a newton kantorovich convergence theorem for the inverse free jarratt method in banach space abstract under the weaken conditions we establish the newton kantorovich convergence theorem of inverse free jarratt method in banach space which is used to solve the nonlinear operator equation finally some examples are provided to show the application of our theorem

fuzzy vectors as a tool for modeling uncertain multidimensional quantities the paper deals with modeling uncertain multidimensional quantities by means of fuzzy vectors in connection with possible interactions among variables the separability of fuzzy vectors is discussed in detail for the case when interconnections among variables are given by a crisp relation a more general form of separability of fuzzy vectors the separability on a given relation e g on a probability simplex is introduced properties of fuzzy vectors separable on a given relation are studied and ways of convenient setting of such fuzzy vectors are proposed finally fuzzy extensions of real valued functions and real vector valued functions to general input fuzzy vectors are investigated

mysovskii type theorem for the secant method under holder continuous frechet derivative abstract the mysovskii type condition is considered in this study for the secant method in banach spaces to solve a nonlinear operator equation we suppose the inverse of divided difference of order one is bounded and the frechet derivative of the nonlinear operator is holder continuous by use of fibonacci generalized sequence a semilocal convergence theorem is established which matches with the convergence order of the method finally two simple examples are provided to show that our results apply where earlier ones fail

a method for finding sharp error bounds for newton s method under the kantorovic assumptions this paper gives a method for finding sharpa posteriori error bounds for newton s method under the assumptions of kantorovich s theorem on the basis of this method new error bounds are derived and comparison is made among the known bounds of dennis 2 doring 4 gragg tapia 5 kantorovich 6 7 kornstaedt 9 lancaster 10 miel 11 13 moret 14 ostrowski 17 18 potra 19 and potra ptak 20

a note on the halley method in banach spaces abstract we provide a convergence theorem under the usual ostrowski kantorovich conditions for an equivalent form of the halley method in banach spaces the best possible upper error bound for the method is obtained some applications to the solution of quadratic operator equations are provided

on the r order of the halley method an r order bound for the halley method is obtained in this work where an analysis of the convergence of the method is also presented under mild differentiability conditions to do this a new technique is developed where the involved operator must satisfy some recurrence relations

on modified newton methods with cubic convergence abstract in this paper we present two new modified newton methods for solving a non linear equation permitting f x 0 in some points these new methods are showed to be cubically convergent from the practical application we show that these new methods are vast superior and possible in global convergence

fuzzy weighted averages and implementation of the extension principle abstract this paper addresses the computational aspect of the extension principle when the principle is applied to algebraic mappings and in particular to weighted average operations in risk and decision analysis a computational algorithm based on the cut representation of fuzzy sets and interval analysis is described the method provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner examples are given to illustrate the method and its relation to other discrete methods and the exact approach by non linear programming the algorithm has been implemented in a computer program which can handle very general extended algebraic operations on fuzzy numbers

non standard cut classification of fuzzy sets abstract several important non standard cut sets of lattice valued fuzzy sets are investigated these are strong cuts not less and neither less nor equal cuts in each case it is proved that collection of all cuts of any lattice valued fuzzy set form a complete lattice under inclusion decomposition theorem representation by cuts is proved for neither less nor equal cuts necessary and sufficient conditions under which two lattice valued fuzzy sets with the same domain have equal families of corresponding cut sets are given

on the comparison of a weak variant of the newton kantorovich and miranda theorems we recently showed a semilocal convergence theorem that guarantees convergence of newton s method to a locally unique solution of a nonlinear equation under hypotheses weaker than those of the newton kantorovich theorem here we first weaken miranda s theorem which is a generalization of the intermediate value theorem then we show that operators satisfying the weakened newton kantorovich conditions satisfy those of the weakened miranda s theorem

a note on halley s method we introduce the degree of logarithmic convexity which provides a measure of the convexity of a function at each point making use of this concept we obtain a new theorem of global convergence for halley s method

modification of the kantorovich assumptions for semilocal convergence of the chebyshev method this study obtains two semilocal convergence results for the well known chebyshev method which is a third order iterative process the hypotheses required are modifications to the normal kantorovich ones the results obtained are applied to the reduction of nonlinear integral equations of the fredholm type and first kind

representing ordered structures by fuzzy sets an overview we present a survey on representations of ordered structures by fuzzy sets any poset satisfying some finiteness condition semilattice lattice belonging to a special class e g distributive noetherian complete and others can be represented by a single function i e by a fuzzy set its domain and co domain are particular subsets of the same structure and consist of irreducible elements the representation is minimal in the sense that another representation could not be obtained by replacing the domain of the former by its proper subset by this approach the structure itself is uniquely represented by the collection of cuts ordered dually to inclusion

