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Abstract: A counterexample is provided in this short note to show that some of local convergence theorems established in [D.K.R. Babajee, M.Z. Dauhoo, An analysis of the properties of the variants of Newton’s method with third order convergence, Appl. Math. Comput. 183 (2006) 659–684] are not always true. Some mistakes in the proofs of these theorems are pointed out. [Copyright &y& Elsevier]

In 1984, Kyurkchiev et al. constructed an infinite sequence of iterative methods for simultaneous approximation of polynomial zeros (with known multiplicity). The first member of this sequence of iterative methods is the famous root-finding method derived independently by Farmer and Loizou (1977) and Gargantini (1978). For every given positive integer N , the N th method of this family has the order of convergence 2 N + 1. In this paper, we prove two new local convergence results for this family of iterative methods. The first one improves the result of Kyurkchiev et al. (1984). We end the paper with a comparison of the computational efficiency, the convergence behavior and the computational order convergence of different methods of the family. [ABSTRACT FROM AUTHOR]

Abstract Twenty years after the classical book of Ortega and Rheinboldt was published, five definitions for the Q -convergence orders of sequences were independently and rigorously studied (i.e., some orders characterized in terms of others), by Potra (1989), resp. Beyer, Ebanks and Qualls (1990). The relationship between all the five definitions (only partially analyzed in each of the two papers) was not subsequently followed and, moreover, the second paper slept from the readers attention. The main aim of this paper is to provide a rigorous, selfcontained, and, as much as possible, a comprehensive picture of the theoretical aspects of this topic, as the current literature has taken away the credit from authors who obtained important results long ago. Moreover, this paper provides rigorous support for the numerical examples recently presented in an increasing number of papers, where the authors check the convergence orders of different iterative methods for solving nonlinear (systems of) equations. Tight connections between some asymptotic quantities defined by theoretical and computational elements are shown to hold. [ABSTRACT FROM AUTHOR]

This paper discusses the range of the eigenvalues of a class of matrices. By using the eigenvalues range of a class of matrices, an extension of the inner product type Cauchy–Schwarz inequality is obtained, the convergence proof of the least squares based iterative algorithm for solving the coupled Sylvester matrix equations is given and the best convergence factor is determined. Moreover, by using the eigenvalues range of this class of matrices, an iterative algorithm for solving linear matrix equation is established. Three numerical examples are offered to illustrate the effectiveness of the results suggested in this paper. [ABSTRACT FROM AUTHOR]

In this paper, an iteration process, referred to in short as MMP, will be considered. This iteration is related to finding the maximum modulus of a complex polynomial over a unit disc on the complex plane creating intriguing images. Kalantari calls these images polynomiographs independently from whether they are generated by the root finding or maximum modulus finding process applied to any polynomial. We show that the images can be easily modified using different MMP methods (pseudo-Newton, MMP-Householder, methods from the MMP-Basic, MMP-Parametric Basic or MMP-Euler–Schröder Families of Iterations) with various kinds of non-standard iterations. Such images are interesting from three points of views: scientific, educational and artistic. We present the results of experiments showing automatically generated non-trivial images obtained for different modifications of root finding MMP-methods. The colouring by iteration reveals the dynamic behaviour of the used root finding process and its speed of convergence. The results of the present paper extend Kalantari’s recent results in finding the maximum modulus of a complex polynomial based on Newton’s process with the Picard iteration to other MMP-processes with various non-standard iterations. [ABSTRACT FROM AUTHOR]

In this paper, we present a new iterative method based on the hybrid viscosity approximation method and the hybrid steepest-descent method for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of common fixed points of a finite family of nonexpansive mappings in Hilbert spaces. Furthermore, we prove that the proposed iterative method has strong convergence under some mild conditions imposed on algorithm parameters. The results presented in this paper improve and extend the corresponding results reported by some authors recently. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new iterative method that is the combination of the proximal point algorithm, viscosity approximation method and alternating resolvent method for finding the common zeros of two accretive operators in Banach spaces. And we will prove the strong convergence theorems for the iterative algorithms and give the example of the main theorems. The results of this paper are improvements and extensions of the corresponding ones announced by many others. [ABSTRACT FROM AUTHOR]

