On the Secant method

We present a new semilocal convergence analysis for the Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis is based on the weaker center-Lipschitz concept instead of the stronger Lipschitz condition which has been ubiquitously employed in other studies such as Amat et al. (2004)  [2], Bosarge and Falb (1969)  [9], Dennis (1971)  [10], Ezquerro et al. (2010)  [11], Hernández et al. (2005, 2000)   and , Kantorovich and Akilov (1982)  [14], Laasonen (1969)  [15], Ortega and Rheinboldt (1970)  [16], Parida and Gupta (2007)  [17], Potra (1982, 1984–1985, 1985)  ,  and , Proinov (2009, 2010)   and , Schmidt (1978) [23], Wolfe (1978)  [24] and Yamamoto (1987)  [25] for computing the inverses of the linear operators. We also provide lower and upper bounds on the limit point of the majorizing sequences for the Secant method. Under the same computational cost, our error analysis is tighter than that proposed in earlier studies. Numerical examples illustrating the theoretical results are also given in this study.     

Improved generalized differentiability conditions for Newton-like methods

We provide a semilocal convergence analysis for Newton-like methods using the ω$\omega$-versions of the famous Newton–Kantorovich theorem (Argyros (2004) [1], Argyros (2007) [3], Kantorovich and Akilov (1982) [13]). In the special case of Newton’s method, our results have the following advantages over the corresponding ones (Ezquerro and Hernaández (2002) [10], Proinov (2010) [17]) under the same information and computational cost: finer error estimates on the distances involved; at least as precise information on the location of the solution, and weaker sufficient convergence conditions.     

Inexact Newton-type methods

We introduce the new idea of recurrent functions to provide a semilocal convergence analysis for an inexact Newton-type method, using outer inverses. It turns out that our sufficient convergence conditions are weaker than in earlier studies in many interesting cases (Argyros, 2004 [5] and [6], Argyros, 2007 [7], Dennis, 1971 [14], Deuflhard and Heindl, 1979 [15], Gutiérrez, 1997 [16], Gutiérrez et al., 1995 [17], Häubler, 1986 [18], Huang, 1993 [19], Kantorovich and Akilov, 1982 [20], Nashed and Chen, 1993 [21], Potra, 1982 [22], Potra, 1985 [23]).     

Weaker conditions for the convergence of Newton’s method

Newton’s method is often used for solving nonlinear equations. In this paper, we show that Newton’s method converges under weaker convergence criteria than those given in earlier studies, such as Argyros (2004) [2, p. 387], Argyros and Hilout (2010)[11, p. 12], Argyros et al. (2011) [12, p. 26], Ortega and Rheinboldt (1970) [26, p. 421], Potra and Pták (1984) [36, p. 22]. These new results are illustrated by several numerical examples, for which the older convergence criteria do not hold but for which our weaker convergence criteria are satisfied.     

Weak convergence conditions for Inexact Newton-type methods

We establish a new semilocal convergence results for Inexact Newton-type methods for approximating a locally unique solution of a nonlinear equation in a Banach spaces setting. We show that our sufficient convergence conditions are weaker and the estimates of error bounds are tighter in some cases than in earlier works [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] and [31]. Special cases and numerical examples are also provided in this study.     

Unified majorizing sequences for Traub-type multipoint iterative procedures

We present a unified approach to generating majorizing sequences for multipoint iterative procedures in order to solve nonlinear equations in a Banach space setting. The semilocal convergence results have the following advantages over earlier work (under the same computational cost): weaker sufficient convergence conditions, more precise error bounds on the distances involved and more precise information on the location of the solution. Special cases and numerical examples are also provided in this study.     

Extending the Newton-Kantorovich hypothesis for solving equations

The famous Newton-Kantorovich hypothesis (Kantorovich and Akilov, 1982 [3], Argyros, 2007 [2], Argyros and Hilout, 2009 [7]) has been used for a long time as a sufficient condition for the convergence of Newton’s method to a solution of an equation in connection with the Lipschitz continuity of the Fréchet-derivative of the operator involved. Here, using Lipschitz and center-Lipschitz conditions, and our new idea of recurrent functions, we show that the Newton-Kantorovich hypothesis can be weakened, under the same information. Moreover, the error bounds are tighter than the corresponding ones given by the dominating Newton-Kantorovich theorem (Argyros, 1998 [1]; [2] and [7]; Ezquerro and Hernández, 2002 [11]; [3]; Proinov 2009, 2010 [16] and [17]).Numerical examples including a nonlinear integral equation of Chandrasekhar-type (Chandrasekhar, 1960 [9]), as well as a two boundary value problem with a Green’s kernel (Argyros, 2007 [2]) are also provided in this study.     

Newton’s method on generalized Banach spaces

We present a weaker convergence analysis of Newton’s method than in Kantorovich and Akilov (1964), Meyer (1987), Potra and Ptak (1984), Rheinboldt (1978), Traub (1964) on a generalized Banach space setting to approximate a locally unique zero of an operator. This way we extend the applicability of Newton’s method. Moreover, we obtain under the same conditions in the semilocal case weaker sufficient convergence criteria; tighter error bounds on the distances involved and an at least as precise information on the location of the solution. In the local case we obtain a larger radius of convergence and higher error estimates on the distances involved. Numerical examples illustrate the theoretical results.     

