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.     

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 weak exponential stability of evolution operators in Banach spaces

The aim of this paper is to give some characterizations for weak exponential stability properties of evolution operators in Banach spaces. Variants for weak exponential stability of some well-known results in uniform stability theory (Bu?e and Niculescu (2009) [1], Daleckij and Krein (1974) [2], Datko (1973) [3], Rolewicz (1986) [7], Stoica and Megan (2009) [8]) are obtained.     

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]).     

Coderivatives of normal cone mappings and Lipschitzian stability of parametric variational inequalities

In this paper, we provide a comprehensive study of coderivative formulas for normal cone mappings. This allows us to derive necessary and sufficient conditions for the Lipschitzian stability of parametric variational inequalities in reflexive Banach spaces. Our development not only gives an answer to the open questions raised in Yao and Yen (2009) [11], but also establishes generalizations and complements of the results given in Henrion et al. (2010) [4] and Yao and Yen (2009) [11] and [12].     

Sperner’s lemma and zero point theorems on a discrete simplex and a discrete simplotope

We show two discrete zero point theorems that are derived from Sperner’s lemma and a Sperner-like theorem (van der Laan and Talman [Math. Oper. Res. (1982)] [5]; Freund [Math. Oper. Res. (1986)]) [3]. Applications to economic and game models are also presented.     

A note on the split common fixed-point problem for quasi-nonexpansive operators

Based on the very recent work by Censor and Segal (2009) [1], and inspired by Xu (2006) [9], Zhao and Yang (2005) [10], and Bauschke and Combettes (2001) [2], we introduce and analyze an algorithm for solving the split common fixed-point problem for the wide class of quasi-nonexpansive operators in Hilbert spaces. Our results improve and develop previously discussed feasibility problems and related algorithms.     

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.     

Fixed point theorems for some generalized multivalued nonexpansive mappings

In this paper, we introduce a condition on multivalued mappings which is a multivalued version of condition (C_λ) defined by Garcia-Falset et al. (2011) [3]. It is shown here that some of the classical fixed point theorems for multivalued nonexpansive mappings can be extended to mappings satisfying this condition. Our results generalize the results in Lim (1974), Lami Dozo (1973), Kirk and Massa (1990), Garcia-Falset et al. (2011), Dhompongsa et al. (2009) and Abkar and Eslamian (2010) [4], [5], [6], [3], [7] and [8] and many others.     

Approximations for nonlinear mappings by the hybrid method in Hilbert spaces

The hybrid method in mathematical programming was introduced by Haugazeau (1968) [1] and he proved a strong convergence theorem for finding a common element of finite nonempty closed convex subsets of a real Hilbert space. Later, Bauschke and Combettes (2001) [2] proposed some condition for a family of mappings (the so-called coherent condition) and established interesting results by the hybrid method. The authors (Nakajo et al., 2009) [10] extended Bauschke and Combettes’s results. In this paper, we introduce a condition weaker than the coherent condition and prove strong convergence theorems which generalize the results of Nakajo et al. (2009) [10]. And we get strong convergence theorems for a family of asymptotically κ-strict pseudo-contractions, a family of Lipschitz and pseudo-contractive mappings and a one-parameter uniformly Lipschitz semigroup of pseudo-contractive mappings.     

A note on the homotopy analysis method

The present work is devoted to using an analytic approach, namely the homotopy analysis method, to obtain convergent series solutions of strongly nonlinear problems. On the basis of the homotopy derivative concept described in Liao (2009) [3], a theorem is proved here which generalizes some lemmas and theorems provided in Liao (2009) [3] and Molabahrami and Khani (2007) [4]. Significant applicability of the theorem obtained here in some practical situations is demonstrated.     

Ramsey numbers for a disjoint union of good graphs

We give the Ramsey number for a disjoint union of some G-good graphs versus a graph G generalizing the results of Stahl (1975) [5] and Baskoro et al. (2006) [1] and the previous result of the author Bielak (2009) [2]. Moreover, a family of G-good graphs with s(G)>1 is presented.     

A new entropy upper bound

Entropy, conditional entropy and mutual information for discrete-valued random variables play important roles in information theory (see Ash (1965) [8] and Cover and Thomas (2006) [9]). Our purpose within this work is to present a strong upper bound for the classical Shannon entropy, refining recent results from the literature. For this purpose we have considered the work of Simic (2009) [4], where new entropy bounds based on a new refinement of Jensen’s inequality are presented. Our work improves the basic result of Simic through a stronger refinement of Jensen’s inequality which is then applied to information theory.     

Comments on: “A family of derivative-free conjugate gradient methods for large-scale nonlinear systems of equations” [J. Comput. Appl. Math. 224 (2009) 11–19]

In this note a simple counter example shows that the proof of Lemma 3.3 in [1, W. Cheng, Y. Xiao and Q. Hu, A family of derivative-free conjugate gradient methods for large-scale nonlinear systems of equations, J. Comput. Appl. Math. 224 (2009) 11–19] is not correct, which implies that Lemma 3.2 in [1] is not enough to ensure Lemma 3.3 in [1]. A new proof is given, which leads to a stronger result than Lemma 3.2 in [1]. And this result not only guarantees that Lemma 3.3 in [1] holds, but also improves the corresponding global convergence Theorem 3.1 in [1].     

Comparison of admissibility conditions for cyclotomic Birman-Wenzl-Murakami algebras

We show the equivalence of admissibility conditions proposed by Wilcox and Yu (in press) [11] and by Rui and Xu (2009) [9] for the parameters of cyclotomic BMW algebras.     

A simplified analytical approach for pricing discretely-sampled variance swaps with stochastic volatility

Pricing variance swaps under stochastic volatility with discretely-sampled realized variance has been a hot subject pursued recently; quite a few papers have already been published (Zhu and Lian (2009, 2011, [11], [4]); Swishchuk and Li (2011) [5]). In this paper, we present a simplified approach to price discretely-sampled variance swaps. Compared with the approach presented by Zhu and Lian (2011) [4], an important feature of our approach is that there is no need for the introduction of a new state variable and the utilization of the generalized Fourier transform. This has significantly simplified the solution procedure and will thus enable researchers to view this type of problems from a different angle.     

Common fixed point results in CAT(0) spaces

Let X be a complete CAT(0) space, T be a generalized multivalued nonexpansive mapping, and t be a single valued quasi-nonexpansive mapping. Under the assumption that T and t commute weakly, we shall prove the existence of a common fixed point for them. In this way, we extend and improve a number of recent results obtained by Shahzad (2009) [7] and [12], Shahzad and Markin (2008) [6], and Dhompongsa et al. (2005) [5].     

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 hydrodynamical model for holes in silicon semiconductors: The case of non-parabolic warped bands

This paper can be considered as the natural prosecution of Mascali and Romano (2009) [5]. Here, we describe the motion of holes in silicon by also taking into account the non-parabolicity of the heavy and light bands. The model is still based on the moment method and the closure of the system of equations is obtained by using the maximum entropy principle. Comparisons are made with the results in [5], in the case of bulk silicon, in order to establish the importance of non-parabolicity.     

On statistical exhaustiveness

We study statistical versions of several types of convergence of sequences of functions between two metric spaces. Special attention is devoted to statistical versions of recently introduced notions of exhaustiveness (Gregoriades and Papanastassiou (2008) [4]) and strong uniform convergence on a bornology (Beer and Levi (2009) [3]). We obtain a few results about the continuity of the statistical pointwise limit of a sequence of functions.