some variants of king s fourth order family of methods for nonlinear equations abstract in this paper we present fourth order families of new variants of king s fourth order family of methods for solving nonlinear equations per iteration the obtained methods require two evaluations of the given function and one of its first derivative the classical traub ostrowski method is obtained as a special variant of king s method several numerical examples are given to illustrate the efficiency and performance of the presented methods

a one parameter fourth order family of iterative methods for nonlinear equations abstract in this paper we present a new one parameter fourth order family of iterative methods for solving nonlinear equations the new family requires two evaluations of the given function and one of its first derivative per iteration the well known traub ostrowski s fourth order method is shown to be part of the family several numerical examples are given to illustrate the efficiency and performance of the presented methods

a family of fourth order steffensen type methods with the applications on solving nonlinear odes abstract in this paper a family of fourth order steffensen type two step methods is constructed to make progress in including ren wu bi s methods h ren q wu w bi a class of two step steffensen type methods with fourth order convergence appl math comput 209 2009 206 210 and liu zheng zhao s method z liu q zheng p zhao a variant of steffensens method of fourth order convergence and its applications appl math comput 216 2010 1978 1983 as its special cases its error equation and asymptotic convergence constant are deduced the family provides the opportunity to obtain derivative free iterative methods varying in different rates and ranges of convergence in the numerical examples the family is not only compared with the related methods for solving nonlinear equations but also applied in the solution of bvps of nonlinear odes by the finite difference method and the multiple shooting method

on the dynamics of a triparametric family of optimal fourth order multiple zero finders with a weight function of the principal mth root of a function to function ratio abstract under the assumption of known root multiplicity m n a triparametric family of two point optimal quartic order methods locating multiple zeros are investigated in this paper by introducing a weight function dependent on a function to function ratio special cases of weight functions with selected free parameters are considered and studied through various test equations and numerical experiments to support the theory developed in this paper in addition we explore the relevant dynamics of proposed methods via mobius conjugacy map when applied to a prototype polynomial z a m z b m the results of such dynamics are visually illustrated through a variety of parameter spaces as well as dynamical planes

coexistence theorem of steady states for nonlinear self cross diffusion systems with competitive dynamics in this paper we discuss the existence of positive solutions to certain nonlinear elliptic systems representing competitive interaction with self cross diffusions between two species the method employed is the fixed point index theory in a positive cone sufficient conditions for the existence of positive solutions are provided

a family of second derivative free fourth order continuation method for solving nonlinear equations in this paper we present a parameter based iterative method free from the second derivative for solving nonlinear equations of type f x 0 ezquerro and hernandez 1999 discussed the convergence analysis of a uniparametric family of an iterative method in banach space based on this idea we propose a uniparametric family of second derivative free iterative method in r we discuss the convergence analysis and observe that it possesses a fourth order convergence for solving nonlinear equations in r several numerical examples are worked out with our proposed methods for parameter 1 2 and the existing fourth order iterative method proposed in king 1973 chun 2007 soleymani et al 2012 and maheshwari 2009 finally from the comparison we observe that our method is more efficient than existing methods finally we compare basins of attraction of our methods with the second derivative free fourth order iterative method proposed in king 1973 chun 2007 soleymani et al 2012 and maheshwari 2009 observe that the proposed scheme is more efficient

factorization in analytic operator algebras abstract a constructive and unified approach is used to obtain the upper lower factorization of positive operators and the outer function factorization of positive operator valued functions on the circle for a projection nest e it is shown that every positive operator admits a canonical factorization c a a with a an outer operator if and only if e is well ordered with new methods we generalize the inner outer factorizations obtained by arveson for nests of order type z and the riesz factorization due to shields for trace class triangular operators weak factorization is obtained in noncommutative h1 spaces associated with general nest subalgebras of a semifinite factor characterizations of a nehari type are given for the associated hankel forms and hankel operators

coexistence in the three species predator prey model with diffusion the three species food chain model is discussed in which the third species is the predator of the second one and the second species is the predator of the first one we consider coexistence states of the associated weakly coupled elliptic problem under the homogeneous neumann boundary conditions it is shown that there are no non constant solutions if the diffusion rates of species are strong or if the intrinsic growth rate of a prey is slow and the intrinsic drop rates of predators are fast it is also shown that the weakly coupled parabolic system has a unique global solution for any non negative initial data

