on the local convergence of a deformed newton s method under argyros type condition abstract for the iteration which was independently proposed by king r f king tangent method for nonlinear equations numer math 18 1972 298 304 and werner w werner uber ein verfarhren der ordnung 1 2 zur nullstellenbestimmung numer math 32 1979 333 342 for solving a nonlinear operator equation in banach space we established a local convergence theorem under the condition which was introduced recently by argyros i k argyros a unifying local semilocal convergence analysis and application for two point newton like methods in banach space j math anal appl 298 2004 374 397

nonlocal theory solution of two collinear cracks in the functionally graded materials in this paper the interaction of two collinear cracks in functionally graded materials subjected to a uniform anti plane shear loading is investigated by means of nonlocal theory the traditional concepts of the nonlocal theory are extended to solve the fracture problem of functionally graded materials to make the analysis tractable it is assumed that the shear modulus varies exponentially with the coordinate vertical to the crack by use of the fourier transform the problem can be solved with the help of a pair of triple integral equations in which the unknown variable is the displacement on the crack surfaces to solve the triple integral equations the displacement on the crack surfaces is expanded in a series of jacobi polynomials unlike the classical elasticity solutions it is found that no stress singularity is present near the crack tips the nonlocal elastic solutions yield a finite hoop stress at the crack tip thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials the magnitude of the finite stress field depends on the crack length the distance between two cracks the parameter describing the functionally graded materials and the lattice parameter of the materials

convergence ball and error analysis of a family of iterative methods with cubic convergence a family of iterative methods with cubic convergence to find approximate solutions of nonlinear equations is proposed the radius of the convergence ball of any iterative method of the family is estimated and quadratic convergence is obtained under the assumption that the first order frechet derivative of the operator involved is lipschitz continuous if the second order frechet derivative of the operator involved is assumed also to be lipschitz continuous cubic convergence of any iterative method of the family is obtained finally some examples are provided to show applications of our theorems

relaxing convergence conditions for newton s method the classical kantorovich theorem on newton s method assumes that the first derivative of the operator involved satisfies a lipschitz condition 0 f x f y l x y in this paper we weaken this condition assuming that 0 f x f x0 x x0 for a given point x0

local convergence analysis of newton s method for solving strongly regular generalized equations abstract in this paper we consider newton s method for solving a generalized equation of the form f x f x 0 where f y is continuously differentiable x and y are banach spaces x is open and f x y has a nonempty closed graph we show that under strong regularity of the equation the method is locally convergent to a solution with superlinear quadratic rate our analysis which is based on general majorant condition enables us to obtain a convergence result under the lipschitz smale s and nesterov nemirovskii s self concordant conditions

convergence analysis of the secant type methods in this paper under the hypothesis that derivative satisfies some kinds of weak lipschitz condition the radius estimates of the convergence balls of secant type methods for operator equations are established in banach space some applications and numerical experiments are also given

free vibration analysis of embedded single layered nanoplates and graphene sheets by using the multiple time scale method an asymptotic approach using the eringen nonlocal elasticity theory and multiple time scale method is developed for the three dimensional 3d free vibration analysis of simply supported single layered nanoplates and graphene sheets gss embedded in an elastic medium in the formulation the small length scale effect is first introduced to the nonlocal constitutive equations by using a nonlocal parameter then the mathematical processes of nondimensionalization asymptotic expansion and successive integration are performed and finally recurrent sets of motion equations for various order problems are obtained the interactions between the nanoplates or gss and their surrounding medium are modeled as a two parameter pasternak foundation nonlocal classical plate theory cpt is derived as a first order approximation of the 3d nonlocal elasticity theory and the motion equations for higher order problems retain the same differential operators as those of nonlocal cpt although with different nonhomogeneous terms some 3d nonlocal elasticity solutions of the natural frequency parameters of nanoplates or gss with and without being embedded in the elastic medium and their corresponding through thickness distributions of modal field variables are given to demonstrate the performance of the 3d asymptotic nonlocal elasticity theory

improved local convergence of newton s method under weak majorant condition we provide a local convergence analysis for newton s method under a weak majorant condition in a banach space setting our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than before 14 special cases and numerical examples are also provided in this study

convergence and uniqueness properties of gauss newton s method abstract the generalized radius and center lipschitz conditions with l average are introduced to investigate the convergence of gauss newton s method for finding the nonlinear least squares solution of nonlinear equations the radii of the convergence ball of gauss newton s method and the uniqueness ball of the solution are estimated some applications are given

on convergence of a new secant like method for solving nonlinear equations in this paper we prove that the order of a new secant like method presented recently with the so called order of 2 618 is only 2 414 some mistakes in the derivation leading to such a conclusion are pointed out meanwhile under the assumption that the second derivative of the involved function is bounded the convergence radius of the secant like method is given and error estimates matching its convergence order are also provided by using a generalized fibonacci sequence

