Moments on Catalan numbers

By combining inverse series relations with binomial convolutions and telescoping method, moments of Catalan numbers are evaluated, which resolves a problem recently proposed by Gutiérrez et al. [J.M. Gutiérrez, M.A. Hernández, P.J. Miana, N. Romero, New identities in the Catalan triangle, J. Math. Anal. Appl. 341 (1) (2008) 52-61].     

Third-order iterative methods with applications to Hammerstein equations: A unified approach

The geometrical interpretation of a family of higher order iterative methods for solving nonlinear scalar equations was presented in [S. Amat, S. Busquier, J.M. Gutiérrez, Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157(1) (2003) 197-205]. This family includes, as particular cases, some of the most famous third-order iterative methods: Chebyshev methods, Halley methods, super-Halley methods, C-methods and Newton-type two-step methods. The aim of the present paper is to analyze the convergence of this family for equations defined between two Banach spaces by using a technique developed in [J.A. Ezquerro, M.A. Hernández, Halley’s method for operators with unbounded second derivative. Appl. Numer. Math. 57(3) (2007) 354-360]. This technique allows us to obtain a general semilocal convergence result for these methods, where the usual conditions on the second derivative are relaxed. On the other hand, the main practical difficulty related to the classical third-order iterative methods is the evaluation of bilinear operators, typically second-order Fréchet derivatives. However, in some cases, the second derivative is easy to evaluate. A clear example is provided by the approximation of Hammerstein equations, where it is diagonal by blocks. We finish the paper by applying our methods to some nonlinear integral equations of this type.     

Convergence analysis of a family of Steffensen-type methods for generalized equations

A class of Steffensen-type algorithms for solving generalized equations on Banach spaces is proposed. Using well-known fixed point theorem for set-valued maps [A.L. Dontchev, W.W. Hager, An inverse function theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994) 481-489] and some conditions on the first-order divided difference, we provide a local convergence analysis. We also study the perturbed problem and we present a new regula-falsi-type method for set-valued mapping. This study follows the works on the Secant-type method presented in [S. Hilout, A uniparametric Secant-type methods for nonsmooth generalized equations, Positivity (2007), submitted for publication; S. Hilout, A. Piétrus, A semilocal convergence of a Secant-type method for solving generalized equations, Positivity 10 (2006) 673-700] and extends the results related to the resolution of nonlinear equations [M.A. Hernández, M.J. Rubio, The Secant method and divided differences Hölder continuous, Appl. Math. Comput. 124 (2001) 139-149; M.A. Hernández, M.J. Rubio, Semilocal convergence of the Secant method under mild convergence conditions of differentiability, Comput. Math. Appl. 44 (2002) 277-285; M.A. Hernández, M.J. Rubio, ω-Conditioned divided differences to solve nonlinear equations, in: Monogr. Semin. Mat. García Galdeano, vol. 27, 2003, pp. 323-330; M.A. Hernández, M.J. Rubio, A modification of Newton's method for nondifferentiable equations, J. Comput. Appl. Math. 164/165 (2004) 323-330].     

On Newton’s method for solving equations containing Fréchet-differentiable operators of order at least two

We provide sufficient conditions for the semilocal convergence of Newton’s method to a locally unique solution of a nonlinear operator equation containing operators that are Fréchet-differentiable of order at least two, in a Banach space setting. Numerical examples are also provided to show that our results apply to solve nonlinear equations in cases earlier ones cannot [J.M. Gutiérrez, A new semilocal convergence theorem for Newton’s method, J. Comput. Appl. Math. 79(1997) 131-145; Z. Huang, A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211-217; F.A. Potra, Sharp error bounds for a class of Newton-like methods, Libertas Mathematica 5 (1985) 71-84].     

