Admissibility of Control Operators in UMD Spaces and the Inverse Laplace Transform

This paper investigates the admissibility of control and observation operators in UMD spaces. Necessary and/or sufficient conditions for unbounded control operators to be admissible and weakly admissible in the Salamon–Weiss sense are presented. This is illustrated by an example which shows that the UMD-property is essential. In particular, we get a direct proof of the known result of Driouich and and El-Mennaoui (Arch Math 72:56–63, 1999) on the validity of the inverse formula of the Laplace transform for C ₀-semigroups on UMD-spaces and in Hilbert spaces, as proved earlier by Yao (SIAM J Math Anal 26(5):1331–1341, 1995). We outline how these results can be used to prove a partial validity of the inverse Laplace transform for semigroups in general Banach spaces. In particular, we obtain the result on the inverse Laplace transform due to Hille and Philllips (Am Math Soc Transl Ser 2, 1957).     

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.     

A Fréchet derivative-free cubically convergent method for set-valued maps

We introduce a new iterative method in order to approximate a locally unique solution of variational inclusions in Banach spaces. The method uses only divided differences operators of order one. An existence–convergence theorem and a radius of convergence are given under some conditions on divided difference operator and Lipschitz-like continuity property of set-valued mappings. Our method extends the recent work related to the resolution of nonlinear equation in Argyros (J Math Anal Appl 332:97–108, 2007) and has the following advantages: faster convergence to the solution than all the previous known ones in Argyros and Hilout (Appl Math Comput, 2008 in press), Hilout (J Math Anal Appl 339:53–761, 2008, Positivity 10:673–700, 2006), and we do not need to evaluate any Fréchet derivative. We provide also an improvement of the ratio of our algorithm under some center-conditions and less computational cost. Numerical examples are also provided.      

Order-Compactifications of Totally Ordered Spaces: Revisited

Order-compactifications of totally ordered spaces were described by Blatter (J Approx Theory 13:56–65, 1975) and by Kent and Richmond (J Math Math Sci 11(4):683–694, 1988). Their results generalize a similar characterization of order-compactifications of linearly ordered spaces, obtained independently by Fedorčuk (Soviet Math Dokl 7:1011–1014, 1966; Sib Math J 10:124–132, 1969) and Kaufman (Colloq Math 17:35–39, 1967). In this note we give a simple characterization of the topology of a totally ordered space, as well as give a new simplified proof of the main results of Blatter (J Approx Theory 13:56–65, 1975) and Kent and Richmond (J Math Math Sci 11(4):683–694, 1988). Our main tool will be an order-topological modification of the Dedekind-MacNeille completion. In addition, for a zero-dimensional totally ordered space X, we determine which order-compactifications of X are Priestley order-compactifications.     

On Second-Order Optimality Conditions for Vector Optimization

In this article, two second-order constraint qualifications for the vector optimization problem are introduced, that come from first-order constraint qualifications, originally devised for the scalar case. The first is based on the classical feasible arc constraint qualification, proposed by Kuhn and Tucker (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 481–492, University of California Press, California, 1951) together with a slight modification of McCormick’s second-order constraint qualification. The second—the constant rank constraint qualification—was introduced by Janin (Math. Program. Stud. 21:110–126, 1984). They are used to establish two second-order necessary conditions for the vector optimization problem, with general nonlinear constraints, without any convexity assumption.     

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.     