Various problems of pure and applied sciences can be studied in the unified frame work of the system of nonlinear equations. In this paper, a new family of iterative methods for solving a system of nonlinear equations is developed by using a new decomposition technique. The convergence of the new methods is proved. Efficiency index of the proposed methods is discussed and compared with some other well-known methods. The upper bounds of the error and the radius of convergence of the methods are also found. For the implementation and performance of the new methods, the combustion problem, streering problem and Van der Pol equation are solved and the results are compared with some existing methods. Several new iterative methods are derived from the general iterative scheme. Using the ideas and techniques of this paper, one may be able to suggest and investigate a wide class of iterative methods for solving the system of nonlinear equations. This is another direction of future research. [ABSTRACT FROM AUTHOR]

In this paper we propose a simple numerical method for approximating solutions of Fredholm–Volterra integral equations of the second kind. The method is based on Picard iteration and uses a suitable quadrature formula. Under certain conditions, we prove the existence and uniqueness of the solution and give error estimates for our approximations. The paper concludes with numerical examples and a discussion of the approximations proposed. [ABSTRACT FROM AUTHOR]

In this paper, the author presents a graphical tool that allows to study the real dynamics of iterative methods whose iterations depends on one parameter in an easy and compact way. This tool gives the information as previous tools such as Feigenbaum diagrams and Lyapunov exponents for every initial point. The convergence plane can be used, inter alia, to find the elements of a family that have good convergence properties, to see how the basins of attraction changes along the elements of the family, to study two-point methods such as Secant method or even to study two-parameter families of iterative methods. To show the applicability of the tool an example of the dynamics of the Damped Newton’s method applied to a cubic polynomial is presented in this paper. [ABSTRACT FROM AUTHOR]

Abstract Nonlinear equations arise in various fields of science and engineering. The present era of computational science – where one needs maximum achievement in minimum time – demands proposal of new and efficient iterative methods for solving nonlinear equations and systems. While the new methods are expected to be higher order convergent, the time efficiency and lesser computational information used are the top priorities. In this paper, we propose a new three-step iterative nonlinear solver for nonlinear equations and systems. The proposed method requires three evaluations of function and two evaluations of the first-order derivative per iteration. The proposed method is sixth order convergent, which is also proved theoretically. The performance of the proposed method is tested against other existing methods on the basis of error distributions, computational efficiency and CPU times. The numerical results on the application of the discussed methods on various nonlinear equations and systems, including an application problem related to combustion for a temperature of 3000 °C, show that the proposed method is comparable with existing methods with the main feature of the proposed method being its time-effectiveness. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a block format of left-looking AINV preconditioner for a nonsymmetric matrix. This preconditioner has block 1 × 1 or 2 × 2 pivot entries. It is introduced based on a block format of Gaussian Elimination process which has been studied in [14]. We have applied the multilevel nested dissection reordering as the preprocessing and have compared this block preconditioner by the plain left-looking AINV preconditioner. If we mix the multilevel nested dissection by the maximum weighted matching process, then the numerical experiments indicate that the number of 2 × 2 pivot entries in the block preconditioner will grow up. In this case, the block preconditioner makes GMRES method convergent in a smaller number of iterations. In the numerical section, we have also compared the ILUT and block left-looking AINV preconditioners. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a numerical solution for a finite time complete tracking problem based on the iterative learning control technique for dynamical systems governed by partial differential inclusions of parabolic type with noninstantaneous impulses. By imposing a standard Lipschitz condition on a set-valued mapping and applying conventional P-type updating laws with an initial iterative learning mechanism, we successfully establish an iterative learning process for the tracking problem and conduct a novel convergence analysis with the help of Steiner-type selectors. Sufficient conditions are presented for ensuring asymptotical convergence of the tracking error to zero. Numerical examples are provided to verify the effectiveness of the proposed method with a suitable selection of set-valued mappings. [ABSTRACT FROM AUTHOR]

Abstract In this paper we show a strategy to devise third order iterative methods based on classic second order ones such as Steffensen's and Kurchatov's. These methods do not require the evaluation of derivatives, as opposed to Newton or other well known third order methods such as Halley or Chebyshev. Some theoretical results on convergence will be stated, and illustrated through examples. These methods are useful when the functions are not regular or the evaluation of their derivatives is costly. Furthermore, special features as stability, laterality (asymmetry) and other properties can be addressed by choosing adequate nodes in the design of the methods. [ABSTRACT FROM AUTHOR]