poset valued convexities we analyze cut properties of lattice and poset valued functions fuzzy sets of n dimensional real variable with respect to convexity in the lattice valued case convexity of a fuzzy set is equivalent with the convexity of its standard cuts dealing with non standard cuts we show that unless the space of the values is a chain such statement in general does not hold we present analogue results in the more general setting of poset valued functions we also prove that there exist lattice and poset valued functions whose cuts are precisely the convex sets of particular families our work is motivated by possible applications of convex functions e g in image processing

modeling uncertain variables of the weighted average operation by fuzzy vectors the paper deals with the fuzzy extension of the weighted average operation first we study the convenient ways how uncertain weights and weighted values can be modeled by fuzzy vectors we show that in comparison to a tuple of fuzzy numbers that have been used for modeling uncertain values of particular weights and weighted values up to now fuzzy vectors extend the possibilities of utilizing the vague expert information concerning the weights and weighted values next we focus on computation of a fuzzy weighted average of a fuzzy vector of weighted values with a fuzzy vector of weights we derive a general formula and we study its special forms the advantage of the approach presented in the paper is that the resulting fuzzy weighted average is not overly imprecise since every available information about its variables is involved in computation this fact is illustrated by several examples finally we briefly discuss the problem of defuzzification of the resulting fuzzy weighted average

on weighted unbalanced linguistic aggregation operators in group decision making we generalize linguistic evaluation values and their weights in group decision making gdm problems based on unbalanced linguistic terms the gdm problems include two types of weights belief degrees of linguistic evaluation values and experts weights due to the time pressure and conflict of multi source of information etc experts may have various evaluations values on alternatives the belief degree is used to represent the confidence level of the values the experts weight is used to represent the differences among experts importance which caused by experts experience or knowledge distinction we propose the weighted unbalanced linguistic aggregation operators to synthesize linguistic evaluation value belief degree and experts weights some desired properties of the operator are then studied these properties show that the operator extends the linguistic weighted averaging operator lwa and the linguistic ordered weighted averaging lowa operator finally an illustrative example of human resource performance appraisal based on the linguistic aggregation operator is provided

fuzzy rule based models with interactive rules and their granular generalization abstract processing realized in fuzzy rule based models is associated with matching individual rules with the existing input data and aggregation of the levels of matching and conclusions of the rules rules are processed as individual entities in this study we introduce an augmentation of fuzzy models by facilitating interaction among the rules leading to more flexible type membership functions of fuzzy sets forming conditions of the rules thus resulting in substantially advanced topology of the partition of the input space different ways of realizing interaction among the rules are studied in the sequel we develop a granular fuzzy model implied by the rule based model showing how granular parameters of the original rule based model enhance its quality expressed in terms of coverage of experimental data the two evaluation criteria of the constructed granular model namely coverage and specificity are studied experimental results are reported for a series of publicly available data

chebyshev method and convexity from convexity we obtain new convergence theorems for the chebyshev method applied to a nonlinear scalar equation f x 0 moreover kantorovich type conditions for the convergence of the chebyshev method in the complex plane are given finally a new type of assumptions are considered to prove the convergence of this method in banach spaces

on lattice valued up sets and down sets isotone and anti isotone mappings from a poset into a complete lattice are investigated as lattice valued up sets and down sets respectively cuts of these are shown to be analogue crisp sub posets of the domain up set or semi filters and down sets or semi ideals the collection of all lattice valued up sets down sets of a poset is a complete lattice under the order inherited from the lattice among other results for a collection of crisp up sets down sets of a poset necessary and sufficient conditions are given under which this collection consists of cuts of a lattice valued up set down sets a generalization in the sense of closed fuzzy sets with respect to fuzzy relations is also carried out

fuzzy weighted average an improved algorithm abstract this paper suggests a modification on the fuzzy weighted average fwa method developed by dong and wong 1987 a steepest descend ascend method is proposed to facilitate steps 3 and 4 of the fwa to find the minimum maximum of f x1 x2 xn w1 w2 wn i 1n wixi i 1n wi an example used by dong and wong is again employed to illustrate the new method the same results are obtained but the new method requires much less evaluations and computations the advantages and disadvantages of the two algorithms are discussed also several relevant theorems are stated and discussed in this paper and the proofs of these theorems are listed in the appendix