One-dimensional chemotaxis kinetic model

In this paper, we study a variation of the equations of a chemotaxis kinetic model and investigate it in one dimension. In fact, we use fractional diffusion for the chemoattractant in the Othmar–Dunbar–Alt system (Othmer in J Math Biol 26(3):263–298, 1988). This version was exhibited in Calvez in Amer Math Soc, pp 45–62, 2007 for the macroscopic well-known Keller–Segel model in all space dimensions. These two macroscopic and kinetic models are related as mentioned in Bournaveas, Ann Inst H Poincaré Anal Non Linéaire, 26(5):1871–1895, 2009, Chalub, Math Models Methods Appl Sci, 16(7 suppl):1173–1197, 2006, Chalub, Monatsh Math, 142(1–2):123–141, 2004, Chalub, Port Math (NS), 63(2):227–250, 2006. The model we study here behaves in a similar way to the original model in two dimensions with the spherical symmetry assumption on the initial data which is described in Bournaveas, Ann Inst H Poincaré Anal Non Linéaire, 26(5):1871–1895, 2009. We prove the existence and uniqueness of solutions for this model, as well as a convergence result for a family of numerical schemes. The advantage of this model is that numerical simulations can be easily done especially to track the blow-up phenomenon.     

On Ulm’s method using divided differences of order one

We provide new sufficient convergence conditions for the semilocal convergence of Ulm’s method (Tzv Akad Nauk Est SSR 16:403–411, 1967) in order to approximate a locally unique solution of an equation in a Banach space setting. We show that in some cases, our hypotheses hold true but the corresponding ones in Burmeister (Z Angew Math Mech 52:101–110, 1972), Kornstaedt (Aequ Math 13:21–45, 1975), Moser (1973), and Potra and Pták (Cas Pest Mat 108:333–341, 1983) do not. We also show that under the same hypotheses and computational cost, finer error bounds can be obtained. Some error bounds are also shown to be sharp. Numerical examples are also provided further validating the results.     

Efficient three-step iterative methods with sixth order convergence for nonlinear equations

In this paper, we present two new three-step iterative methods for solving nonlinear equations with sixth convergence order. The new methods are obtained by composing known methods of third order of convergence with Newton’s method and using an adequate approximation for the derivative, that provides high order of convergence and reduces the required number of functional evaluations per step. The first method is obtained from Potra-Pták’s method and the second one, from Homeier’s method, both reaching an efficiency index of 1.5651. Our methods are comparable with the method of Parhi and Gupta (Appl Math Comput 203:50–55, 2008). Methods proposed by Kou and Li (Appl Math Comput 189:1816–1821, 2007), Wang et al. (Appl Math Comput 204:14–19, 2008) and Chun (Appl Math Comput 190:1432–1437, 2007) reach the same efficiency index, although they start from a fourth order method while we use third order methods and simpler arithmetics. We prove the convergence results and check them with several numerical tests that allow us to compare the convergence order, the computational cost and the efficiency order of our methods with those of the original methods.     

RETRACTED ARTICLE: Convergence of Weighted Sums for Arrays of Negatively Dependent Random Variables and Its Applications

We discuss the complete convergence of weighted sums for arrays of rowwise negatively dependent random variables (ND r.v.’s) to linear processes. As an application, we obtain the complete convergence of linear processes based on ND r.v.’s which extends the result of Li et al. (Stat. Probab. Lett. 14:111–114, 1992), including the results of Baum and Katz (Trans. Am. Math. Soc. 120:108–123, 1965), from the i.i.d. case to a negatively dependent (ND) setting. We complement the results of Ahmed et al. (Stat. Probab. Lett. 58:185–194, 2002) and confirm their conjecture on linear processes in the ND case.     

Discrete Approximation of Spaces of Homogeneous Type

In this note we combine the dyadic families introduced by M. Christ in (Colloq. Math. 60/61(2):601–628, 1990) and the discrete partitions introduced by J.M. Wu in (Proc. Am. Math. Soc. 126(5):1453–1459, 1998) to get approximation of a compact space of homogeneous type by a uniform sequence of finite spaces of homogeneous type. The convergence holds in the sense of a metric built on the Hausdorff distance between compact sets and on the Kantorovich-Rubinshtein metric between measures. The authors were supported by CONICET, CAI+D (UNL) and ANPCyT.     

