Search

Abstract: We show that the main results in the recent paper [4] by Sadiq Basha (2012) are not real generalizations but particular cases of existing fixed point theorems in the literature. [Copyright &y& Elsevier]

In this paper we propose and analyze Banach spaces-based fully-mixed approaches yielding new finite element methods for numerically solving the coupled partial differential equations describing the pseudostress-assisted diffusion of a solute into an elastic material. Two mixed formulations employing the diffusive flux as an additional variable are introduced for the diffusion equation, and the concentration gradient is considered as an auxiliary unknown of the second one of them. The resulting coupled systems are rewritten as equivalent fixed point operator equations, so that the respective unique solvabilities are proved by applying the classical Banach theorem along with the Babuška-Brezzi theory. The nonlinear dependency on the elastic variables of the diffusion coefficient and its source term, as well as the nonlinear dependency on the concentration of the elastic source term, suggest, for appropriate continuous and discrete analyses, that the unknowns be sought in suitable Lebesgue spaces. The associated Galerkin schemes are addressed similarly, and the Brouwer theorem yields the existence of discrete solutions. A priori error estimates are derived for both approaches, and rates of convergence for specific finite element subspaces satisfying the required discrete inf-sup conditions, are established in 2D. Finally, several numerical examples illustrating the performance of the two methods and confirming the theoretical findings, are reported. [ABSTRACT FROM AUTHOR]

In this paper, the existence of mild solution and controllability of fractional semilinear mixed Volterra–Fredholm integro differential equations of order 1 < γ < 2 with nonlocal conditions are investigated. First we discuss the existence of mild solution and then controllability of the proposed system by using the concept of Banach fixed point theorem and semigroup theory. For the authenticity and applicability of the presented theory some examples are also provided. [ABSTRACT FROM AUTHOR]

The Billaud Conjecture, which has been open since 1993, is a fundamental problem on finite words w and their heirs, i.e., the words obtained by deleting every occurrence of a given letter from w. It posits that every morphically primitive word, i.e., a word which is a fixed point of the identity morphism only, has at least one morphically primitive heir. In this paper, we introduce and investigate the related class of so-called Billaud words, i.e., words whose all heirs are morphically imprimitive. We provide a characterisation of morphically imprimitive Billaud words, using a new concept. We show that there are two phenomena through which words can have morphically imprimitive heirs, and we highlight that only one of those occurs in morphically primitive words. Finally, we examine our concept further, and we use it to rephrase and study the Billaud Conjecture in more detail. [ABSTRACT FROM AUTHOR]

In this paper a new definition for Menger probabilistic inner product spaces is presented. In the new setting a Menger probabilistic inner product space can naturally become a Menger probabilistic normed space, and a classical inner product space can be considered as a special case of Menger probabilistic inner product spaces. An example is given to illustrate that, the new definition is a nontrivial generalization for classical inner product spaces, and so it has rich contents in probability. By virtue of this definition, the topological structure is discussed and some elementary properties are described in terms of the families of semi-inner products. Also, some convergence theorems and probabilistic Pythagorean theorem are given. As applications, a new fixed point theorem for nonlinear contractions in Menger probabilistic inner product spaces is established. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of solutions for the following fractional boundary value problem c D α u (t) = f (t , u (t)) , α ∈ (n , n + 1) , 0 ≤ t < + ∞ , u (0) = 0 , u ′ ′ (0) = 0 , ... , u (n) (0) = 0 , lim t → + ∞ c D α − 1 u (t) = β u (ξ). The considerations of this paper are based on the concept of a new family of measures of noncompactness in the space of functions C n , γ (R +) satisfying the Hölder condition and a fixed point theorem of Darbo type. We also provide an illustrative example in support of our existence theorems. Finally, to credibility, we apply successive approximation and homotopy perturbation method to find solution of the above problem with high accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we propose the concept of fuzzy Meir–Keeler contractive mappings in fuzzy metric spaces, which covers fuzzy ψ -contractive mappings and fuzzy H -contractive mappings as special cases. With the aid of the proposed concept, we obtain some Meir–Keeler type fixed point theorems. The derived theorems improve and extend the corresponding ones developed by D. Mihet, D. Wardowski and V. Gregori, et al. And an example in the end of the paper illustrates the validity of our results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish some fixed point results for a pair of fuzzy dominated mappings satisfying contractive conditions on closed ball in multiplicative metric space. Some new fixed point results with graphic contraction on closed ball for a pair of fuzzy graph dominated mappings on multiplicative metric space have been established. Furthermore, we find a unique common solution for a system of non linear Voltera type integral equations and lastly we give an application to ensure the existence of common bounded solution of a functional equation in dynamic programming. [ABSTRACT FROM AUTHOR]