on the newton kantorovich hypothesis for solving equations 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 in connection with the lipschitz continuity of the frechet derivative of the operator involved here using lipschitz and center lipschitz conditions we show that the newton kantorovich hypothesis can be weakened the error bounds obtained under our semilocal convergence result are more precise than the corresponding ones given by the dominating newton kantorovich theorem

some variant of newton s method with third order convergence we present a new modification of the newton s method which produces iterative methods with order of convergence three a general error analysis providing the higher order of convergence is given and the best efficiency in term of function evaluations of two of this new methods is provided

convergence of the variants of the chebyshev halley iteration family under the holder condition of the first derivative the present paper is concerned with the convergence problem of the variants of the chebyshev halley iteration family with parameters for solving nonlinear operator equations in banach spaces under the assumption that the first derivative of the operator satisfies the holder condition of order p a convergence criterion of order 1 p for the iteration family is established an application to a nonlinear hammerstein integral equation of the second kind is provided

a modification of the super halley method under mild differentiability conditions a new two point iteration of order three is introduced to approximate a solution of a nonlinear operator equation in banach spaces under the same assumptions as for newton s method we provide a result on the existence of a unique solution for the nonlinear equation which is based on a technique consisting of a new system of recurrence relations

on interactive fuzzy numbers in this paper we will introduce a measure of interactivity between marginal distributions of a joint possibility distribution c as the expected value of the interactivity relation between the level sets of its marginal distributions

on the normalization of interval and fuzzy weights the normalization of interval and fuzzy weights is often necessary in multiple criteria decision analysis mcda under uncertainty especially in analytic hierarchy process ahp with interval or fuzzy judgements the existing normalization methods based on interval arithmetic and fuzzy arithmetic are found flawed and need to be revised this paper presents the correct normalization methods for interval and fuzzy weights and offers relevant theorems in support of them numerical examples are examined to show the correctness of the proposed normalization methods and their differences from those existing normalization methods

indices of convexity and concavity application to halley method we define an index to measure the convexity of a convex function f at each point we use this index to establish conditions on the convergence of the halley method in the complex plane and in banach spaces

on weighted possibilistic mean and variance of fuzzy numbers dubois and prade defined an interval valued expectation of fuzzy numbers viewing them as consonant random sets carlsson and fuller defined an interval valued mean value of fuzzy numbers viewing them as possibility distributions in this paper we shall introduce the notation of weighted interval valued possibilistic mean value of fuzzy numbers and investigate its relationship to the interval valued probabilistic mean we shall also introduce the notations of crisp weighted possibilistic mean value variance and covariance of fuzzy numbers which are consistent with the extension principle furthermore we show that the weighted variance of linear combination of fuzzy numbers can be computed in a similar manner as in probability theory

convergence of the family of the deformed euler halley iterations under the holder condition of the second derivative the convergence problem of the family of the deformed euler halley iterations with parameters for solving nonlinear operator equations in banach spaces is studied under the assumption that the second derivative of the operator satisfies the holder condition a convergence criterion of the order 2 p of the iteration family is established an application to a nonlinear hammerstein integral equation of the second kind is provided

on various approaches to normalization of interval and fuzzy weights abstract the paper deals with the problem of fuzzification of the procedure of normalization of weights first the existing methods for normalization of interval and fuzzy weights are reviewed second we study the problem of normalization of a fuzzy vector of weights that expresses the joint possibility distribution of initial weights we show that a correct way is to apply the extension principle proposed by zadeh since the result of such normalization is the fuzzy vector of normalized weights that expresses the true joint possibility distribution of normalized weights further we establish some properties of this approach to normalization that are important from the point of view of real applications finally since an n tuple of non interactive interval or fuzzy weights can be viewed as a fuzzy vector of weights of a special kind we investigate normalization of such kind of fuzzy vectors of weights according to the extension principle we show that from the point of view of the way of modelling uncertain normalized weights the result of this approach can be directly compared only with the result of normalization proposed by wang and elhag 2006 35 we find out that it is not sufficient to express the result of normalization only by an n tuple of normalized interval or fuzzy weights together with the constraint that the sum of the weights is equal to 1 since it can cause a false increase of uncertainty in the model this fact is illustrated by an example

on existence of p valued fuzzy sets with a given collection of cuts a representation theorem of families of subsets by poset valued fuzzy sets is presented namely necessary and sufficient conditions are given under which for a given family of subsets f of a set x and a fixed poset p there is a fuzzy set m x p such that the collection of cuts of m coincides with f