On Borell-Brascamp-Lieb Inequalities on Metric Measure Spaces

In this paper we introduce the notion of a Borell-Brascamp-Lieb inequality for metric measure spaces (M,d,m) denoted by BBL(K,N) for two numbers K,N ∈ ℝ with N ≥ 1. In the first part we prove that BBL(K,N) holds true on metric measure spaces satisfying a curvature-dimension condition CD(K,N) developed and studied by Lott and Villani in (Ann Math 169:903–991, 2007) as well as by Sturm in (Acta Math 196(1):133–177, 2006). The aim of the second part is to show that BBL(K,N) is stable under convergence of metric measure spaces with respect to the L ₂-transportation distance.     

A few remarks on recurrence relations for geometrically continuous piecewise Chebyshevian B-splines

This works complements a recent article (Mazure, J. Comp. Appl. Math. 219(2):457–470, 2008) in which we showed that T. Lyche’s recurrence relations for Chebyshevian B-splines (Lyche, Constr. Approx. 1:155–178, 1985) naturally emerged from blossoms and their properties via de Boor type algorithms. Based on Chebyshevian divided differences, T. Lyche’s approach concerned splines with all sections in the same Chebyshev space and with ordinary connections at the knots. Here, we consider geometrically continuous piecewise Chebyshevian splines, namely, splines with sections in different Chebyshev spaces, and with geometric connections at the knots. In this general framework, we proved in (Mazure, Constr. Approx. 20:603–624, 2004) that existence of B-spline bases could not be separated from existence of blossoms. Actually, the present paper enhances the powerfulness of blossoms in which not only B-splines are inherent, but also their recurrence relations. We compare this fact with the work by G. Mühlbach and Y. Tang (Mühlbach and Tang, Num. Alg. 41:35–78, 2006) who obtained the same recurrence relations via generalised Chebyshevian divided differences, but only under some total positivity assumption on the connexion matrices. We illustrate this comparison with splines with four-dimensional sections. The general situation addressed here also enhances the differences of behaviour between B-splines and the functions of smaller and smaller supports involved in the recurrence relations.     

On a class of secant-like methods for solving nonlinear equations

We provide a semilocal convergence analysis for a certain class of secant-like methods considered also in Argyros (J Math Anal Appl 298:374–397, 2004, 2007), Potra (Libertas Mathematica 5:71–84, 1985), in order to approximate a locally unique solution of an equation in a Banach space. Using a combination of Lipschitz and center-Lipschitz conditions for the computation of the upper bounds on the inverses of the linear operators involved, instead of only Lipschitz conditions (Potra, Libertas Mathematica 5:71–84, 1985), we provide an analysis with the following advantages over the work in Potra (Libertas Mathematica 5:71–84, 1985) which improved the works in Bosarge and Falb (J Optim Theory Appl 4:156–166, 1969, Numer Math 14:264–286, 1970), Dennis (SIAM J Numer Anal 6(3):493–507, 1969, 1971), Kornstaedt (1975), Larsonen (Ann Acad Sci Fenn, A 450:1–10, 1969), Potra (L’Analyse Numérique et la Théorie de l’Approximation 8(2):203–214, 1979, Aplikace Mathematiky 26:111–120, 1981, 1982, Libertas Mathematica 5:71–84, 1985), Potra and Pták (Math Scand 46:236–250, 1980, Numer Func Anal Optim 2(1):107–120, 1980), Schmidt (Period Math Hung 9(3):241–247, 1978), Schmidt and Schwetlick (Computing 3:215–226, 1968), Traub (1964), Wolfe (Numer Math 31:153–174, 1978): larger convergence domain; weaker sufficient convergence conditions, finer error bounds on the distances involved, and a more precise information on the location of the solution. Numerical examples further validating the results are also provided.     

A Little More on Coz-Unique Frames

Coz-unique frames were defined and characterized by Banaschewski and Gilmour (J Pure Appl Algebra 157:1–22, 2001). In this note we give further characterizations of these frames along the lines of characterizations of absolutely z-embedded spaces obtained by Blair and Hager (Math Z 136:41–52, 1974) on the one hand, and by Hager and Johnson (Canad J Math 20:389–393, 1968) on the other. We also extend to frames certain characterizations of z-embedded spaces; namely, we give a characterization of coz-onto frame homomorphisms in terms of normal covers.      

