existence of positive solutions for non positive higher order bvps abstract we shall provide conditions on non positive function f t u 1 u n 1 so that the boundary value problem has at least one positive solution then we shall apply this result to establish several existence theorems which guarantee the multiple positive solutions

positive solutions of difference equations with two point right focal boundary conditions abstract we consider the following boundary value problem formula formula where 1 p n 1 is fixed using a fixed point theorem for operators on a cone we develop criteria for the existence of two positive solutions of the boundary value problem in addition upper and lower bounds for these positive solutions are established for special cases we also include several examples to dwell upon the importance of the results obtained

multiplicity of positive solutions of a class of nonlinear fractional differential equations this paper is concerned with the nonlinear fractional differential equation l d u f x u u 0 0 0 x 1 where l d dsn an 1dsn 1 a1ds11 0 j 1 2 n 1 some results are obtained for the existence nonexistence and multiplicity of positive solutions of the above equation by using krasnoselskii s fixed point theorem in a cone in particular it is proved that the above equation has n positive solutions under suitable conditions where n is an arbitrary positive integer

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

the rate of convergence for the method of alternating projections ii abstract the purpose of the paper is threefold 1 to develop a useful error bound for the method of alternating projections which is relatively easy to compute and remember 2 to exhibit a counterexample to a conjecture of kayalar and weinert 3 to show that in the case of at least three subspaces any error bound which only depends on the angles between the various subspaces involved can never be sharp

triple positive solutions and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem y 2 m t f y t y 2 j t y 2 m 1 t 0 t 1 y 2 i 0 0 y 2 i 1 0 i m 1 where 1 m f 0 growth conditions are imposed on f and inequalities involving an associated green s function are employed which enable us to apply the leggett williams fixed point theorem to cones in ordered banach spaces this in turn yields the existence of at least three positive symmetric concave solutions the emphasis here is that f depends on higher order derivatives

a note on the convergence of linear stationary iterative processes abstract this note investigates the convergence of a linear stationary iterative process of the form x k 1 h x k f it is shown that if h is a weakly convergent matrix then the sequence i h x k converges for any choice of x0 and f there are relaxation schemes for solving a symmetric positive semidefinite system of linear equations g x b whose iteration matrix is weakly convergent e g the sor method in such a case the above result implies that the sequence g x k b always converges in other words the sequence of residuals converges even when the system g x b is inconsistent special attention is given to the case when h is weakly convergent and symmetric here the sequence g x k b is monotonic decreasing and the sequence gxk converges to gx where x is a least squares solution furthermore if x0 belongs to the column space of g and the system g x b is solvable then the sequence xk converges to the minimum norm solution

existence of positive solutions of bvps for third order discrete nonlinear difference systems in this paper we consider the following boundary value problem of discrete email protected 3u 1 k f 1 k u 1 k u 2 k 0 email protected 0 t d 3u 2 k f 2 k u 1 k u 2 k 0 email protected 0 t with the dirichlet boundary conditionu 1 0 u 1 1 u 1 t 3 0 u 2 0 u 2 1 u 2 t 3 some new results of the existence nonexistence and multiplicity are obtained by using krasnosel skii s fixed point theorem in a cone in particular it is proved that the above boundary value problem has n positive solutions under suitable conditions where n is an arbitrary positive integer

on the rate of convergence of the alternating projection method in finite dimensional spaces abstract using the results of smith solmon and wagner k smith d solomon s wagner practical and mathematical aspects of the problem of reconstructing objects from radiographs bull amer math soc 83 1977 1227 1270 and nelson and neumann s nelson m neumann generalizations of the projection method with application to sor theory for hermitian positive semidefinite linear systems numer math 51 1987 123 141 we derive new estimates for the speed of the alternating projection method and its relaxed version in r m these estimates can be computed in at most o m 3 arithmetic operations unlike the estimates in papers mentioned above that require spectral information the new and old estimates are equivalent in many practical cases in cases when the new estimates are weaker the numerical testing indicates that they approximate the original bounds in papers mentioned above quite well

positive solutions of boundary value problems for third order functional difference equations the existence of positive solutions are established for the third order functional difference equation 3u n a n f n u w n 0 0 n t satisfying u n n n1 n 1 and u n n t 3 n n2 with 0 1 t 3 0 the results in this paper generalize and substantially improve recent work by agarwal and henderson on boundary value problems related to third order difference equations