mathematical modeling of vibration problem of nano sized annular sector plates using the nonlocal continuum theory via eight node discrete singular convolution transformation abstract in the present study free vibration analysis of nano sized annular sector plate is analyzed using the nonlocal continuum theory the method of discrete singular convolution dsc is used for numerical computations firstly equation of motion of thin plates is formulated via nonlocal elasticity then irregular physical domain is transformed into a rectangular domain by using geometric coordinate transformation the dsc procedures are then applied to discretization of the transformed set of governing equations and related boundary conditions the effects of nonlocal parameter mode numbers sector angle and radius ratio on the vibration frequencies are investigated in detail it is seen that the size effects are significant in vibration analysis of nano scaled annular sector plates and need to be included in the mechanical model

local convergence of the gauss newton method for injective overdetermined systems of equations under a majorant condition a local convergence analysis of the gauss newton method for solving injective overdetermined systems of nonlinear equations under a majorant condition is provided the convergence as well as results on its rate are established without a convexity hypothesis on the derivative of the majorant function the optimal convergence radius the biggest range for uniqueness of the solution along with some other special cases are also obtained

a newton conditional gradient method for constrained nonlinear systems in this paper we consider the problem of solving constrained systems of nonlinear equations we propose an algorithm based on a combination of newton and conditional gradient methods and establish its local convergence analysis our analysis is set up by using a majorant condition technique allowing us to prove in a unified way convergence results for two large families of nonlinear functions the first one includes functions whose derivative satisfies a holder like condition and the second one consists of a substantial subclass of analytic functions some preliminary numerical experiments are reported

enlarging the convergence ball of the method of parabola for finding zero of derivatives we present a new technique for enlarging the convergence ball of the method of parabola in order to approximate a zero of derivatives this approach also leads to more precise error estimates on the distances involved than in earlier studies such as hua 1974 ren and wu 2009 and wand 1975 these advantages are obtained under the same computational cost on the lipschitz constants involved as in the earlier studies numerical examples are also given to show the advantages over the earlier work

convergence of gauss newton s method and uniqueness of the solution abstract in this paper we study the convergence of gauss newton s method for nonlinear least squares problems under the hypothesis that derivative satisfies some kinds of weak lipschitz condition we obtain the exact estimates of the radii of convergence ball of gauss newton s method and the uniqueness ball of the solution new results can be used to determinate approximation zero of gauss newton s method

local convergence of newton s method under majorant condition a local convergence analysis of newton s method for solving nonlinear equations under a majorant condition is presented in this paper without assuming convexity of the derivative of the majorant function which relaxes the lipschitz condition on the operator under consideration convergence the biggest range for uniqueness of the solution the optimal convergence radius and results on the convergence rate are established besides two special cases of the general theory are presented as applications

mathematical modelling of surface vibration with volume constraint and its analysis abstract a hyperbolic problem with volume constraint is investigated by means of a minimizing method called the discrete morse flow a weak solution is introduced in a constructive way and results of numerical computations are shown

newton s method for overdetermined systems of equations complexity theoretic aspects of continuation methods for the solution of square or underdetermined systems of polynomial equations have been studied by various authors in this paper we consider overdetermined systems where there are more equations than unknowns we study newton s method for such a system

an efficient computational approach for size dependent analysis of functionally graded nanoplates abstract in this paper an efficient computational approach based on refined plate theory rpt including the thickness stretching effect namely quasi 3d theory in conjunction with isogeometric formulation iga is proposed for the size dependent bending free vibration and buckling analysis of functionally graded nanoplate structures the present novel quasi 3d theory not only possesses 4 variables as refined plate theory but also accounts for both shear deformation and stretching effect without any requirement of shear correction factors scfs the size dependent effect is taken into account by nonlocal elasticity theory isogeometric analysis shows a great advantage in dealing with the high continuity and high order derivative requirements of the displacement fields used in quasi 3d and nonlocal theory the reliability and accuracy of the present method are ascertained by comparing the obtained results with other published ones numerical examples are also performed to show the significance of nonlocal effect material distribution profile aspect ratios and boundary conditions on the behaviour of fgm nanoplates

convergence radius of the modified newton method for multiple zeros under holder continuous derivative abstract in recent years a lot of iterative methods for finding multiple zeros of nonlinear equations have been presented and analyzed however almost all these studies give no information for the convergence radius of the corresponding method in this paper we give an estimate of the convergence radius of the well known modified newton s method for multiple zeros when the involved function satisfies a holder and center holder continuity condition