On the Existence of Min-Max Minimal Torus

In this paper, we will study the existence problem of min-max minimal torus. We use classical conformal invariant geometric variational methods. We prove a theorem about the existence of min-max minimal torus in Theorem 5.1. First we prove a strong uniformization result (Proposition 3.1) using the method of Ahlfors and Bers (Ann. Math. 72(2):385–404, 1960). Then we use this proposition to choose good parameterization for our min-max sequences. We prove a compactification result (Lemma 4.1) similar to that of Colding and Minicozzi (Width and finite extinction time of Ricci flow, [math.DG], 2007), and then give bubbling convergence results similar to that of Ding et al. (Invent. math. 165:225–242, 2006). In fact, we get an approximating result similar to the classical deformation lemma (Theorem 1.1).     

Aggregation in age and space structured population models: an asymptotic analysis approach

In this paper we describe how techniques of asymptotic analysis can be used in a systematic way to perform ‘aggregation’ of variables, based on a separation of different time scales, in a population model with age and space structure. The main result of the paper is proving the convergence of the formal asymptotic expansion to the solution of the original equation. This result improves and clarifies earlier results of Arino et al. (SIAM J Appl Math 60(2):408–436, 1999), Auger et al. (Structured population models in biology and epidemiology. Springer Verlag, Berlin, 2008), Lisi and Totaro (Math Biosci 196(2):153–186, 2005).     

Solution Methods for Pseudomonotone Variational Inequalities

We extend some results due to Thanh-Hao (Acta Math. Vietnam. 31: 283–289, [2006]) and Noor (J. Optim. Theory Appl. 115:447–452, [2002]). The first paper established a convergence theorem for the Tikhonov regularization method (TRM) applied to finite-dimensional pseudomonotone variational inequalities (VIs), answering in the affirmative an open question stated by Facchinei and Pang (Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, [2003]). The second paper discussed the application of the proximal point algorithm (PPA) to pseudomonotone VIs. In this paper, new facts on the convergence of TRM and PPA (both the exact and inexact versions of PPA) for pseudomonotone VIs in Hilbert spaces are obtained and a partial answer to a question stated in (Acta Math. Vietnam. 31:283–289, [2006]) is given. As a byproduct, we show that the convergence theorem for inexact PPA applied to infinite-dimensional monotone variational inequalities can be proved without using the theory of maximal monotone operators. This research was supported in part by a grant from the National Sun Yat-Sen University, Kaohsiung, Taiwan. It has been carried out under the agreement between the National Sun Yat-Sen University, Kaohsiung, Taiwan and the University of Pisa, Pisa, Italy. The authors thank the anonymous referee for useful comments and suggestions.     

Some new common fixed point theorems in fuzzy metric spaces

The purpose of this paper is to prove some new common fixed point theorems in (GV)-fuzzy metric spaces. While proving our results, we utilize the idea of compatibility due to Jungck (Int J Math Math Sci 9:771–779, 1986) together with subsequentially continuity due to Bouhadjera and Godet-Thobie (arXiv: 0906.3159v1 [math.FA] 17 Jun 2009) respectively (also alternately reciprocal continuity due to Pant (Bull Calcutta Math Soc 90:281–286, 1998) together with subcompatibility due to Bouhadjera and Godet-Thobie (arXiv:0906.3159v1 [math.FA] 17 Jun 2009) as patterned in Imdad et al. (doi:) wherein conditions on completeness (or closedness) of the underlying space (or subspaces) together with conditions on continuity in respect of any one of the involved maps are relaxed. Our results substantially generalize and improve a multitude of relevant common fixed point theorems of the existing literature in metric as well as fuzzy metric spaces which include some relevant results due to Imdad et al. (J Appl Math Inform 26:591–603, 2008), Mihet (doi:), Mishra (Tamkang J Math 39(4):309–316, 2008), Singh (Fuzzy Sets Syst 115:471–475, 2000) and several others.