Abstract: Multipoint iterative methods belong to the class of the most efficient methods for solving nonlinear equations. Recent interest in the research and development of this type of methods has arisen from their capability to overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency. This survey paper is a mixture of theoretical results and algorithmic aspects and it is intended as a review of the most efficient root-finding algorithms and developing techniques in a general sense. Many existing methods of great efficiency appear as special cases of presented general iterative schemes. Special attention is devoted to multipoint methods with memory that use already computed information to considerably increase convergence rate without additional computational costs. Some classical results of the 1970s which have had a great influence to the topic, often neglected or unknown to many readers, are also included not only as historical notes but also as genuine sources of many recent ideas. To a certain degree, the presented study follows in parallel main themes shown in the recently published book (Petković et al., 2013) [53], written by the authors of this paper. [Copyright &y& Elsevier]

Abstract: In this paper we provide an extension of the classical Chaos game for IFSP. The paper is divided into two parts: in the first one, we discuss how to determine the integral with respect to a measure which is a combination of a self-similar measure from an IFSP along with a density given by an IFSM. In the second part, we prove a version of the Ergodic Theorem for the integration of a continuous multifunction with respect to the invariant measure of an IFSP. These results are in line with some recent extensions of IFS theory to multifunctions. [Copyright &y& Elsevier]

Abstract: In this paper, an iterative scheme is proposed to find the roots of a nonlinear equation. It is shown that this iterative method has fourth order convergence in the neighborhood of the root. Based on this iterative scheme, we propose the main contribution of this paper as a new high-order computational algorithm for finding an approximate inverse of a square matrix. The analytical discussions show that this algorithm has fourth-order convergence as well. Next, the iterative method will be extended by theoretical analysis to find the pseudo-inverse (also known as the Moore–Penrose inverse) of a singular or rectangular matrix. Numerical examples are also made on some practical problems to reveal the efficiency of the new algorithm for computing a robust approximate inverse of a real (or complex) matrix. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study the strong convergence of the implicit and composite viscosity iteration schemes to a unique common fixed point of nonexpansive semigroups , which is a solution of some variational inequality under certain conditions. As a application, we apply the proposed iterative algorithm to the minimization problem of finding a optimality condition. The results of this paper extend some recent results. [Copyright &y& Elsevier]

Abstract: Very recently, Yao et al. (2010) [4] proposed a hybrid iterative algorithm. Under the parameter sequences satisfying some quite restrictive conditions, they derived a strong convergence theorem in a Hilbert space. In this paper, under the weaker conditions, we prove the strong convergence of the sequence generated by their iterative algorithm to a common fixed point of an infinite family of nonexpansive mappings, which solves a variational inequality. An appropriate example, such that all conditions of this result that are satisfied and that other conditions are not satisfied, is provided. Moreover, the method of our paper is different from the method in Yao et al. (2010) [4]. [Copyright &y& Elsevier]

Abstract: The standard approximate dynamic programming has only one action output. It’s applied to single control variable system, such as inverted pendulum. For multi-input multi-output system, approximate dynamic programming needs a complex scheme. Few papers have derived its iterative convergence calculation, or the presented algorithm lacks rigorous mathematical basis. This paper fist researches matrix analysis foundation for the derivation of multi-input multi-output approximate dynamic programming. The research finds flaws in mathematics of a typical algorithm of its derivation. Hence, we promote approximate dynamic programming to multi-input multi-output form. The detailed iterative convergence calculation of it is derived. An experiment shows its effect. This algorithm is proved to be rigorous in mathematics and not complicated. It is effective for the iterative convergence calculation of multi-input multi-output approximate dynamic programming. [Copyright &y& Elsevier]