a biparametric extension of king s fourth order methods and their dynamics a class of two point quartic order simple zero finders and their dynamics are investigated in this paper by extending king s fourth order family of methods with the introduction of an error corrector having a weight function dependent on a function to function ratio higher order convergence is obtained through a variety of test equations numerical experiments strongly support the theory developed in this paper in addition relevant dynamics of the proposed methods is successfully explored for a prototype quadratic polynomial as well as parameter spaces and dynamical planes

dynamics of prey predator n species models with density dependent diffusion this paper is concerned with the n species predator prey models in a bounded domain in two cases one of which involves n 1 cooperative preys and the other involves n 1 competitive preys in these models the diffusion coefficients may be density dependent and the reaction functions include ratio dependent terms some simple conditions are obtained to ensure the dynamical behaviors of the time dependent solutions in relation to some positive solutions or quasi solutions of the steady state problems which include the existence of these solutions this dynamical behaviors lead to the coexistence and global attractors of the predator prey systems some numerical simulations are given in the end

fourth order iterative methods free from second derivative in this paper we present a family of new iterative methods with order of convergence four for solving non linear equations per iteration these methods require one evaluation of the function and two of its first derivative analysis of efficiency in term of function evaluations shows that this family of methods has great superiority which is also demonstrated by numerical examples

dynamics of food chain models with density dependent diffusion and ratio dependent reaction function abstract this paper is concerned with a 3 species and a 2 species food chain reaction diffusion systems in a bounded domain where the diffusion coefficients may be density dependent and the reaction functions are ratio dependent these equations are quasilinear where the diffusion coefficients may be degenerate on the boundary of the domain three basic types of dirichlet neumann and robin boundary conditions are considered and in each case some very simple conditions are obtained to ensure the dynamical behavior of the time dependent solution in relation to some positive solutions or quasi solutions of the steady state problem including the existence of these solutions this dynamical behavior leads to the coexistence and global attractor of the food chain systems in the case of neumann boundary condition sufficient conditions are given to ensure that the steady state problem has a unique positive constant solution which is a global attractor of the time dependent system

a diffusive one prey and two competing predator system with a ratio dependent functional response i long time behavior and stability of equilibria abstract we study a ratio dependent reaction diffusion system incorporating one prey and two competing predator species under homogeneous neumann boundary conditions in this paper we examine the global attractor and persistence of the system which characterize the long time behavior of the time dependent solution and the stability of all non negative equilibria of the system we classify the relations between two competing predators into four categories via domination of one predator over another and weak strong competition states these competition states will be the criteria which influence the outcomes of the system the results include total extinction competitive exclusion behavior of a predator driving the extinction of another predator and its prey behavior of a predator saving its competing predator and prey on the verge of extinction

positive solutions for ratio dependent predator prey interaction systems abstract in this paper we study the dynamics of predator prey interaction systems between two species with ratio dependent functional responses first we provide sufficient and necessary conditions for positive steady state solutions and then we investigate the relationships between positive equilibria and positive solutions of the system over a large domain furthermore we deal with the uniqueness and the stability of positive steady states solutions with some assumptions in addition we discuss the extinction and the persistence results of time dependent positive solutions to the system

derivative free two point methods with and without memory for solving nonlinear equations abstract two families of derivative free two point iterative methods for solving nonlinear equations are constructed these methods use a suitable parametric function and an arbitrary real parameter it is proved that the first family has the convergence order four requiring only three function evaluations per iteration in this way it is demonstrated that the proposed family without memory supports the kung traub hypothesis 1974 on the upper bound 2n of the order of multipoint methods based on n 1 function evaluations further acceleration of the convergence rate is attained by varying a free parameter from step to step using information available from the previous step this approach leads to a family of two step self accelerating methods with memory whose order of convergence is at least 2 5 4 236 and even 2 6 4 449 in special cases the increase of convergence order is attained without any additional calculations so that the family of methods with memory possesses a very high computational efficiency numerical examples are included to demonstrate exceptional convergence speed of the proposed methods using only few function evaluations

a variant of super halley method with accelerated fourth order convergence in this paper we present a new variant of super halley method for solving non linear equations analysis of convergence shows that the method has the order of convergence four per iteration the new method requires the same evaluations of the function the first derivative and second derivative as its classical predecessor super halley method numerical results show that the method has definite practical utility

on the indices of fixed points of mappings in cones and applications abstract assume that k is a cone in a banach space and a k k is completely continuous we obtain a formula for the index in k of a fixed point of a under the assumption that a linearization exists and satisfies an invertibility condition we then use this formula to generalize some results of amann on the number of fixed points of a to the case where k has empty interior