Abstract As one of the main factors affecting the dynamic behaviors of the ultra-large spatial structure, the non-sphere perturbation effects have been considered in the dynamic analysis of the rigid spatial structures recently. In this paper, employing the structure-preserving method developed in our previous job, the non-sphere perturbation effects on the orbit dynamic behavior, the attitude dynamic behavior as well as the transverse vibration of the spatial flexible damping beam are investigated in detail. The remarkable effects of the non-sphere perturbation on the dynamic behaviors of the beam obtained numerically in this paper include: The non-sphere perturbation effects on the evolution of the orbit radius should not be neglected when the attitude angle is far away from the stable attitude state; The non-sphere perturbation accelerates the dissipation of the transverse vibration of the spatial flexible damping beam; It is more important that, the consideration of the non-sphere perturbation results in a new stable attitude of the beam when the initial attitude angle is close to π / 2 and the initial attitude angle velocity is small enough. The above novel findings give some good advice on the attitude adjustment scheme design and the vibrational control strategy design for the ultra-large spatial structures. Highlights • A coupling dynamic model of spatial flexible damping beam with non-sphere perturbation is developed. • The non-sphere perturbation effects on the orbit are obvious when attitude angle is far away from stable attitude state. • Neglecting the non-sphere perturbation effects results in the overestimation of the transverse vibrational amplitude. • A new stable attitude is found when initial attitude angle is close to π / 2 and initial attitude angle velocity is tiny. [ABSTRACT FROM AUTHOR]

This paper presents NAR-RM, a method for learning robot reaching motions from a set of demonstrations using Nonlinear AutoRegressive (NAR) polynomial models. Reaching motions are modeled as solutions to autonomous discrete-time nonlinear dynamical systems, so that the movements started near the data of the demonstrations follow the trained trajectories and always reach and stop at the target. Since NAR models obtained using standard system identification techniques do not always adequately model the reaching motions, in this paper we present a method that uses a least-squares estimator with constraints to impose the location of fixed points in the model. With the imposition of new fixed points it is possible to change the location of the original fixed points of the model, thus allowing the learning of stable reaching motions. We evaluate our method using a library of human handwriting motions, a mobile robot and an industrial manipulator. [ABSTRACT FROM AUTHOR]

Let Π be a finite lattice of integer points in a box of R n and f an increasing mapping in terms of the componentwise ordering from Π to itself. The well-known Tarski's fixed point theorem asserts that f has a fixed point in Π. A simple expansion of f from Π to a larger lattice C of integer points in a box of R n yields that the smallest point in C is always a fixed point of f (an expanded Tarski's fixed point problem). By introducing an integer labeling rule and applying a cubic triangulation of the Euclidean space, we prove in this paper that the expanded Tarski's fixed point problem is in the class PPA when f is given as an oracle. It is shown in this paper that Nash equilibria of a bimatrix game can be reformulated as fixed points different from the smallest point in C of an increasing mapping from C to itself. This implies that the expanded Tarski's fixed point problem has at least the same complexity as that of the Nash equilibrium problem. As a byproduct, we also present a homotopy-like simplicial method to compute a Tarski fixed point of f . The method starts from an arbitrary lattice point and follows a finite simplicial path to a fixed point of f . [ABSTRACT FROM AUTHOR]