Abstract Given a vertex-weighted graph G = (V , E) and a positive integer k ≥ 2, the minimum weight vertex cover P k (MWVCP k) problem is to find a vertex subset F ⊆ V with minimum total weight such that every path of order k in G contains at least one vertex in F. For any integer k ≥ 2, the MWVCP k problem for general graphs is NP-hard. In this paper, we restrict our attention to the MWVCP 3 problem and present a multi-start iterated greedy algorithm to solve the MWVCP 3 problem. The experiments are carried out on randomly generated instances with up to 1000 vertices and 250000 edges. Our work is the first one to adopt heuristic algorithms to solve the MWVCP 3 problem, and the experimental results show that our algorithm performs reasonably well in practice. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a preconditioned two-step modulus-based matrix splitting iteration method for linear complementarity problems associated with an M -matrix is proposed. The convergence analysis of the presented method is given. In particular, we provide a comparison theorem between preconditioned two-step modulus-based Gauss–Seidel (PTMGS) iteration method and two-step modulus-based Gauss–Seidel (TMGS) iteration method, which shows that PTMGS method improves the convergence rate of original TMGS method for linear complementarity problem. Numerical tested examples are used to illustrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a numerical method for solving a class of nonlinear fractional differential equation with proportional delays is proposed. In order to overcome the strongly nonlinear case, we propose the piecewise Picard iteration method(PPIM). The convergence proof and error estimations of the Picard and the PPIM are obtained. Meanwhile, a sufficient condition for the stability of the PPIM is also given. Some numerical examples confirm the validity of the PPIM. It's worth noting that the PPIM is quite effective for solving linear, weakly nonlinear and some strongly nonlinear fractional differential equations with proportional delays. [ABSTRACT FROM AUTHOR]

Abstract Research on the theories and efficient numerical methods of M-matrix algebraic Riccati equation (MARE) has become a hot topic in recent years. In this paper, we consider numerical solution of M-matrix algebraic Riccati equation and propose a modified alternately linearized implicit iteration method (MALI) for computing the minimal nonnegative solution of MARE. Convergence of the MALI method is proved by choosing proper parameters for the nonsingular M-matrix or irreducible singular M-matrix. Theoretical analysis and numerical experiments show that the MALI method is effective and efficient in some cases. [ABSTRACT FROM AUTHOR]

Abstract This paper analyzes the local convergence of several iterative methods for approximating a locally unique solution of a nonlinear equation in a Banach space. It is shown that the local convergence of these methods depends of hypotheses requiring the first-order derivative and the Lipschitz condition. The new approach expands the applicability of previous methods and formulates their theoretical radius of convergence. Several numerical examples originated from real world problems illustrate the applicability of the technique in a wide range of nonlinear cases where previous methods can not be used. [ABSTRACT FROM AUTHOR]

Abstract This paper pertains to the area of shape preservation and sets a theoretical foundation for the applications of preserving constrained nature of a given constraining data in fractal interpolation functions (FIFs) techniques. We construct a new class of rational cubic spline FIFs (RCSFIFs) with a preassigned quadratic denominator with two shape parameters, which includes classical rational cubic interpolant [Appl. Math. Comp., 216 (2010), pp. 2036–2049] as special case and improves the sufficient conditions for positivity. Convergence analysis of RCSFIF to the original function in C 1 is studied. In order to meet the needs of practical design or overcome the drawback of the tension effect in the proposed RCSFIFs, we improve our method by introducing a new tension parameter w i and construct a new class of rational cubic spline FIFs with three shape parameters. The scaling factors and shape parameters have a predictable adjusting role on the shape of curves. The elements of the rational iterated function system in each subinterval are identified befittingly so that the graph of the resulting C 1 -rational cubic spline FIF constrained (i) within a prescribed rectangle (ii) above a prescribed straight line (iii) between two piecewise straight lines. These parameters include, in particular, conditions on the positivity of the C 1 -rational cubic spline FIF. Several numerical examples are presented to ascertain the correctness and usability of developed scheme and to suggest how these schemes outperform their classical counterparts. [ABSTRACT FROM AUTHOR]

Abstract The main objective of this paper is study the existence, uniqueness and other properties of solution of some iterated dynamic integrodifferential on time scales. The main tools employed are Banach Fixed Point theorem and an inequality with explicit estimates are used for proving our results. [ABSTRACT FROM AUTHOR]

Abstract Integral-algebraic equations (IAEs) are coupled systems of Volterra integral equations of the first and second kind which naturally arise in many applications in mathematical physics. In this paper, we solve the IAEs of index-1 by Sinc-collocation discretization and prove that the discrete solutions converge to the true solutions of the IAEs exponentially. The discrete solutions are determined by linear systems which can be effectively solved by suitable iteration methods. Numerical examples are given to illustrate the effective performance of our method. [ABSTRACT FROM AUTHOR]