numerical linear algebra for reconstruction inverse problems our goal in this paper is to discuss various issues we have encountered in trying to find and implement efficient solvers for a boundary integral equation bie formulation of an iterative method for solving a reconstruction problem we survey some methods from numerical linear algebra which are relevant for the solution of this class of inverse problems we motivate the use of our constructing algorithm discuss its implementation and mention the use of preconditioned krylov methods

positive solutions of nonlinear functional difference equations abstract in this paper we apply a cone theoretic fixed point theorem and obtain sufficient conditions for the existence of positive solutions to some boundary value problems for a class of functional difference equations we consider analogues of sublinear or superlinear growth in the nonlinear terms

spectral theory of second order vector difference equations abstract this paper is concerned with spectral problems of second order vector difference equations with two point boundary value conditions where the matrix valued coefficient of the leading term may be singular a concept of self adjointness of the boundary value conditions is introduced the self adjointness of the corresponding difference operator is discussed on a suitable admissible function space and fundamental spectral results are obtained the dual orthogonality of eigenfunctions is shown in a special case rayleigh s principles and the minimax theorems in two linear spaces are given as an application a comparison theorem for eigenvalues of two sturm liouville problems is presented

positive solutions of bvps for third order nonlinear difference systems abstract in this paper we consider the following discrete system boundary value problems 3 u 1 k f 1 k u 1 k u 2 k 0 k o t 3 u 2 k f 2 k u 1 k u 2 k 0 k o t with the dirichlet boundary condition u 1 0 u 1 1 u 1 t 3 0 u 2 0 u 2 1 u 2 t 3 some sufficient conditions are obtained for the existence of one or two positive solutions to the above system by using a nonlinear alternative of leray schauder type and krasnosel skii s fixed point theorem in a cone

randomized block kaczmarz method with projection for solving least squares abstract the kaczmarz method is an iterative method for solving overcomplete linear systems of equations a x b the randomized version of the kaczmarz method put forth by strohmer and vershynin iteratively projects onto a randomly chosen solution space given by a single row of the matrix a and converges linearly 1 in expectation to the solution of a consistent system in this paper we analyze two block versions of the method each with a randomized projection designed to converge in expectation to the least squares solution often faster than the standard variants our approach utilizes both a row and column paving of the matrix a to guarantee linear convergence when the matrix has consistent row norms called nearly standardized and a single column paving when the row norms are unbounded the proposed methods are an extension of the block kaczmarz method analyzed by needell and tropp and the randomized extended kaczmarz method of zouzias and freris the contribution is thus two fold unlike the standard kaczmarz method our results demonstrate convergence to the least squares solution of inconsistent systems both methods in the nearly standardized case and the second method in other cases by using appropriate blocks of the matrix this convergence can be significantly accelerated as is demonstrated by numerical experiments

equivalence of regularization and truncated iteration for general ill posed problems abstract we prove that solutions by direct regularization of linear systems are equivalent to truncated iterations of certain type of iterative methods our proofs extend previous results of h e fleming to the rank deficient case we give a unified approach that includes the undetermined and overdetermined problems

a minimum norm approach for low rank approximations of a matrix the problems of calculating a dominant eigenvector or a dominant pair of singular vectors arise in several large scale matrix computations in this paper we propose a minimum norm approach for solving these problems given a matrix a the new method computes a rank one matrix that is nearest to a regarding the frobenius matrix norm this formulation paves the way for effective minimization techniques the methods proposed in this paper illustrate the usefulness of this idea the basic iteration is similar to that of the power method but the rate of convergence is considerably faster numerical experiments are included

triple positive solutions of conjugate boundary value problems ii we consider the following boundary value problems 1 n py n a t h y n 2 t 0 1 y i 0 0 0 i p 1 y i 1 0 0 i n p 1 and 1 n p ny f k y y n 1y n 2 0 k m iy 0 0 i p 1 iy m p 1 0 0 i n p 1 where 1 p n 1 is fixed by employing fixed point theorems for operators on a cone existence criteria are developed for multiple at least three positive solutions of the boundary value problems as an application we also establish the existence of radial solutions of certain partial difference equations several examples are included to dwell upon the importance of the results obtained

on the semiconvergence of additive and multiplicative splitting iterations for singular linear systems in this paper we investigate the additive multiplicative and general splitting iteration methods for solving singular linear systems when the coefficient matrix is hermitian the semiconvergence conditions are proposed which generalize some results of bai z z bai on the convergence of additive and multiplicative splitting iterations for systems of linear equations j comput appl math 154 2003 195 214 for the nonsingular linear systems to the singular systems