a constructive method for solving the equation xa b in rn a generalization of division in rn abstract in this paper we present a new approach of defining division of vectors in r n for arbitrary dimension n our approach is based on constructing a class of solutions of the equation x a b for any two known vectors a b r n for arbitrary dimension n by using some basic properties of octonions the defined procedure differs depending on the dimension of the vectors being analyzed the cases 1 n 7 and n 7 we present an algorithm for computing divisions of vectors in multidimension space and some numerical examples are given to confirm the presented theoretical results lastly our algorithm is applied to extend several known methods of scalar case to multidimensional cases and some numerical tests are made to compare the performance of the nonlinear system solvers

certain improvements of chebyshev halley methods with accelerated fourth order convergence abstract in this paper we present a new one parameter family of methods which improves the classical chebyshev halley methods with cubic convergence the convergence analysis shows that each method of the family is quartically convergent and per iteration it requires one evaluation of the given function and two of its first derivative the well known jarratt fourth order method is shown to be part of the family several numerical examples are given to illustrate the efficiency and performance of the presented methods

construction of fourth order optimal families of iterative methods and their dynamics in this paper we propose a general class of fourth order optimal multi point methods without memory for obtaining simple roots this class requires only three functional evaluations viz two evaluations of function f xn f yn and one of its first order derivative f xn per iteration further we show that the well known ostrowski s method and king s family of fourth order procedures are special cases of our proposed schemes one of the new particular subclasses is a biparametric family of iterative methods by using complex dynamics tools its stability is analyzed showing stable members of the family further on one of the parameters is fixed and the stability of the resulting class is studied on the other hand the accuracy and validity of new schemes is tested by a number of numerical examples by comparing them with recent and classical optimal fourth order methods available in the literature it is found that they are very useful in high precision computations

construction of newton like iteration methods for solving nonlinear equations in this paper we present a simple and yet powerful and easily applicable scheme in constructing the newton like iteration formulae for the computation of the solutions of nonlinear equations the new scheme is based on the homotopy analysis method applied to equations in general form equivalent to the nonlinear equations it provides a tool to develop new newton like iteration methods or to improve the existing iteration methods which contains the well known newton iteration formula in logic those all improve the newton method the orders of convergence and corresponding error equations of the obtained iteration formulae are derived analytically or with the help of maple some numerical tests are given to support the theory developed in this paper

coexistence stability and limiting behavior in a one predator two prey model abstract this paper considers a reaction diffusion system that models the situation in which a predator feeds on two prey species the existence of a positive steady state solution as well as the existence and stability of various semitrivial steady state solutions is obtained in terms of the natural growth rates of the three species the present paper also gives some explicit information on the limiting behavior of the time dependent solution to these steady states

stability in a diffusive food chain model with michaelis menten functional response abstract this paper deals with the behavior of positive solutions to a reaction diffusion system with homogeneous neumann boundary conditions describing a three species food chain a sufficient condition for the local asymptotical stability is given by linearization and also a sufficient condition for the global asymptotical stability is given by a lyapunov function our result shows that the equilibrium solution is globally asymptotically stable if the net birth rate of the first species is big enough and the net death rate of the third species is neither too big nor too small

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

qualitative analysis of a ratio dependent predator prey system with diffusion ratio dependent predator prey models are favoured by many animal ecologists recently as they better describe predator prey interactions where predation involves a searching process when densities of prey and predator are spatially homogeneous the so called michaelis menten ratio dependent predator prey system which is an ordinary differential system has been studied by many authors the present paper deals with the case where densities of prey and predator are spatially inhomogeneous in a bounded domain subject to the homogeneous neumann boundary condition its main purpose is to study qualitative properties of solutions to this reaction diffusion partial differential system in particular we will show that even though the unique positive constant steady state is globally asymptotically stable for the ordinary differential equation dynamics non constant positive steady states exist for the partial differential equation model this demonstrates that stationary patterns arise as a result of diffusion

a variant of steffensen king s type family with accelerated sixth order convergence and high efficiency index first it is attempted to derive an optimal derivative free steffensen king s type family without memory for computing a simple zero of a nonlinear function with efficiency index 4 1 3 1 587 next since our without memory family includes a parameter in which it is still possible to increase the convergence order without any new function evaluations therefore we extract a new method with memory so that the convergence order rises to six without any new function evaluation and therefore reaches efficiency index 6 1 3 1 817 consequently derivative free and high efficiency index would be the substantial contributions of this work as opposed to the classical steffensen s and king s methods finally we compare some of the convergence planes with different weight functions in order to show which are the best ones