convergence criterion of newton s method for singular systems with constant rank derivatives abstract the present paper is concerned with the convergence problem of newton s method to solve singular systems of equations with constant rank derivatives under the hypothesis that the derivatives satisfy a type of weak lipschitz condition a convergence criterion based on the information around the initial point is established for newton s method for singular systems of equations with constant rank derivatives applications to two special and important cases the classical lipschitz condition and the smale s assumption are provided the latter in particular extends and improves the corresponding result due to dedieu and kim in j p dedieu m kim newton s method for analytic systems of equations with constant rank derivatives j complexity 18 2002 187 209

a variational scheme for hyperbolic obstacle problems abstract we consider an obstacle problem for possibly non local wave equations and we prove existence of weak solutions through a convex minimization approach based on a time discrete approximation scheme we provide the corresponding numerical implementation and raise some open questions

a variational approach to a constrained hyperbolic free boundary problem abstract this paper concerns the construction of a weak solution to a constrained hyperbolic equation in one space dimension the problem includes a nonlocal term corresponding to a volume constraint and the presence of a free boundary we approach our problem by means of a variational method known as the discrete morse flow

the convergence ball of newton s method and the uniqueness ball of equations under holder type continuous derivatives abstract under the hypotheses that nonlinear operators have k p holder type continuous derivatives exact estimates of the radius of the convergence ball of newton s method and of the uniqueness ball of solution of equations are obtained

nonlinear thermo elastic bending behavior of graphene sheets embedded in an elastic medium based on nonlocal elasticity theory this paper investigates the large deflection behavior of orthotropic single layered graphene sheet slgs embedded in a winkler pasternak elastic medium under a uniform transverse load in thermal environments using the nonlocal differential constitutive relations of eringen the slgs is modeled as a nonlocal orthotropic plate using the principle of virtual work the coupled nonlinear equilibrium equations are obtained based on first order shear deformation theory fsdt and the von karman geometrical model the differential quadrature dq discretized form of the governing equations with clamped and simply supported boundary conditions is derived the newton raphson iterative scheme is used to solve the resulting system of nonlinear algebraic equations effects of small scale parameter thermal environment width to length elasticity ratio aspect ratio thickness elastic foundation load value and boundary conditions are considered in detail the results show that unlike the simply supported boundary conditions increase of small scale parameter plays a decreasing role in effect of thermal environment on the deflections of nanoplates with clamped edges it is also observed that increasing the plate thickness thermal load effects decline noticeably and depending on the small scale value the thermal loads do not have any significant effect on the results beyond a specified thickness

convergence ball and error analysis of muller s method abstract under the hypotheses that the second order and third order derivative of function f are bounded an estimate of the radius of the convergence ball of muller s method is obtained an error analysis is given which matches the convergence order of the method

on convergence of the modified newton s method under holder continuous frechet derivative in this paper the upper and lower estimates of the radius of the convergence ball of the modified newton s method in banach space are provided under the hypotheses that the frechet derivative of the nonlinear operator are center holder continuous for the initial point and the solution of the operator the error analysis is given which matches the convergence order of the modified newton s method the uniqueness ball of solution is also established numerical examples for validating the results are also provided including a two point boundary value problem

atomistic viewpoint of the applicability of microcontinuum theories abstract microcontinuum field theories including micromorphic theory microstructure theory micropolar theory cosserat theory nonlocal theory and couple stress theory are the extensions of the classical field theories for the applications in microscopic space and time scales they have been expected to overlap atomic model at microscale and encompass classical continuum mechanics at macroscale this work provides an atomic viewpoint to examine the physical foundations of those well established microcontinuum theories and to justify their applicability through lattice dynamics and molecular dynamics

convergence ball of a modified secant method for finding zero of derivatives under the hypotheses that second order derivatives of function f is lipschitz continuous estimate of the radius of the convergence ball of a modified secant method to find a zero of derivatives of function f is obtained error analysis is given which matches with the convergence order of the modified secant method

comment on nonlocal third order shear deformation plate theory with application to bending and vibration of plates by r aghababaei and j n reddy journal of sound and vibration 326 2009 277 289 abstract this paper aims to prove the inaccuracy of the navier solution presented by aghababaei and reddy 1 for the bending analysis of nanoplates based on the nonlocal theory of eringen according to the derived relations for bending of the nonlocal plate model the main cause of this inaccuracy is attributed to an incorrect approximation of the navier solution for a uniform transverse load of course this problem does not inherently occur for the navier solution in cases such as free vibration or the buckling of a nonlocal plate model in which the amount of transverse load is zero in order to obtain further verification the results reported based on the navier solution by aghababaei and reddy 2009 1 for the bending analysis of a nanoplate are compared with those computed by the differential quadrature dq and finite difference fd methods as shown the results obtained by both the fd and dq methods are consistently alike and unlike the solutions reported by aghababaei and reddy 2009 1 they are independent from small scale effect