multiple positive solutions of bvps for third order discrete difference systems this paper is concerned with the following email protected 3u 1 k f 1 k u 1 k u 2 k 0 email protected 0 t d 3u 2 k f 2 k u 1 k u 2 k 0 email protected 0 t with the dirichlet boundary conditionu 1 0 u 1 1 u 1 t 3 0 u 2 0 u 2 1 u 2 t 3 by using leggett williams fixed point theorem sufficient conditions are obtained for the existence of three positive solutions to the above system

eigenvalues and eigenfunctions of discrete conjugate boundary value problems abstract we consider the following boundary value problem 1 n p n y f k y y n 1 y n 2 0 k m i y 0 0 0 i p 1 i y m n i 0 0 i n p 1 where 1 p n 1 is fixed and 0 a characterization of the values of is carried out so that the boundary value problem has a positive solution next for 1 criteria are developed for the existence of two positive solutions of the boundary value problem in addition for particular cases we also offer upper and lower bounds for these positive solutions several examples are included to dwell upon the importance of the results obtained

algebraic and numerical techniques for the computation of matrix determinants we review modify and combine together several numerical and algebraic techniques in order to compute the determinant of a matrix or the sign of such a determinant the resulting algorithms enable us to obtain the solution by using a lower precision of computations and relatively few arithmetic operations the problem has important applications to computational geometry

extended m matrices and subtangentiality the concept of a singular m matrix a with respect to a proper cone x is extended by replacing the usual regularity condition a al b for a xnonnegative matrix b with the weaker condition exponential nonnegativity of a as in earlier work which dealt with the nonsingular case in the present characterizations the lack of regularity is overcome by employing subtangentiality

a generalization of concavity for finite differences abstract the concept of concavity is generalized to discrete functions u satisfying the n th order difference inequality 1 n k n u m 0 m 0 1 n and the homogeneous boundary conditions u 0 u k 1 0 u n k 1 u n n 0 for some k 1 n 1 a piecewise polynomial is constructed which bounds u below the piecewise polynomial is employed to obtain a positive lower bound on u m for m k n k where the lower bound is proportional to the supremum of u an analogous bound is obtained for a related green s function

semiconvergence criteria of iterations and extrapolated iterations and constructive methods of semiconvergent iteration matrices in this paper we simultaneously study iterations and extrapolated iterations for the consistent rectangular or sigular linear systems 1 1 ax b in the case that a is rectangular let a m n be a subproper splitting of a and let the generalized inverse m t s 1 2 exist and in the case that a is singular let a m n is a usual splitting of a consider the iteration 2 3 x j 1 gx j c where g m t s 1 2 nandc m t s 1 2 b if a is rectangular and g m 1n and c m 1b if a is singular and the extrapolated iteration 2 19 x j 1 g wx email protected where email protected r w 0 1 and 2 20 g w email protected email protected we establish the semiconcovergence criteria of the iterations 2 3 and 2 19 and present the efficient and convenient construction methods of the semiconvergent iteration matrices g and g w and the choices of the extrapolated parameter w

the importance of diagonal dominance in the iterative solution of equations generated from the boundary element method the unsymmetric matrix equations generated from the boundary element method bem can be solved iteratively with convergence to the correct solution guaranteed if the boundary element system of equations can be first transformed into an equivalent diagonally dominant system a transformation is presented which selectively annihilates terms in the coefficient matrix of the system ax b until an equivalent row diagonally dominant system if available is obtained the new row diagonally dominant system is well suited for use with jacobi and gauss seidel point iterative equation solvers the diagonal dominizing transformation presented in this work is not suitable for large systems of equations but is useful as a research tool for studying the importance of diagonal dominance in the iterative solution of equations generated from the bem a simple laplacian problem is used to examine the structure of the bem equations and to introduce the diagonal dominizing transformation the importance of diagonal dominance is shown by comparing iterative convergence of positive definite symmetric positive definite and diagonally dominant systems of bem equations obtained from a plane strain elasticity problem

new methods for linear inequalities abstract the iterative method of cimmino for solving linear equations is generalized to linear inequalities we also present a richardson type iterative method for solving the inequality problem which includes the generalized cimmino scheme convergence proofs are provided

