition coefficients of the eigenfunction corrections in some basis the number of summands in these decompositions depends on the degree of the potential coefficients and the correction number our method uses only the algebraic operations and basic operations on 2 1 column vectors and 2 2 matrices the proposed approach does not require solving any boundary value problems and computations of any integrals unlike the previous variants of fd method by i p gavrilyuk v l makarov a m popov and n m romaniuk in 2010 and 2017 the corrections to eigenpairs are computed exactly as analytical expressions the numerical examples illustrate the theoretical results the numerical results obtained with the fd method are compared with the numerical test results obtained with other existing numerical techniques

strong convergence theorems on an iterative method for a family of finite nonexpansive mappings in reflexive banach spaces in this paper by using some new analysis techniques we study the approximation problems of common fixed points of halpern s iterative sequence for a class of finite nonexpansive mappings in strictly convex and reflexive banach spaces by using banach s limit the main results presented in this paper generalize extend and improve the corresponding results of bauschke the approximation of fixed points of compositions of nonexpansive mappings in hilbert spaces j math anal appl 202 1996 150 159 halpern fixed points of nonexpansive maps bull am math soc 73 1967 957 961 shioji and takahashi strong convergence of approximated sequences for nonexpansive mappings in banach spaces proc am math soc 125 1997 3641 3645 takahashi et al approximation of common fixed points of a family of finite nonexpansive mappings in banach spaces sci math jpn 56 2002 475 480 wittmann approximation of fixed points of nonexpansive mappings arch math 58 1992 486 491 xu another control condition in an iterative method for nonexpansive mappings bull austral math soc 65 2002 109 113 remarks on an iterative method for nonexpansive mappings commun appl nonlinear anal 10 2003 67 75 and others

correction of finite element estimates for stum liouville eigenvalues it is shown that a simple asymptotic correction technique of paine de hoog and anderssen reduces the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by the finite element method with linear hat functions and mesh lengthh fromo k 4 h 2 too k h 2 the result still holds when the matrix elements are evaluated by simpson s rule but if the trapezoidal rule is used the error iso k 2 h 2 numerical results demonstrate the usefulness of the correction even for low values ofk

an efficient method for computing eigenelements of sturm liouville fourth order boundary value problems in this paper an efficient method based on the adomian decomposition for computing the eigenelements of fourth order sturm liouville boundary value problems is developed numerical examples show that the method proposed is easy to implement and produces accurate results

strong convergence of viscosity approximation methods for finding zeros of accretive operators in banach spaces abstract in this paper we introduce a composite iterative scheme by viscosity approximation method for finding a zero of an accretive operator in banach spaces then we establish strong convergence theorems for the composite iterative scheme the main theorems improve and generalize the recent corresponding results of kim and xu t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 2005 51 60 qin and su x qin y su approximation of a zero point of accretive operator in banach spaces j math anal appl 329 2007 415 424 and xu h k xu strong convergence of an iterative method for nonexpansive and accretive operators j math anal appl 314 2006 631 643 as well as aoyama et al k aoyama y kimura w takahashi m toyoda approximation of common fixed points of a countable family of nonexpansive mappings in banach spaces nonlinear anal 67 2007 2350 2360 benavides et al t d benavides g l acedo h k xu iterative solutions for zeros of accretive operators math nachr 248 249 2003 62 71 chen and zhu r chen z zhu viscosity approximation fixed points for nonexpansive and m accretive operators fixed point theory and appl 2006 2006 1 10 and kamimura and takahashi s kamimura w takahashi approximation solutions of maximal monotone operators in hilberts spaces j approx theory 106 2000 226 240

the approximation of fixed points of compositions of nonexpansive mappings in hilbert space determining fixed points of nonexpansive mappings is a frequent problem in mathematics and physical sciences an algorithm for finding common fixed points of nonexpansive mappings in hilbert space essentially due to halpern is analyzed the main theorem extends wittmann s recent work and partially generalizes a result by lions algorithms of this kind have been applied to the convex feasibility problem

some iterative methods for finding fixed points and for solving constrained convex minimization problems the present paper is divided into two parts in the first part we introduce implicit and explicit iterative schemes for finding the fixed point of a nonexpansive mapping defined on the closed convex subset of a real hilbert space we establish results on the strong convergence of the sequences generated by the proposed schemes to a fixed point of a nonexpansive mapping such a fixed point is also a solution of a variational inequality defined on the set of fixed points in the second part we propose implicit and explicit iterative schemes for finding the approximate minimizer of a constrained convex minimization problem and prove that the sequences generated by our schemes converge strongly to a solution of the constrained convex minimization problem such a solution is also a solution of a variational inequality defined over the set of fixed points of a nonexpansive mapping the results of this paper extend and improve several results presented in the literature in the recent past