This paper discusses the classifier ensemble problem with sparsity and diversity learning, which is a central issue in machine learning. The current approach for reducing the size and increasing the accuracy of a classifier ensemble is to formulate it as a convex quadratic programming problem, which is a relaxation problem, and then solve it by using the existing methods for convex quadratic programming or by computing closed-form solutions. This paper presents a novel computational approach for solving the classifier ensemble problem with sparsity and diversity learning without any recourse to relaxation problems and their associated methods. We first show that the classifier ensemble problem can be expressed as a minimization problem for the sum of certain convex functions over the intersection of fixed point sets of quasi-nonexpansive mappings. Next, we propose fixed point optimization algorithms for solving the minimization problem and show that the algorithms converge to the solution of the minimization problem. It is shown that the proposed algorithms can directly solve the classifier ensemble problem with sparsity and diversity learning. Finally, we compare the performance of the proposed sparsity and diversity learning methods against an existing method in classification experiments using data sets from the UCI machine learning repository and the LIBSVM. The experimental results show that the proposed methods have higher classification accuracies than the existing method. [ABSTRACT FROM AUTHOR]

Considering the well-known shuffling operation in x - and in y -direction yields so-called double shuffles of bivariate copulas. We study continuity properties of the double shuffle operator S T induced by pairs T = (T 1 × T 2) of measure preserving transformations on ([ 0 , 1 ] , B ([ 0 , 1 ]) , λ) on the family C of all bivariate copulas, analyze its interrelation with the star/Markov product, and show that for each left- and for each right-invertible copula A the set of all possible double shuffles of A is dense in C with respect to the uniform metric d ∞. After deriving some general properties of the set Ω T of all S T -invariant copulas we focus on the situation where T 1 , T 2 are strongly mixing and show that in this case the product copula Π is an extreme point of Ω T. Moreover, motivated by a recent paper by Horanská and Sarkoci (Fuzzy Sets and Systems 378, 2018) we then study double shuffles induced by pairs of so-called Lüroth maps and derive various additional properties of Ω T , including the surprising fact that Ω T contains uncountably many extreme points which (interpreted as doubly stochastic measures) are pairwise mutually singular with respect to each other and which allow for an explicit construction. [ABSTRACT FROM AUTHOR]

In this paper we study a class of quasi-variational-hemivariational inequalities in reflexive Banach spaces. The inequalities contain a convex potential, a locally Lipschitz superpotential, and a implicit obstacle set of constraints. Solution existence and compactness of the solution set to the inequality problem are established based on the Kakutani–Ky Fan fixed point theorem. The applicability of the results is illustrated by the steady-state Oseen model of a generalized Newtonian incompressible fluid with mixed boundary conditions. The latter involve a unilateral boundary condition, the Navier slip condition, a nonmonotone version of the nonlinear Navier-Fujita slip condition, and a generalization of the threshold slip and leak condition of frictional type. [ABSTRACT FROM AUTHOR]

In this paper, we present the existence criteria for the solutions of boundary value problems involving generalized fractional integro-Langevin equation and inclusion supplemented with nonlocal fractional boundary conditions. The main idea of the current research is to combine the integro-differential and Langevin structures together. The main tools of our study include certain inequalities and well-known fixed point theorems. Numerical examples are constructed to demonstrate the application of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we modify the general iterative method to approximate a common element of the set of solutions of split variational inclusion problem and the set of common fixed points of a finite family of k -strictly pseudo-contractive nonself mappings. Strong convergence theorem is established under some suitable conditions in a real Hilbert space, which also solves some variational inequality problems. Results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors. Finally, some examples to study the rate of convergence and some illustrative numerical examples are presented. [ABSTRACT FROM AUTHOR]

The purpose of this work is to fill a gap in the article Priya and Uthayakumar (2022) [13]. To prove fractals (fixed points), the authors of that article tried to connect two independent conceptual domains. This results in their main theorem being wrong and becoming a conjecture. This paper scrutinizes and presents both the concepts above to demonstrate fractals in the settings posed in that article. Using a proper definition, the correct theorem to achieve the purpose of that article is discussed. [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]