Semilocal convergence of multi-point improved super-Halley-type methods without the second derivative under generalized weak condition

In this paper, we consider the semilocal convergence of multi-point improved super-Halley-type methods in Banach space. Different from the results of super-Halley method studied in reference Gutiérrez, J.M. and Hernández, M.A. (Comput. Math. Appl. 36,1–8, 1998) these methods do not require second derivative of an operator, the R-order is improved and the convergence condition is also relaxed. We prove a convergence theorem to show existence and uniqueness of the solution.     

Reduced Recurrence Relations for the Chebyshev Method

In this paper, we give sufficient conditions ensuring the convergence of the Chebyshev method in Banach spaces. We use a new system of recurrence relations which simplifies those given by Kantorovich for the Newton method or those given by Candela and Marquina for the Chebyshev and Halley methods.     

On the semilocal convergence of inexact Newton methods in Banach spaces

We provide two types of semilocal convergence theorems for approximating a solution of an equation in a Banach space setting using an inexact Newton method [I.K. Argyros, Relation between forcing sequences and inexact Newton iterates in Banach spaces, Computing 63 (2) (1999) 134–144; I.K. Argyros, A new convergence theorem for the inexact Newton method based on assumptions involving the second Fréchet-derivative, Comput. Appl. Math. 37 (7) (1999) 109–115; I.K. Argyros, Forcing sequences and inexact Newton iterates in Banach space, Appl. Math. Lett. 13 (1) (2000) 77–80; I.K. Argyros, Local convergence of inexact Newton-like iterative methods and applications, Comput. Math. Appl. 39 (2000) 69–75; I.K. Argyros, Computational Theory of Iterative Methods, in: C.K. Chui, L. Wuytack (Eds.), in: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Co., New York, USA, 2007; X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242]. By using more precise majorizing sequences than before [X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242; Z.D. Huang, On the convergence of inexact Newton method, J. Zheijiang University, Nat. Sci. Ed. 30 (4) (2003) 393–396; L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982; X.H. Wang, Convergence on the iteration of Halley family in weak condition, Chinese Sci. Bull. 42 (7) (1997) 552–555; T.J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21 (3) (1984) 583–590], we provide (under the same computational cost) under the same or weaker hypotheses: finer error bounds on the distances involved; an at least as precise information on the location of the solution. Moreover if the splitting method is used, we show that a smaller number of inner/outer iterations can be obtained.     

Local convergence and the dynamics of a two-point four parameter Jarratt-like method under weak conditions

We present a local convergence analysis of a two-point four parameter Jarratt-like method of high convergence order in order to approximate a locally unique solution of a nonlinear equation. In contrast to earlier studies such us (Amat et al. Aequat. Math. 69(3), 212–223 2015; Amat et al. J. Math. Anal. Appl. 366(3), 24–32 2010; Behl, R. 2013; Bruns and Bailey Chem. Eng. Sci. 32, 257–264 1977; Candela and Marquina. Computing 44, 169–184 1990; Candela and Marquina. Computing 45(4), 355–367 1990; Chun. Appl. Math. Comput. 190(2), 1432–1437 2007; Cordero and Torregrosa. Appl. Math. Comput. 190, 686–698 2007; Deghan. Comput. Appl Math. 29(1), 19–30 2010; Deghan. Comput. Math. Math. Phys. 51(4), 513–519 2011; Deghan and Masoud. Eng. Comput. 29(4), 356–365 15; Cordero and Torregrosa. Appl. Math. Comput. 190, 686–698 2012; Deghan and Masoud. Eng. Comput. 29(4), 356–365 2012; Ezquerro and Hernández. Appl. Math. Optim. 41(2), 227–236 2000; Ezquerro and Hernández. BIT Numer. Math. 49, 325–342 2009; Ezquerro and Hernández. J. Math. Anal. Appl. 303, 591–601 2005; Gutiérrez and Hernández. Comput. Math. Appl. 36(7), 1–8 1998; Ganesh and Joshi. IMA J. Numer. Anal. 11, 21–31 1991; González-Crespo et al. Expert Syst. Appl. 40(18), 7381–7390 2013; Hernández. Comput. Math. Appl. 41(3-4), 433–455 2001; Hernández and Salanova. Southwest J. Pure Appl. Math. 1, 29–40 1999; Jarratt. Math. Comput. 20(95), 434–437 1966; Kou and Li. Appl. Math. Comput. 189, 1816–1821 2007; Kou and Wang. Numer. Algor. 60, 369–390 2012; Lorenzo et al. Int. J. Interact. Multimed. Artif. Intell. 1(3), 60–66 2010; Magreñán. Appl. Math. Comput. 233, 29–38 2014; Magreñán. Appl. Math. Comput. 248, 215–224 2014; Parhi and Gupta. J. Comput. Appl. Math. 206(2), 873–887 2007; Rall 1979; Ren et al. Numer. Algor. 52(4), 585–603 2009; Rheinboldt Pol. Acad. Sci. Banach Ctr. Publ. 3, 129–142 1978; Sicilia et al. J. Comput. Appl. Math. 291, 468–477 2016; Traub 1964; Wang et al. Numer. Algor. 57, 441–456 2011) using hypotheses up to the fifth derivative, our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. The dynamics of the family for choices of the parameters such that it is optimal is also shown. Numerical examples are also provided in this study     