fuzzy weighted average the linear programming approach via charnes and cooper s rule abstract to improve the method of dong and wong fuzzy sets and systems 21 1987 183 199 for obtaining the fuzzy weighted average a steepest descent ascent method was proposed by liou and wang fuzzy sets and systems 49 1992 307 315 in this paper we propose to replace the steepest descent ascent method by linear programming which is a much more powerful approach for large problems the reduction of the original nonlinear programming problem into a linear one is achieved by the use of the charnes and cooper s linear transformation method the example used by dong and wong is again used to illustrate the approach

recurrence relations for the super halley method in this paper we give sufficient conditions in order to assure the convergence of the super halley method in banach spaces we use a system of recurrence relations analogous to those given in the classical newton kantorovich theorem or those given for chebyshev and halley methods by different authors 1998 elsevier science ltd all rights reserved

third order convergence theorem by using majorizing function for a modified newton method in banach space abstract following the ideas of frontini and sormani we present a modified newton method in banach space which is used to solve the nonlinear operator equation we establish the newton kantorovich convergence theorem for the modified newton method with third order convergence in banach space by using majorizing function we also get the error estimate finally two examples are provided to show the application of our theorem

novel weighted averages versus normalized sums in computing with words in this paper some properties of novel weighted averages that are related to the concepts of possibility theory are examined it is shown that novel weighted averages have certain interpretations in terms of addition of interactive interval or fuzzy constraints to do this alternative forms of novel weighted averages are provided in particular an alternative form of novel weighted averages represented by the extension principle is determined it is shown that when fuzzy set models of words are obtained by collecting data from subjects in a computing with words setting interactive addition of fuzzy sets is not a well defined method and the optimization problems related to it may have no solutions although interactive addition is recommended in the literature for solving multicriteria decision making problems and for dealing with uncertain probabilities on the other hand novel weighted averages perform a specific normalization that guarantees that they always exist

additions of interactive fuzzy numbers this paper provides an account of an approach to modeling unknown data by means of fuzzy set theory and addresses the problem of deriving the uncertainty on a sum of variables whose values lie within fuzzy intervals the first part is an extensive presentation of the theoretical background of the approach the extension principle is stated in terms of possibility of an event the concept of variable interaction is investigated at length section ii gives new results regarding the effective practical computation of additions of fuzzy numbers its originality lies in the introduction of interaction which enables to control the growth of uncertainty in calculations moreover the problem of computing mathematical expectations with fuzzy probabilities is solved the results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities more generally fuzzy arithmetic can be an important tool for sophisticated computationally tractable sensitivity analysis in systems modeling computer aided design and operations research

on newton s method under mild differentiability conditions and applications in this study we are concerned with the problem of approximating a locally unique solution of a nonlinear operator equation in a banach space we examine the convergence of the newton kantorovich method by assuming that the frechet derivative of the nonlinear operator involved is only holder continuous many results exist already in the literature to cover the stronger case when the operator is lipschitz continuous moreover we use this study to counter some claims made by galperin and waksman on related work of ours furthermore we use our results to solve a two point boundary value problem appearing in kinetic theory of gasses elasticity and other applied areas

modified halley s method free from second derivative in this paper we derive a class of new third order methods free from second derivative from halley s method per iteration the methods require two evaluations of the function and one evaluation of its first derivative analysis of efficiency in term of function evaluations of this class of methods shows that they have definite practical utility which is also demonstrated by numerical examples

concerning the terra incognita between convergence regions of two newton methods abstract the majorizing principle is used to show local and semilocal convergence of newton methods to a locally unique solution of a nonlinear operator in a banach space when the frechet derivative of the operator involved satisfies a center holder and a holder continuity condition then we investigate an unknown area terra incognita between the convergence regions of newton s method and the corresponding modified newton s method our approach compares favorably with other corresponding ones in this direction

estimates of majorizing sequences in the newton kantorovich method a further improvement let f b x0 r x y be an operator with x and y banach spaces and f be holder continuous with exponent the convergence of the sequence of newton kantorovich approximations xn xn 1 f xn 1 1f xn 1 n n is a classical tool to solve the equation f x 0 the convergence of xn is often reduced to the study of the majorizing sequence rn defined by r0 0 r1 a rn 1 rn bk rn rn 1 1 1 1 bkrn n n with a b k parameters related to f and f in the paper f cianciaruso e de pascale estimates of majorizing sequences in the newton kantorovich method submitted for publication we proved that if a bk 1 1 1 1 1 then the following estimates for rn hold rn bk 1 1 1 1 1 1 1 n n n in the present paper we give a stronger at least asymptotically estimates on rn under a weaker condition on the techniques employed in the paper are similar to the ones used in f cianciaruso e de pascale estimates of majorizing sequences in the newton kantorovich method submitted for publication finally we make a comparison with previous results