This paper studies a class of global fractional-order projective dynamical systems involving set-valued perturbations in real separable Hilbert spaces. We prove that the set of solutions for this type of systems is nonempty and closed under some suitable conditions. Furthermore, we show that the set of solutions is continuous with respect to initial value in the sense of Hausdorff metric. Finally, an interesting numerical example is given to illustrate the validity of the main theorem presented in this paper. [ABSTRACT FROM AUTHOR]

In the first half of the paper, we are concerned with the problems of existence (and searching) of coincidence points and the common preimage of a closed subset (in particular, a common root) in the case of a finite system of mappings of one metric space to another one. The second half of the paper is devoted to fiberwise variants of Arutyunov's theorem on coincidence points. Obtaining the main results of the paper is based on the use of the class of almost exactly ( α , β ) -search functionals that is wider than Fomenko's class of ( α , β ) -search functionals. [ABSTRACT FROM AUTHOR]

Let X be a compact polyhedron that is homotopy equivalent to the figure-eight. This is a survey paper for the WYK algorithm which provides computations for the Nielsen number of self-maps of X . The WYK algorithm consists of three parts and each part was introduced separately by Wagner [11] , Yi and Kim [12] , and Kim [8] . In this paper, we present the whole algorithm so that people can more easily use the algorithm. [ABSTRACT FROM AUTHOR]

Due to its possible applications, Fixed Point Theory has become one of the most useful branches of Nonlinear Analysis. In a very recent paper, Khojasteh et al. introduced the notion of simulation function in order to express different contractivity conditions in a unified way, and they obtained some fixed point results. In this paper, we slightly modify their notion of simulation function and we investigate the existence and uniqueness of coincidence points of two nonlinear operators using this kind of control functions. [ABSTRACT FROM AUTHOR]

This research paper investigates and proves some theorems of the fixed point for self–mapping T : X → X under (ϕ , ψ) –contractive mappings and (ϕ , φ) –contractive mappings in Menger probabilistic 2–metric space. These theorems are used as an essential tool to convert the probabilistic metric to 2–metric space and are employed to prove the uniqueness and existence the fixed point of the a mapping T from a complete 2–Menger probabilistic space into itself. The used definitions and theorems show effective and possibility of our main idea. [ABSTRACT FROM AUTHOR]

Variational time integrators are derived in the context of discrete mechanical systems. In this area, the governing equations for the motion of the mechanical system are built following two steps: (a) Postulating a discrete action; (b) Computing the stationary point of the discrete action. The former is formulated by considering Lagrangian (or Hamiltonian) systems with the discrete action being constructed through numerical approximations of the action integral. The latter derives the discrete Euler–Lagrange equations whose solutions give the variational time integrator. In this paper, we build variational time integrators in the context of smoothed particle hydrodynamics (SPH). So, we start with a variational formulation of SPH for fluids. Then, we apply the generalized midpoint rule, which depends on a parameter α , in order to generate the discrete action. Then, the step (b) yields a variational time integration scheme that reduces to an explicit approach if α ∈ { 0 , 1 } but it is implicit otherwise. Hence, we design a fixed point iterative method to approximate the solution and prove its convergence condition. Besides, we show that the obtained discrete Euler–Lagrange equations preserve linear momentum. In the experimental results, we simulate a bubble flow and a dam breaking set up and consider viscosity as well as boundary interaction effects. We compare standard and implicit SPH solutions. We analyze linear momentum conservation and other benchmark quantities to conclude that the proposed algorithm is accurate and preserves the linear momentum better than the counterpart one for dam breaking set up. [ABSTRACT FROM AUTHOR]

In this note, we show that although the main result Theorem 21 in the paper published by Roldán-López-de-Hierro and Sintunavarat is valid, its proof has a small gap, which needs a revision. The main contribution of the note is to detect the gap, and give an appropriate proof of Theorem 21. [ABSTRACT FROM AUTHOR]