Majorizing sequences for iterative methods

We provide convergence results for very general majorizing sequences of iterative methods. Using our new concept of recurrent functions, we unify the semilocal convergence analysis of Newton-type methods (NTM) under more general Lipschitz-type conditions. We present two very general majorizing sequences and we extend the applicability of (NTM) using the same information before Chen and Yamamoto (1989) [13], Deuflhard (2004) [16], Kantorovich and Akilov (1982) [19], Miel (1979) [20], Miel (1980) [21] and Rheinboldt (1968) [30]. Applications, special cases and examples are also provided in this study to justify the theoretical results of our new approach.     

A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space

We provide a local as well as a semilocal convergence analysis for two-point Newton-like methods in a Banach space setting under very general Lipschitz type conditions. Our equation contains a Fréchet differentiable operator F and another operator G whose differentiability is not assumed. Using more precise majorizing sequences than before we provide sufficient convergence conditions for Newton-like methods to a locally unique solution of equation F(x)+G(x)=0. In the semilocal case we show under weaker conditions that our error estimates on the distances involved are finer and the information on the location of the solution at least as precise as in earlier results. In the local case a larger radius of convergence is obtained. Several numerical examples are provided to show that our results compare favorably with earlier ones. As a special case we show that the famous Newton-Kantorovich hypothesis is weakened under the same hypotheses as the ones contained in the Newton-Kantorovich theorem.     

On Newton-like methods of ��bounded deterioration�� using recurrent functions

We provide a semilocal convergence analysis for Newton-like methods of ??bounded deterioration?? in a Banach space setting. We establish tighter error bounds on the distances involved, and a more precise information on the location of the solution, under the same or weaker hypotheses than before (Argyros, Acta Math. Sin. (Engl. Ser.), 23:2087?C2096, 2007; Deuflhard, Newton methods for nonlinear problems. Affine invariance and adaptive algorithms, Springer Series in Computational Mathematics, vol. 35. Springer, Berlin, 2004; Ezquerro and Hern??ndez, IMA J. Numer. Anal., 22:187?C205, 2002) using recurrent functions. Numerical examples are also provided involving polynomial, integral, and differential equations.     

Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces

In this paper, we study the semilocal convergence for a fifth-order method for solving nonlinear equations in Banach spaces. The semilocal convergence of this method is established by using recurrence relations. We prove an existence-uniqueness theorem and give a priori error bounds which demonstrates the R-order of the method. As compared with the Jarratt method in Hernández and Salanova (Southwest J Pure Appl Math 1:29–40, 1999) and the Multi-super-Halley method in Wang et al. (Numer Algorithms 56:497–516, 2011), the differentiability conditions of the convergence of the method in this paper are mild and the R-order is improved. Finally, we give some numerical applications to demonstrate our approach.     

Convergence of Directional Methods under mild differentiability and applications

A semilocal convergence analysis for Directional Methods under mild differentiability conditions is provided in this study. Using our idea of recurrent functions, we provide sufficient convergence conditions as well as the corresponding errors bounds. The results are extended to hold in a Hilbert space setting and a favorable comparison is provided with earlier works [6], [7], [8], [9], [10], [11] and [20]. Numerical examples are also provided in this study.     

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].     

On strong convergence in real Banach spaces obtained by dropping the Lipshitz condition

We consider an implicit iterative process for two finite families of mappings in a real Banach space and prove strong convergence results without using the Lipschitz condition on mappings. Our results mainly improve and extend the recent results of Chang et al. (2001, 2009, 2007) [1], [2], [3], Cho et al. (2005) [4], Gu (2008) [14], Ofoedu (2006) [9], Schu (1991) [13], Zeng (2003, 2005) [20], [21], and Qin et al. (2008) [11], [12].     

Kantorovich-type semilocal convergence analysis for inexact Newton methods

We provide a new semilocal convergence analysis for generating an inexact Newton method converging to a solution of a nonlinear equation in a Banach space setting. Our analysis is based on our idea of recurrent functions. Our results are compared favorably to earlier ones by others and us (Argyros (2007, 2009) [5] and [6], Argyros and Hilout (2009) [7], Guo (2007) [15], Shen and Li (2008) [18], Li and Shen (2008) [19], Shen and Li (2009) [20]). Numerical examples are provided to show that our results apply, but not earlier ones [15], [18], [19] and [20].     

On recent developments in the theory of abstract differential equations with fractional derivatives

This note is motivated from some recent papers treating the problem of the existence of a solution for abstract differential equations with fractional derivatives. We show that the existence results in [Agarwal et al. (2009) [1], Belmekki and Benchohra (2010) [2], Darwish et al. (2009) [3], Hu et al. (2009) [4], Mophou and N’Guérékata (2009) [6] and [7], Mophou (2010) [8] and [9], Muslim (2009) [10], Pandey et al. (2009) [11], Rashid and El-Qaderi (2009) [12] and Tai and Wang (2009) [13]] are incorrect since the considered variation of constant formulas is not appropriate. In this note, we also consider a different approach to treat a general class of abstract fractional differential equations.     

Rotundity structure of local nature for generalized Orlicz-Lorentz function spaces

Criteria for extreme points and rotund points in generalized Orlicz-Lorentz function spaces, which were introduced in Foralewski (2011) [27] are given. Some examples show that in these spaces the notion of rotund point is essentially stronger than the notion of extreme point. Finally, the criteria obtained in this paper are interpreted in the case of classical Orlicz-Lorentz spaces. Results of this paper are related to the results from Carothers et al. (1992) [9], Kamińska (1990) [4], Foralewski et al. (2008) [26].     

Weaker convergence conditions for the secant method

We use tighter majorizing sequences than in earlier studies to provide a semilocal convergence analysis for the secant method. Our sufficient convergence conditions are also weaker. Numerical examples are provided where earlier conditions do not hold but for which the new conditions are satisfied.