the jarratt method in banach space setting abstract in this note we extend the jarratt method of order four into banach spaces we also establish a kantorovich type convergence theorem see the references and give an explicit expression for the error bound of the method jarratt 1966 1969

coexistence of a one prey two predators model with ratio dependent functional responses abstract we examine the positive coexistence of one prey and two competing predators in an interacting system with ratio dependent functional responses under a hostile environment furthermore we examine the extinction conditions for all three interacting species and for one or two of the three species in addition we present the biological interpretation and simulations based on the result the methods employed are a comparison argument for the elliptic problem and the fixed point theory applied to a positive cone on a banach space

a diffusive one prey and two competing predator system with a ratio dependent functional response ii stationary pattern formation abstract this paper is a continuation of ko and ahn 2013 1 which investigates the stability at all non negative equilibria and long time behavior of solutions for a ratio dependent reaction diffusion system incorporating one prey and two competing predator species under homogeneous neumann boundary conditions we examine the nonexistence and the appearance of stationary patterns in the time independent system in achieving these we deal with the system only when the competition state between two competing predators is weak strong in particular the results explain the phenomenon of a stationary pattern being induced by the introduction of a new predator species in the one prey and one predator system with no stationary pattern

fourth order convergent iterative method for nonlinear equation in this paper we suggest and analyze a new iterative method for finding approximate solution of the nonlinear equation f x 0 it is shown that proposed method has fourth order convergence several numerical examples are given to illustrate that the method developed in this paper give better results than the other methods including newton method

analysis of ratio dependent food chain model abstract in this paper a food chain model with ratio dependent functional response is studied under homogeneous neumann boundary conditions the large time behavior of all non negative equilibria in the time dependent system is investigated i e conditions for the stability at equilibria are found moreover non constant positive steady states are studied in terms of diffusion effects namely turing patterns arising from diffusion driven instability turing instability are demonstrated the employed methods are comparison principle for parabolic problems and leray schauder theorem

operator theoretic approach to the optimal two disk problem the nonstandard two disk problem plays a fundamental role in robust feedback optimization here it is shown via banach space duality theory that its solutions satisfy an extremal identity and may be viewed as a dual extremal kernel of a particular l sup 1 optimization problem a novel operator theoretic framework to characterize explicitly its solutions is developed in particular the two disk optimization is shown to be equal to the induced norm of a specific operator defined on a projective tensor product space involving a non hilbert version of a vector valued h sup 2 space moreover this operator is shown to be a combination of multiplication and toeplitz operators under certain conditions existence of maximal vectors is established leading to an explicit formula for the optimal controller an infinite matrix representation with respect to a canonical basis is derived together with an algorithm to compute it the norm of the relevant operator is approximated by special finite dimensional optimizations whose solutions lead to solving semi definite programming problems involving the computation of a matrix projective tensor norm

time varying optimal disturbance minimization in presence of plant uncertainty the optimal robust disturbance rejection problem plays an important role in feedback control theory here its time varying version is solved explicitly in terms of duality and operator theory in particular the optimum is shown to satisfy a time varying allpass property moreover optimal performance is given in terms of the norm of a bilinear form the latter depends on a lower triangular projection and a multiplication operator defined on special versions of spaces of compact operators

coexistence theorems of steady states for predator prey interacting systems on donne des conditions necessaires et suffisantes pour l existence des solutions positives des etats stationnaires pour des systemes predateur proie sous des conditions aux limites de dirichlet sur r n

coexistence and stability of a competition diffusion system in population dynamics abstract the coexistence and stability of the population densities of two competing species in a bounded habitat are investigated in the present paper where the effect of dispersion transportation is taken into consideration the mathematical problem involves a coupled system of lotka volterra type reaction diffusion equations together with some initial and boundary conditions including the dirichlet neumann and third type necessary and sufficient conditions for the coexistence and competitive exclusion are established and the effect of diffusion is explicitly given for the stability problem general criteria for the stability and instability of a steady state solution are established and then applied to various situations depending on the relative magnitude among the physical parameters also given are necessary and sufficient conditions for the existence of multiple steady state solutions and the stability or instability of each of these solutions special attention is given to the neumann boundary condition with respect to which some threshold results for the coexistence and stability or instability of the four uniform steady states are characterized it is shown in this situation that only one of the four constant steady states is asymptotically stable while the remaining three are unstable the stability or instability of these states depends solely on the relative magnitude among the various rate constants and is independent of the diffusion coefficients