In this paper we present an example of a fuzzy ψ -contractive sequence in the sense of D. Mihet, which is not Cauchy in a fuzzy metric space in the sense of George and Veeramani. To overcome this drawback we introduce and study a concept of strictly fuzzy contractive sequence. Then, we also make an appropriate correction to Lemma 3.2 of Gregori and Miñana (2016) [5]. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of elliptic equations in annular domains involving derivative terms. Using nonlinear analysis methods, some results regarding existence of solutions are established. More precisely, at least one radial solution is obtained basing on Schauder's fixed point theorem and contraction mapping theorem, respectively. Some iterative technique is also applied, which allows us to overcome difficulties that the nonlinearity is related to derivative terms. [ABSTRACT FROM AUTHOR]

In topological fixed point theory, the Reidemeister trace is an invariant associated to a selfmap of a polyhedron which combines information from the Lefschetz and Nielsen numbers. In this paper we define the Reidemeister trace in the context of n -valued selfmaps of compact polyhedra. We prove several properties of the Reidemeister trace which generalize properties from the single-valued theory, and prove an averaging formula. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, existence results are derived for the unique, smallest, greatest, minimal, and/or maximal solutions of finite systems of nonlinear distributional Cauchy problems. The dependence of solutions on the data is also studied. Applications to systems of distributional differential equations with impulses and to higher-order distributional Cauchy problems are presented. The main tools are fixed point results in function spaces and a primitive integral of distributions defined in the paper, including recently introduced regulated and continuous primitive integrals on compact intervals. [Copyright &y& Elsevier]

Abstract: Following John Kennison, a flow (or discrete dynamical system) in a category C is a couple , where X is an object of C and is a morphism, called the iterator. If and are flows in C, then is a morphism of flows from to if . We let denote the resulting category of flows in C. This paper deals with and , where Set and Top denote respectively the categories of sets and topological spaces. By a Gottschalk flow, we mean a flow in Top satisfying the following conditions: [(i)] If is any almost periodic point of f, then the closure is a minimal set of f; [(ii)] All points in any minimal set of f are almost periodic points. As proven by Gottschalk, if X is a compact Hausdorff space and is a continuous function, then is a Gottschalk flow. In this paper, we prove that for any flow of Set, there is a topology on X for which is a Gottschalk flow in Top. This, actually, defines a covariant functor from into . The main result of this paper provides a characterization of spaces in the image of the functor in order-theoretical terms. Some categorical properties of and are also given. [Copyright &y& Elsevier]

Abstract: The purpose of this article is to propose a new hybrid algorithm with variable coefficients and prove convergence theorems for asymptotically pseudocontractive mappings in the intermediate sense on unbounded domains. The results of the paper improve and extend the recent results of Qin et al. [X.L. Qin, S.Y. Cho, J.K. Kim, Convergence theorems on asymptotically pseudocontractive mappings in the intermediate sense, Fixed Point Theory Appl. 2010, doi:10.1155/2010/186874, Article ID 186874, 14 pages] and several others. The algorithm with variable coefficients introduced in this paper is of independent interest. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to present some new fixed point theorems for mixed monotone operators with perturbation by using the properties of cones and a fixed point theorem for mixed monotone operators. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems. [Copyright &y& Elsevier]

Abstract: In this paper, we study a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of variational inequalities for two inverse strongly accretive mappings and the set of common fixed points of an infinite family of nonexpansive mappings in a real 2-uniformly smooth and uniformly convex Banach space. We prove some strong convergence theorems under suitable conditions. The results obtained in this paper improve and extend the corresponding results announced by many others. [Copyright &y& Elsevier]

Abstract In this paper, we study a new fractional-order initial value problem involving a Caputo-type generalized fractional derivative and a Steiltjes type fractional integral. Extremal solutions for the given problem are obtained by monotone iterative method. An example illustrating the main result is presented. [ABSTRACT FROM AUTHOR]