Abstract Recently, Njeru and Guo presented an accelerated SOR-like (ASOR) method for solving the large and sparse augmented systems. In this paper, we establish an accelerated symmetric SOR-like (ASSOR) method, which is an extension of the ASOR method. Furthermore, the convergence properties of the ASSOR method for augmented systems are studied under suitable restrictions, and the functional equation between the iteration parameters and the eigenvalues of the relevant iteration matrix is established in detail. Finally, numerical examples show that the ASSOR is an efficient iteration method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose an iterative method based on reduced-space approximations for unconstrained optimization problems. The method works as follows: among iterations, samples are taken about the current solution by using, for instance, a Normal distribution; for all samples, gradients are computed (approximated) in order to build reduced-spaces onto which descent directions of cost functions are estimated. By using such directions, intermediate solutions are updated. The overall process is repeated until some stopping criterion is satisfied. The convergence of the proposed method is theoretically proven by using classic assumptions in the line search context. Experimental tests are performed by using well-known benchmark optimization problems and a non-linear data assimilation problem. The results reveal that, as the number of sample points increase, gradient norms go faster towards zero and even more, in the data assimilation context, error norms are decreased by several order of magnitudes with regard to prior errors when the assimilation step is performed by means of the proposed formulation. [ABSTRACT FROM AUTHOR]

Abstract The restricted version of the overlapping Schur complement (SchurRAS) preconditioner was introduced by Li and Saad (2006) for the solution of linear system A x = b , and numerical results have shown that the SchurRAS method outperforms the restricted additive Schwarz (RAS) method both in terms of iteration count and CPU time. In this paper, based on meticulous derivation, we give an algebraic representation of the SchurRAS preconditioner, and prove that the SchurRAS method is convergent under the condition that A is an M -matrix and it converges faster than the RAS method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, two iterative algorithms are proposed to solve stochastic algebraic Riccati matrix equations arising in the linear quadratic optimal control problem of linear stochastic systems with state-dependent noise. In the first algorithm, a standard Riccati matrix equation needs to be solved at each iteration step, and in the second algorithm a standard Lyapunov matrix equation needs to be solved at each iteration step. In the proposed algorithms, a weighted average of the estimates in the last and the previous steps is used to update the estimate of the unknown variable at each iteration step. Some properties of the sequences generated by these algorithms under appropriate initial conditions are presented, and the convergence properties of the proposed algorithms are analyzed. Finally, two numerical examples are employed to show the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The PageRank algorithm plays an important role in determining the importance of Web pages. The multi-step splitting iteration (MSPI) method for calculating the Pagerank problem is an iterative framework of combining the multi-step classical power method with the inner-outer method. In this paper, we present a preprocessed MSPI method called the Arnoldi-MSPI iteration, which is the MSPI method modified with the thick restarted Arnoldi algorithm. The implementation and convergence of the new method are discussed in detail. Numerical experiments are given to show that our method has a good computational effect when the damping factor is close to 1. [ABSTRACT FROM AUTHOR]

The aim of the paper is to give a method for modeling and practical realization of iterative fractional variable-type and -order difference operator. Based on already known serial switching scheme, it was unable to obtain practical realization of such an operator. Therefore, a new parallel switching scheme is introduced. The equivalence between proposed switching scheme and variable-type operator is proved as well. Using proposed method an analog realization of fractional variable-type and -order difference operator is presented and comparison of experimental and numerical results are given. [ABSTRACT FROM AUTHOR]

In this paper, we extend the improved pointwise iteration-complexity estimation of a dynamic regularized alternating direction method of multipliers (ADMM) for a new stepsize domain. In this complexity analysis, the stepsize parameter can be chosen in the interval (0,2) instead of interval ( 0 , ( 1 + 5 ) / 2 ) . We illustrate, by means of a numerical experiment, that the enlargement of this stepsize domain can lead to better performance of the method in some applications. Our complexity study is established by interpreting this ADMM variant as an instance of a hybrid proximal extragradient framework applied to a specific monotone inclusion problem. [ABSTRACT FROM AUTHOR]