the convergence of linear stationary iterative processes for solving singular unstructured systems of linear equations this paper reviews the convergence properties of linear stationary iterative processes for solving large unstructured systems of linear equations special attention is given to the case where the system to be solved is singular and possibly inconsistent the review starts by investigating the iteration bf x k 1 h bf x k bf f k 0 1 2 cdots when the system i h bf x bf f is inconsistent in this case the sequence left bf x k right always diverges nevertheless when h is convergent there exists a vector bf q in text null i h such that the sequence hat bf x k bf x k k bf q converges the limit of this sequence hat bf x solves a certain least squares problem and the sequence of residual vectors bf r k i h bf x k bf f converges to i h hat bf x bf f furthermore if h is similar to a real diagonal matrix then the corresponding norms of bf r k and hat bf x k hat bf x decr

multiple positive solutions of a discrete difference system this paper is concerned with the following system d 2u 1 k f 1 k u 1 k u 2 k 0 k 0 t d 2u 2 k f 2 k u 1 k u 2 k 0 k 0 t u 1 0 u 1 t 2 0 u 2 0 u 2 t 2 by using leggett williams fixed point theorem sufficient conditions are obtained for the existence of three positive solutions to the above system

generalizations of the projection methods with application to sor theory for hermitian positive semidefinite linear systems ann n complex matrixb is calledparacontracting if b 2 1 and 0 x n i b bx 2 x 2 we show that a productb b k b k 1 b1 ofk paracontracting matrices is semiconvergent and give upper bounds on the subdominant eigenvalue ofb in terms of the subdominant singular values of theb i s and in terms of the angles between certain subspaces our results here extend earlier results due to halperin and due to smith solomon and wagner we also determine necessary and sufficient conditions forn numbers in the interval 0 1 to form the spectrum of a product of two orthogonal projections and hence characterize the subdominant eigenvalue of such a product in the final part of the paper we apply the upper bounds mentioned earlier to provide an estimate on the subdominant eigenvalue of the sor iteration matrix associated with ann n hermitian positive semidefinite matrixa none of whose diagonal entries vanish

positive solutions and nonlinear eigenvalue problems for third order difference equations first solutions that lie in a cone intersected with an annular type region are obtained for the third order difference equation 3u t a t f u t 0 2 t t 2 satisfying the boundary conditions u 0 u 1 u t 3 0 in the cases that f is either superlinear or sublinear the second part is devoted to determining values of for which there exist positive solutions of 3u t a t f u t 0 2 t t 2 satisfying the boundary conditions u 0 u 1 u t 3 0 when both limx 0 f x x and limx f x x exist as positive real numbers

convergent block iterative algorithms for image reconstruction from inconsistent data it has been shown that convergence to a solution can be significantly accelerated for a number of iterative image reconstruction algorithms including simultaneous cimmino type algorithms the expectation maximization method for maximizing likelihood emml and the simultaneous multiplicative algebraic reconstruction technique smart through the use of rescaled block iterative bi methods these bi methods involve partitioning the data into disjoint subsets and using only one subset at each step of the iteration one drawback of these methods is their failure to converge to an approximate solution in the inconsistent case in which no image consistent with the data exists they are always observed to produce limit cycles lcs of distinct images through which the algorithm cycles no one of these images provides a suitable solution in general the question that arises then is whether or not these lc vectors retain sufficient information to construct from them a suitable approximate solution we show that they do to demonstrate that we employ a feedback technique in which the lc vectors are used to produce a new data vector and the algorithm restarted convergence of this nested iterative scheme to an approximate solution is then proven preliminary work also suggests that this feedback method may be incorporated in a practical reconstruction method

row action methods for huge and sparse systems and their applications this paper brings together and discusses theory and applications of methods identified and labelled as row action methods for linear feasibility problems find x in bf r n such that ax leqq b linearly constrained optimization problems minimize f x subject to ax leqq b and some interval convex programming problems minimize f x subject to c leqq ax leqq b the main feature of row action methods is that they are iterative procedures which without making any changes to the original matrix a use the rows of a one row at a time such methods are important and have demonstrated effectiveness for problems with large or huge matrices which do not enjoy any detectable or usable structural pattern apart from a high degree of sparaseness fields of application where row action methods are used in various ways include image reconstruction from projection operations research and game theory learning theory pattern recognition and transportation theory a row action method for the