large deflection of thermo mechanical loaded bilayer orthotropic graphene sheet in on polymer matrix based on nonlocal elasticity theory abstract in this paper large deflection behavior of a bilayer graphene sheet blgs is investigated under uniformly transverse load in thermal environment with various boundary conditions using the principle of virtual work the non linear formulations are derived based on first order shear deformation plate theory fsdt and large deflection von karman equations that includes the thermal effects differential quadrature dq method and the newton raphson iterative technique is employed to solve the system of nonlinear algebraic equations to study the effect of elastic foundation on the results two types of embedded in polymer matrix eipm and resting on polymer matrix ropm are considered besides the influences of temperature rise material properties boundary conditions small scale effect thickness to length ratio aspect ratio and polymer matrix properties are studied in detail

variational approach to evolutionary free boundary problems abstract the analysis of evolutionary nonlocal problems represented by volume constrained motion is discussed in the physical sense these problems can describe the motion of droplets on a solid plane the variational approach called the discrete morse flow combined with a fluid dynamics model appears to be a suitable tool

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

block hybrid method for the numerical solution of fourth order boundary value problems abstract a linear multistep block hybrid method with four off grid points is presented for the direct approximation of the solution of fourth order boundary value problems multiple finite difference formulas are derived and grouped into a unique block to form a numerical integrator to solve directly the fourth order problem without the need to reduce it previously to a first order system the convergence of the proposed method is discussed the superiority of this method over existing methods is established numerically considering different problems that have appeared in the literature

non local elastic plate theories a non local plate model is proposed based on eringen s theory of non local continuum mechanics the basic equations for the non local kirchhoff and the mindlin plate theories are derived these non local plate theories allow for the small scale effect which becomes significant when dealing with micro nanoscale plate like structures as illustrative examples the bending and free vibration problems of a rectangular plate with simply supported edges are solved and the exact non local solutions are discussed in relation to their corresponding local solutions

a non local asymptotic theory for thin elastic plates the three dimensional dynamic non local elasticity equations for a thin plate are subject to asymptotic analysis assuming the plate thickness to be much greater than a typical microscale size the integral constitutive relations incorporating the variation of an exponential non local kernel across the thickness are adopted long wave low frequency approximations are derived for both bending and extensional motions boundary layers specific for non local behaviour are revealed near the plate faces it is established that the effect of the boundary layers leads to the first order corrections to the bending and extensional stiffness in the classical two dimensional plate equations

nonlocal third order shear deformation plate theory with application to bending and vibration of plates abstract the third order shear deformation plate theory of reddy a simple higher order theory for laminated composite plates j appl mech 51 1984 745 752 is reformulated using the nonlocal linear elasticity theory of eringen this theory has ability to capture the both small scale effects and quadratic variation of shear strain and consequently shear stress through the plate thickness analytical solutions of bending and free vibration of a simply supported rectangular plate are presented using this theory to illustrate the effect of nonlocal theory on deflection and natural frequency of the plates finally the relations between nonlocal third order first order and classical theories are discussed by numerical results

reply to the comments of m e golmakani and j rezatalab comment on nonlocal third order shear deformation plate theory with application to bending and vibration of plates by r aghababaei and j n reddy journal of sound and vibration 326 2009 277 289 journal of sound vibration 333 2014 3831 3835 abstract golmakani and rezatalab 1 suggested in their paper that the deflection of a simply supported nonlocal elastic plate under uniform load is not affected by the small length scale terms they based their proof on the use of navier s method using a sinusoidal based deflection solution this insensitivity of the deflection solution of a simply supported nonlocal elastic plate with respect to the small length terms of eringen s model is not correct as already detailed in the literature for example see 2 for beam problems in fact the deflection of the nonlocal plate in the eringen sense is larger than the one of the local case as shown in many papers available in the literature we prove in this reply to the authors that the navier s method has to be correctly applied for highlighting the specific sensitivity phenomenon of the deflection solution as compared to exact analytical solution the problem is exactly the same as for the bending of a beam

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

convergence ball of a modified secant method with convergence order 1 839 abstract a local convergence theorem on a modified secant method with convergence order 1 839 for solving nonlinear equations is established under the hypotheses that the second order and third order derivative of the function involved are bounded an estimate of the radius of the convergence ball of this method is obtained moreover an error estimate is provided which matches the convergence order of the method

local convergence analysis of inexact gauss newton like methods under majorant condition in this paper we present a local convergence analysis of inexact gauss newton like methods for solving nonlinear least squares problems under the hypothesis that the derivative of the function associated with the least squares problem satisfies a majorant condition we obtain that the method is well defined and converges our analysis provides a clear relationship between the majorant function and the function associated with the least squares problem it also allows us to obtain an estimate of convergence ball for inexact gauss newton like methods and some important special cases