Motivated by a specific application in seismic reflection, the goal of this paper is to present a modified version of the Castillo–Grone mimetic gradient operators that allows for a high-order accurate solution of the Eikonal equation with Soner boundary conditions. The modified gradient operators utilize a non-staggered grid. In dimensions other than 1D, the modified gradient operators are expressed as Kronecker products of their corresponding 1D versions and some identity matrices. It is shown, that these modified 1D gradient operators are as accurate as the original gradient operators in terms of approximating first-order partial derivatives. It turns out, that in 1D one requires to solve two linear systems for finding a numerical solution of the Eikonal equation. Some examples show that the solution obtained by utilizing the modified operators increases its accuracy when incrementing the order of their approximation, something that does not occur when using the original operators. An iterative scheme is presented for the nonlinear 2D case. The method is of a quasi-Newton-like nature. At each iteration a linear system is built, with progressively higher-order stencils. The solution of the Fast Marching method is the initial guess. Numerical evidence indicates that high-order accurate solutions can be achieved. [ABSTRACT FROM AUTHOR]

In this paper, by transforming the original problem equivalently, we propose a new preconditioned iterative method for solving saddle point problem. We call the new method as PTU (preconditioned transformative Uzawa) method. And we study the convergence of the PTU method under suitable restrictions on the iteration parameters. Moreover, we show the choices of the optimal parameters and the spectrum of the preconditioned matrix deriving from the PTU method. Based on the PTU iterative method, we propose another iterative method – nonlinear inexact PTU method – for solving saddle point problem. We also prove its convergence and study the choices of the optimal parameters. In addition, we present some numerical results to illustrate the behavior of the considered algorithms. [ABSTRACT FROM AUTHOR]

An efficient iterative method is developed for the static analysis of large deflections of an infinite beam with variable cross-section resting on a nonlinear foundation. A pseudo spring constant is added and explicit matrix operators are introduced to perform differentiation through Green’s function. The nonlinearity of the problem is handled with quasilinearization. To compute the solution of the quasilinear differential equation with prescribed accuracy, a new discretization method for solving quasilinear differential equations involving up to the 4th order derivative is used. The discretization method is based on relating discretizations of up to the fourth order derivative of the solution with a discretization of the solution by using a suitable Green function. Numerical experiments show that the error incurred by the discretization can be made small for the two first derivatives and that the method proposed in the paper converges fast and has good accuracy. [ABSTRACT FROM AUTHOR]

The main goal of this paper is the numerical computation of solutions of the so-called Yang–Baxter-like matrix equation A X A = X A X , where A is a given complex square matrix. Two novel matrix iterations are proposed, both having second-order convergence. A sign modification in one of the iterations gives rise to a third matrix iteration. Strategies for finding starting approximations are discussed as well as a technique for estimating the relative error. One of the methods involves a very small cost per iteration and is shown to be stable. Numerical experiments are carried out to illustrate the effectiveness of the new methods. [ABSTRACT FROM AUTHOR]

In this paper, for solving systems of nonlinear equations, we develop a family of two-step third order methods and introduce a technique by which the order of convergence of many iterative methods can be improved. Given an iterative method of order p  ≥ 2 which uses the extended Newton iteration as a predictor, a new method of order p + 2 is constructed by introducing only one additional evaluation of the function. In addition, for an iterative method of order p  ≥ 3 using the Newton iteration as a predictor, a new method of order p + 3 can be extended. Applying this procedure, we develop some new efficient methods with higher order of convergence. For comparing these new methods with the ones from which they have been derived, we discuss the computational efficiency in detail. Several numerical examples are given to justify the theoretical results by the asymptotic behaviors of the considered methods. [ABSTRACT FROM AUTHOR]

Abstract: In the paper [A. Rababah, S. Mann, Iterative process for G 2-multi degree reduction of Bézier curves, Applied Mathematics and Computation 217 (2011) 8126–8133], Rababah and Mann proposed an iterative method for multi-degree reduction of Bézier curves with C 1 and G 2-continuity at the endpoints. In this paper, we provide a theoretical proof for the existence of the unique solution in the first step of the iterative process, while the proof in their paper applies only in some special cases. Also, we give a complete convergence proof for the iterative method. We solve the problem by using convex quadratic optimization. [Copyright &y& Elsevier]