an algorithm for prescribed multiple domination in arbitrary graphs abstract let g v e be a graph with node set v edge set e and ivi n a set s c v is an open h dominating set if every node v 6 v is adjacent to at least h nodes in s consider a vector p pl p2 p with p positive integers and pi deg vi a set s c v is an open p dominating set if every vi 6 v is adjacent to at least pl nodes in s we develop a backtracking algorithm that finds the open p domination number of g which is the smallest cardinality of an open p dominating set a comparative computational experiment is presented on arbitrary graphs of different sizes keywords multiple domination backtracking 1 introduction the notion of domination in graphs was introduced formally by k6nig 1 and ore 2 since then a tremendous volume of research work related to the concept of domination has been published for an extensive bibliography see 3 here we follow in general the notation and terminology given by harary 4 let g v e be a graph with iv n and el m the

comparison theorems for weak splittings of bounded operators comparison theorems for weak splittings of bounded operators are presented these theorems extend the classical comparison theorem for regular splittings of matrices by varga the less known result by wo nicki and the recent results for regular and weak regular splittings of matrices by neumann and plemmons elsner and lanzkron rose and szyld the hypotheses of the theorems presented here are weaker and the theorems hold for general banach spaces and rather general cones hypotheses are given which provide strict inequalities for the comparisons it is also shown that the comparison theorem by alefeld and volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators

necessary and sufficient conditions for the convergence of orthomin k on singular and inconsistent linear systems we consider the convergence of orthomin k on singular and inconsistent linear systems criteria for the breakdown of orthomin k are discussed and analyzed moreover necessary and sufficient conditions for the convergence of orthomin k for any right hand side are given and a rate of convergence is provided as well finally numerical experiments are shown to confirm the convergence theorem

finite precision behavior of stationary iteration for solving singular systems a stationary iterative method for solving a singular system ax b converges for any starting vector if limi gi exists where g is the iteration matrix and the solution to which it converges depends on the starting vector we examine the behavior of stationary iteration in finite precision arithmetic a pertubation bound for singular systems is derived and used to define forward stability of a numerical method a rounding error analysis enables us to deduce conditions under which a stationary iterative method is forward stable or backward stable the component of the forward error in the null space of a can grow linearly with the number of iterations but it is innocuous as long as the iteration converges reasonably quickly as special cases we show that when a is symmetric positive semidefinite the richardson iteration with optimal parameter is forward stable and if a also has unit diagonal and property a then the gauss seidel method is both forward and backward stable two numerical examples are given to illustrate the analysis

on parallel multisplitting iterative methods for singular linear systems in this paper the semiconvergence of parallel multisplitting iterative methods and extrapolated parallel methods for singular linear systems is discussed some sufficient conditions for semiconvergence are presented whenever the multisplittings are p regular sub regular or nonnegative as special cases the semiconvergence of parallel generalized aor block aor aor parallel generalized sor block sor sor extrapolated parallel generalized block aor and extrapolated parallel generalized block sor methods are proved

a splitting preconditioner for a block two by two linear system with applications to the bidomain equations we construct an alternating splitting iteration scheme for solving and preconditioning a block two by two linear system arising from numerical discretizations of the bidomain equations the convergence theory of this class of splitting iteration methods is established and some useful properties of the preconditioned matrix are analyzed the potential of this approach is illustrated by numerical experiments

further results on fixed sign solutions for a system of higher order difference equations abstract we consider the following system of n th order boundary value problems download full size image where u i k u i k u i k n 2 u i k 1 i m criteria are offered for the existence of single and double fixed sign solutions of the system we also include examples to illustrate the importance of the results obtained

generalized m matrices abstract let r n be a proper cone from the theory of m matrices see e g 1 it is known that if there exist 0 and a matrix b such that a b i then the following conditions are equivalent i a is monotone ii a is seminegative iii re spectrum a e ta t 0 is more general than the structural assumption a b i conditions i iii are nevertheless all equivalent to iv x ax 0 when e holds

generalized m matrices and ordered banach algebras abstract a matrix a r n n is an m matrix if and only if the mapping x ax is quasimonotone increasing qmi and if the right spectral bound of a is negative here qmi is meant with respect to the natural cone k x r n x k 0 one possibility of generalizing m matrices is to consider qmi linear mappings on r n with respect to other cones k r n we will present results on such mappings in the banach algebra setting and discuss some special cones moreover by means of one sided estimates it is possible to get informations on the right spectral bound of qmi mappings