On the convergence of Newton-type methods under mild differentiability conditions

We introduce the new idea of recurrent functions to provide a new semilocal convergence analysis for Newton-type methods, under mild differentiability conditions. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in some interesting cases (Chen, Ann Inst Stat Math 42:387–401, 1990; Chen, Numer Funct Anal Optim 10:37–48, 1989; Cianciaruso, Numer Funct Anal Optim 24:713–723, 2003; Cianciaruso, Nonlinear Funct Anal Appl 2009; Dennis 1971; Deuflhard 2004; Deuflhard, SIAM J Numer Anal 16:1–10, 1979; Gutiérrez, J Comput Appl Math 79:131–145, 1997; Hernández, J Optim Theory Appl 109:631–648, 2001; Hernández, J Comput Appl Math 115:245–254, 2000; Huang, J Comput Appl Math 47:211–217, 1993; Kantorovich 1982; Miel, Numer Math 33:391–396, 1979; Miel, Math Comput 34:185–202, 1980; Moret, Computing 33:65–73, 1984; Potra, Libertas Mathematica 5:71–84, 1985; Rheinboldt, SIAM J Numer Anal 5:42–63, 1968; Yamamoto, Numer Math 51: 545–557, 1987; Zabrejko, Numer Funct Anal Optim 9:671–684, 1987; Zinc̆ko 1963). Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, and a differential equation are also provided in this study.     

On Halley-type iterations with free second derivative

In this paper, we relax the convergence conditions required in Ezquerro and Hernández (Int. J. Pure Appl. Math. 6(1) (2003) 103) for a multipoint third-order iteration of Halley type, where the conditions provided are the known ones for methods of order three.     

Complexity analysis of an interior-point algorithm for linear optimization based on a new proximity function

In this paper, we focus on a family of modified Chebyshev methods and study the semilocal convergence for these methods. Different from the results in reference (Hernández and Salanova, J. Comput. Appl. Math. 126:131–143, 2000), the Hölder continuity of the second derivative is replaced by its generalized continuity condition, and the latter is weaker than the former. Using the recurrence relations, we establish the semilocal convergence of these methods and prove a convergence theorem to show the existence-uniqueness of the solution. The R-order of these methods is also analyzed. Especially, when the second derivative of the operator is Hölder continuous, the R-order of these methods is at least 3 + 2p, which is higher than the one of Chebyshev method considered in reference (Hernández and Salanova, J. Comput. Appl. Math. 126:131–143, 2000) under the same condition. Finally, we give some numerical results to show our approach.     