Abstract: This paper deals with the first order nonlinear neutral delay differential equationwhere and with lim t→+∞ σ l (t)=+∞ for l ∈{1,2,…, n}. By using the Banach fixed point theorem, we prove the global existence of uncountably many bounded positive solutions for the above equation relative to all ranges of the function p, construct some Mann type iterative algorithms with errors to approximate these positive solutions and discuss several error estimates between the sequences generated by the iterative algorithms and these positive solutions. Seven examples are presented to illuminate the results obtained in this paper. [Copyright &y& Elsevier]

Abstract: In this paper, some existence theorems for the mixed quasi-variational-like inequalities problem in a reflexive Banach space are established. The auxiliary principle technique is used to suggest a novel and innovative iterative algorithm for computing the approximate solution for the mixed quasi-variational-like inequalities problem. Consequently, not only the existence of theorems of the mixed quasi-variational-like inequalities is shown, but also the convergence of iterative sequences generated by the algorithm is also proven. The results proved in this paper represent an improvement of previously known results. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to propose a modified block iterative algorithm for find a common element of the set of common fixed points of an infinite family of quasi-ϕ-asymptotically nonexpansive mappings and the set of an equilibrium problem. Under suitable conditions, some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with the Kadec–Klee property. As an application, at the end of the paper a numerical example is given. The results presented in the paper improve and extend the corresponding results in Qin et al. [Convergence theorems of common elements for equilibrium problems and fixed point problem in Banach spaces, J. Comput. Appl. Math., 225, 2009, 20–30], Zhou et al. [Convergence theorems of a modified hybrid algorithm for a family of quasi-ϕ-asymptotically nonexpansive mappings, J. Appl. Math. Compt., 17 March, 2009, doi:10.1007/s12190-009-0263-4], Takahashi and Zembayshi [Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal., 70, 2009, 45–57], Wattanawitoon and Kumam [Strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasi-nonexpansive mappings, Nonlinear Anal. Hybrid Syst., 3, 2009, 11–20] and Matsushita and Takahashi [A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theory, 134, 2005, 257–266] and others. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study a strong convergence of multi-step iterative scheme to a common solution for a finite family of uniformly continuous ϕ-strongly accretive operator equations in an arbitrary Banach space. As a consequence, the strong convergence theorem for the multi-step iterative sequence to a common fixed point for finite family of ϕ-strongly pseudocontractive mappings is also obtained. The results presented in this paper thus improve and extend the corresponding results of Inchan , Kang and many others. [Copyright &y& Elsevier]

Abstract: In this paper, the minimal and maximal nonnegative solutions for second order m-point boundary value problems at resonance are investigated by using a new fixed point theorem of increasing operators. And the monotone iterative sequences which converge to solutions are given. The method is different essentially from the previous papers used by coincidence degree continuation theorem of Mawhin. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to study the Ishikawa type iterative scheme to approximate a common fixed point of infinite families of uniformly quasi-Lipschitzian mappings and nonexpansive mappings in convex metric spaces. Under appropriate conditions, some convergence theorems are proved. The results presented in the paper generalize, improve and unify some recent results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new two-step iterative scheme for a finite family of -asymptotically nonexpansive nonself-mappings is constructed in a uniformly convex Banach space. Weak and strong convergence theorems of this iterative scheme to a common fixed point of and are proved in a uniformly convex Banach space. The results of this paper improve and extend the corresponding results of Temir []. [Copyright &y& Elsevier]

Abstract: In this paper, a kind of Ishikawa type iterative scheme with errors for approximating a common fixed point of two sequences of uniformly quasi-Lipschitzian mappings is introduced and studied in convex metric spaces. Under some suitable conditions, the convergence theorems concerned with the Ishikawa type iterative scheme with errors to approximate a common fixed point of two sequences of uniformly quasi-Lipschitzian mappings were proved in convex metric spaces. The results presented in the paper generalize and improve some recent results of Wang and Liu (C. Wang, L.W. Liu, Convergence theorems for fixed points of uniformly quasi-Lipschitzian mappings in convex metric spaces, Nonlinear Anal., TMA 70 (2009), 2067–2071). [Copyright &y& Elsevier]