on the construction of adjunctions between a fuzzy preposet and an unstructured set in this work we focus on adjunctions also called isotone galois connections in the framework of fuzzy preordered sets hereafter fuzzy preposets specifically we present necessary and sufficient conditions so that given a mapping f a b from a fuzzy preposet a into an unstructured set b it is possible to construct a suitable fuzzy preorder relation on b for which there exists a mapping g b a such that the pair f g constitutes an adjunction

fuzzy weighted average a max min paired elimination method by the use of max min paired elimination an efficient method for obtaining the fuzzy weighted average fwa was developed this approach is much more efficient than either the approach developed by dong and wong 1 or the improved approach developed by liou and wang 2 the computational requirements for the method of dong and wong are 22n and the largest computational requirements for the improved method of liou and wang are 2 n n 1 the computational requirements for the proposed approach are proportional to the problem size n and the total computational requirements are 2 n 1 the same example used by dong and wong is used to illustrate the approach in addition a more complicated problem is also solved

correspondence note on the lack of equality between fuzzy weighted average and fuzzy convex sum the paper shows that the equality of the interval valued weighted average based on interval weights and the convex sum based on normalized interval weights presented in dubois the role of fuzzy sets in decision sciences old techniques and new directions fuzzy sets syst 184 2011 3 28 is generally false for n 3 and discusses its consequences

completion of ordered structures by cuts of fuzzy sets an overview the aim of the paper is to present a role of fuzzy sets in the theory of ordered structures main algebraic properties of cuts of fuzzy sets are given and a completion of partially ordered sets to complete lattices is described it turns out that this completion is equivalent with the famous dedekind macneille completion but the algorithm presented here is much simpler

an acceleration of newton s method super halley method from a study of the convexity we give an acceleration for newton s method and obtain a new third order method then we use this method for solving non linear equations in banach spaces establishing conditions on convergence existence and uniqueness of solution as well as error estimates

the convergence ball of the secant method under holder continuous divided differences under the hypothesis that nonlinear operators have holder continuous divided differences of order one an estimate of the radius of the convergence ball of the secant method is obtained error estimate is also established

newton kantorovich theorem for a family of modified halley s method under holder continuity conditions in banach space abstract in this study under p holder continuity conditions a family of new modified halley s method with 2 p order is proposed in banach space which is used to solve the nonlinear operator equations the newton kantorovich convergence theorem for the family of new modified halley s method is established under holder continuity conditions the error estimate also is gotten finally two examples are provided to show the application of our theorem

cuts of poset valued functions in the framework of residuated maps abstract we analyze cuts of poset valued functions relating them to residuated mappings dealing with the lattice valued case we prove that a function x l induces a residuated map f l p x whose values are the cuts of and we describe the corresponding residual conversely it turns out that every residuated map f from l to the power set of x determines a lattice valued function x l whose cuts coincide with the values of f for general poset valued functions we give conditions under which the map sending an element of a poset to the corresponding cut is quasi residuated and then conditions under which it is also residuated we prove that without additional conditions the map analogue to the residual is a partial function hence we get particular weakly residuated maps which on the power set of the domain generate centralized systems instead of closures we show that the main properties of residuated maps are preserved in this generalized case we apply these results to the canonical representation of poset valued and lattice valued functions using the corresponding closures and centralized systems

a modified newton method with cubic convergence the multivariate case recently a modification of the newton method for finding a zero of a univariate function with local cubic convergence has been introduced here we extend this modification to the multi dimensional case i e we introduce a modified newton method for vector functions that converges locally cubically without the need to compute higher derivatives the case of multiple roots is not treated per iteration the method requires one evaluation of the function vector and solving two linear systems with the jacobian as coefficient matrix where the jacobian has to be evaluated twice since the additional computational effort is nearly that of an additional newton step the proposed method is useful especially in difficult cases where the number of iterations can be reduced by a factor of two in comparison to the newton method this much better convergence is indeed possible as shown by a numerical example also the modified newton method can be advantageous in cases where the evaluation of the function is more expensive than solving a linear system with the jacobian as coefficient matrix an example for this is given where numerical quadrature is involved finally we discuss shortly possible extensions of the method to make it globally convergent