Abstract In this paper, some new theorems for various cyclic contractions are established in G b -metric spaces to discuss the existence and uniqueness of the solutions for a class of the operator equations, which generalize many known results in corresponding literatures. On the other hand, we introduce the notion of the cyclic α - ψ φ-contractive mappings in G b -metric spaces and establish a new fixed point theorem, which is applied to consider the existence of the solutions of the integral equations and ordinary differential equations. Moreover, some examples are given to support our main results. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the existence of solutions for Caputo type sequential fractional differential equations and inclusions supplemented with semi-periodic and nonlocal integro-multipoint boundary conditions involving Riemann-Liouville integral. We make use of standard fixed point theorems for single-valued and multivalued maps to obtain the desired results. Examples are constructed for the illustration of the main results. [ABSTRACT FROM AUTHOR]

Abstract In the present paper we prove some new fixed point theorems for self-mappings defined on a complete metric space with a w -distance. These results extend some previous fixed point theorems in this field to more general contractive conditions in the setting of w -distances for selfmappings which satisfy certain weaker Meir–Keeler conditions. [ABSTRACT FROM AUTHOR]

Abstract In the present paper we study the class of all those copulas that are invariant under a special class of transformations, called reflections. In particular, we focus on the special role played by the independence copula within this class. For this purpose, we introduce a bijective transformation which turns every copula into a reflection invariant copula and enforces a strong geometric property: The n -fold composition of this transformation to a d -dimensional copula coincides with the independence copula on the mesh { 0 , 1 2 n ,... , 2 n − 1 2 n , 1 } d. It turns out that the independence copula is the unique fixed point of the introduced transformation. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, by studying the modified Ishikawa''s iteration method, we establish a weak convergence theorem for the asymptotically strict pseudo-contraction in Hilbert spaces. We also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which can generate a strongly convergent sequence. The results presented in this paper improve and extend the previous known results. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce hybrid pseudo-viscosity approximation schemes with strongly positive bounded linear operators for finding a common element of the set of solutions to a system of equilibrium problems, the set of fixed points of an infinite family and left amenable semigroup of non-expansive mappings in the frame work of Hilbert spaces. Our goal is to prove a result of strong convergence for hybrid pseudo-viscosity approximation schemes to approach a solution of systems of equilibrium problems which is also a common fixed point of an infinite family and left amenable semigroup of non-expansive mappings. The results presented in this paper can be treated as an extension and improvement of the corresponding results announced by Ceng et al. [L.C. Ceng, Q.H. Ansari, and J.C. Yao, Hybrid pseudo-viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many non-expansive mappings, Nonlinear Analysis 4 (2010) 743–754] and many others. [Copyright &y& Elsevier]

Abstract: The present paper is divided into two parts. In the first part, we introduce implicit and explicit iterative schemes for finding the fixed point of a nonexpansive mapping defined on the closed convex subset of a real Hilbert space. We establish results on the strong convergence of the sequences generated by the proposed schemes to a fixed point of a nonexpansive mapping. Such a fixed point is also a solution of a variational inequality defined on the set of fixed points. In the second part, we propose implicit and explicit iterative schemes for finding the approximate minimizer of a constrained convex minimization problem and prove that the sequences generated by our schemes converge strongly to a solution of the constrained convex minimization problem. Such a solution is also a solution of a variational inequality defined over the set of fixed points of a nonexpansive mapping. The results of this paper extend and improve several results presented in the literature in the recent past. [Copyright &y& Elsevier]

Abstract: In 2007, Huang and Zhang [L.G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476] rediscovered normal cone metric spaces and obtained the Banach contraction principle for this setting. Later on, Rezapour and Hamlbarani [Sh. Rezapour, R. Hamlbarani, Some notes on the paper: Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 345 (2008) 719–724] showed that there are non-normal cones and that the assumption of normality is redundant. In this paper, we obtain a generalization of the Banach contraction principle to the class of convex contractions on non-normal cone metric spaces. Our result includes, as special cases, the recent results of Huang and Zhang (2007) and Rezapour and Hamlbarani (2008) . [Copyright &y& Elsevier]