Kantorovich's type theorems for systems of equations with constant rank derivatives

The famous Newton–Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton's method to a solution of an equation. Here we present a “Kantorovich type” convergence analysis for the Gauss–Newton's method which improves the result in [W.M. Häußler, A Kantorovich-type convergence analysis for the Gauss–Newton-method, Numer. Math. 48 (1986) 119–125.] and extends the main theorem in [I.K. Argyros, On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math. 169 (2004) 315–332]. Furthermore, the radius of convergence ball is also obtained.     

Properties of the complex bimatrix variate beta distribution

In this article, we derive several properties such as marginal distribution, moments involving zonal polynomials, and asymptotic expansion of the complex bimatrix variate beta type 1 distribution introduced by D?´az-Garc?´a and Gutiérrez Jáimez [José A. D?´az-Garc?´a, Ramón Gutiérrez Jáimez, Complex bimatrix variate generalised beta distributions, Linear Algebra Appl. 432 (2010) 571-582]. We also derive distributions of several matrix valued functions of random matrices jointly distributed as complex bimatrix variate beta type 1.     

Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces

Viscosity approximation methods for a family of finite nonexpansive mappings are established in Banach spaces. The main theorems extend the main results of Moudafi [Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46–55] and Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291] to the case of finite mappings. Our results also improve and unify the corresponding results of Bauschke [The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 202 (1996) 150–159], Browder [Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces, Archiv. Ration. Mech. Anal. 24 (1967) 82–90], Cho et al. [Some control conditions on iterative methods, Commun. Appl. Nonlinear Anal. 12 (2) (2005) 27–34], Ha and Jung [Strong convergence theorems for accretive operators in Banach spaces, J. Math. Anal. Appl. 147 (1990) 330–339], Halpern [Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957–961], Jung [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509–520], Jung et al. [Iterative schemes with some control conditions for a family of finite nonexpansive mappings in Banach space, Fixed Point Theory Appl. 2005 (2) (2005) 125–135], Jung and Kim [Convergence of approximate sequences for compositions of nonexpansive mappings in Banach spaces, Bull. Korean Math. Soc. 34 (1) (1997) 93–102], Lions [Approximation de points fixes de contractions, C.R. Acad. Sci. Ser. A-B, Paris 284 (1977) 1357–1359], O’Hara et al. [Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417–1426], Reich [Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 287–292], Shioji and Takahashi [Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (12) (1997) 3641–3645], Takahashi and Ueda [On Reich's strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984) 546–553], Wittmann [Approximation of fixed points of nonexpansive mappings, Arch. Math. 59 (1992) 486–491], Xu [Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2) (2002) 240–256], and Zhou et al. [Strong convergence theorems on an iterative method for a family nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput., in press] among others.     

Topological mesh for shell structures

Recently a new technique for the meshing of tension structures has been developed by Hernández-Montes et al. [E. Hernández Montes, R. Jurado-Piña, E. Bayo, Topological mapping for tension structures, J. Struct. Eng. ASCE 132 (6) (2006) 970–977]. This new procedure was based on topology and it was specially designed for tension structures. In this paper an extension of this technique to shell structures is developed. Topological meshing may constitute the first step in the common iterative process of discretization to facilitate the resolution of a partial differential equation.     

A Kantorovich-type analysis for a fast iterative method for solving nonlinear equations

We revisit a fast iterative method studied by us in [I.K. Argyros, On a two-point Newton-like method of convergent order two, Int. J. Comput. Math. 88 (2) (2005) 219-234] to approximate solutions of nonlinear operator equations. The method uses only divided differences of order one and two function evaluations per step. This time we use a simpler Kantorovich-type analysis to establish the quadratic convergence of the method in the local as well as the semilocal case. Moreover we show that in some cases our method compares favorably, and can be used in cases where other methods using similar information cannot [S. Amat, S. Busquier, V.F. Candela, A class of quasi-Newton generalized Steffensen's methods on Banach spaces, J. Comput. Appl. Math. 149 (2) (2002) 397-406; D. Chen, On the convergence of a class of generalized Steffensen's iterative procedures and error analysis, Int. J. Comput. Math. 31 (1989) 195-203]. Numerical examples are provided to justify the theoretical results.     

General convergence conditions of Newton’s method for m-Fréchet differentiable operators

We present a local as well as a semilocal convergence analysis for Newton’s method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. Our hypotheses involve m-Fréchet-differentiable operators and general Lipschitz-type hypotheses, where m≥2 is a positive integer. The new convergence analysis unifies earlier results; it is more flexible and provides a finer convergence analysis than in earlier studies such as Argyros in J. Comput. Appl. Math. 131:149–159, 2001, Argyros and Hilout in J. Appl. Math. Comput. 29:391–400, 2009, Argyros and Hilout in J. Complex. 28:364–387, 2012, Argyros et al. Numerical Methods for Equations and Its Applications, CRC Press/Taylor & Francis, New York, 2012, Gutiérrez in J. Comput. Appl. Math. 79:131–145, 1997, Ren and Argyros in Appl. Math. Comput. 217:612–621, 2010, Traub and Wozniakowski in J. Assoc. Comput. Mech. 26:250–258, 1979. Numerical examples are presented further validating the theoretical results.     

Semilocal convergence for a class of improved multi-step Chebyshev–Halley-like methods under extended conditions

Semilocal convergence for a class of improved multi-step Chebyshev–Halley-like methods is considered in this paper. Compared with the results for the Chebyshev method in Hernández (J Comput Appl Math 126:131–143, 2000), the R-order of convergence is heightened and the Hölder continuity of second derivative is also relaxed. Moreover, an existence-uniqueness theorem is proved under the extended conditions.     

The preconditioned Gauss–Seidel method faster than the SOR method

In recent years, a number of preconditioners have been applied to linear systems [A.D. Gunawardena, S.K. Jain, L. Snyder, Modified iterative methods for consistent linear systems, Linear Algebra Appl. 154–156 (1991) 123–143; T. Kohno, H. Kotakemori, H. Niki, M. Usui, Improving modified Gauss–Seidel method for Z-matrices, Linear Algebra Appl. 267 (1997) 113–123; H. Kotakemori, K. Harada, M. Morimoto, H. Niki, A comparison theorem for the iterative method with the preconditioner (I+_Smax)$(I+S_{\mathit{max}})$, J. Comput. Appl. Math. 145 (2002) 373–378; H. Kotakemori, H. Niki, N. Okamoto, Accelerated iteration method for Z$Z$-matrices, J. Comput. Appl. Math. 75 (1996) 87–97; M. Usui, H. Niki, T.Kohno, Adaptive Gauss-Seidel method for linear systems, Internat. J. Comput. Math. 51(1994)119–125 [10]]. Since these preconditioners are constructed from the elements of the upper triangular part of the coefficient matrix, the preconditioning effect is not observed on the n$n$th row of matrix A. In the present paper, in order to deal with this drawback, we propose a new preconditioner. In addition, the convergence and comparison theorems of the proposed method are established. Simple numerical examples are also given, and we show that the convergence rate of the proposed method is better than that of the optimum SOR.     

A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for α$\alpha$-inverse-strongly monotone mappings in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. As applications, at the end of the paper we utilize our results to study the optimization problem and some convergence problem for strictly pseudocontractive mappings. The results presented in the paper extend and improve some recent results of Yao and Yao [Y.Y. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186 (2) (2007) 1551–1558], Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonlinear mappings and monotone mappings, Appl. Math. Comput. (2007) doi:10.1016/j.amc.2007.07.075], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for Equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006) 506–515], Su, Shang and Qin [Y.F. Su, M.J. Shang, X.L. Qin, An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. (2007) doi:10.1016/j.na.2007.08.045] and Chang, Cho and Kim [S.S. Chang, Y.J. Cho, J.K. Kim, Approximation methods of solutions for equilibrium problem in Hilbert spaces, Dynam. Systems Appl. (in print)].