on the steepest descent approximation method for the zeros of generalized accretive operators abstract in this paper we present certain characteristic conditions for the convergence of the generalized steepest descent approximation process to a zero of a generalized strongly accretive operator defined on a uniformly smooth banach space our study is based on an important result of reich s reich an iterative procedure for constructing zeros of accretive sets in banach spaces nonlinear anal 2 1978 85 92 and given results extend and improve some of the earlier results which include the steepest descent approximation method

the solution by iteration of nonlinear equations involving psi strongly accretive operators let e be an arbitrary real banach space and let k c e be a proximal subset of e suppose t k i e is a lipschitz strongly accretive operator such that for any given g e the equation tx f has a solution x e d t it is proved that modified iteration processes of the mann and ishikawa types converge strongly to x related results deal with the solution of the equation x tx f 0 for a strongly accretive or strongly dissipative t and the iterative approximation of fixed points to a strongly pseudocontractive t we also discuss the case where the operator t is only uniformly continuous miramare trieste december 1996 permanent address department of mathematics and computer science nnamdi azikiwe university p m b 5025 awka anambra state nigeria

approximating fixed points of φ hemicontractive mappings by the ishikawa iteration process with errors in uniformly smooth banach spaces abstract let e be a uniformly smooth banach space and t e e be a continuous and strongly o hemicontractive mapping this paper proves that under suitable conditions the ishikawa iterative sequence with errors strongly converges to the unique fixed point of t the related result deals with the strong convergence of the ishikawa iterative sequence with errors to the unique solution of the equation t x f when t e e is o strongly accretive these results generalize the results of ding 1 into more general o hemicontractive operators and extend a recent paper written by osilike 2 in two ways 1 i the lipschitzian continuity is replaced by the continuity on mapping t 2 ii if the errors u n v n 0 for all n n our theorems of this paper extend results of osilike 2 to the more general class of real uniformly smooth banach spaces

symmetric and arbitrarily high order birkhoff hermite time integrators and their long time behaviour for solving nonlinear klein gordon equations abstract the klein gordon equation with nonlinear potential occurs in a wide range of application areas in science and engineering its computation represents a major challenge the main theme of this paper is the construction of symmetric and arbitrarily high order time integrators for the nonlinear klein gordon equation by integrating birkhoff hermite interpolation polynomials to this end under the assumption of periodic boundary conditions we begin with the formulation of the nonlinear klein gordon equation as an abstract second order ordinary differential equation ode and its operator variation of constants formula we then derive a symmetric and arbitrarily high order birkhoff hermite time integration formula for the nonlinear abstract ode accordingly the stability convergence and long time behaviour are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi definite matrix subject to suitable temporal and spatial smoothness a remarkable characteristic of this new approach is that the requirement of temporal smoothness is reduced compared with the traditional numerical methods for pdes in the literature numerical results demonstrate the advantage and efficiency of our time integrators in comparison with the existing numerical approaches

a decomposition method for solving the nonlinear klein gordon equation in this paper adomian s decomposition scheme is presented as an alternate method for solving the nonlinear klein gordon equation the method is demonstrated by several examples comparing the scheme with existing collocation finite difference and finite element techniques shows that the present approach is highly accurate and converges rapidly

analysis and comparison of numerical methods for the klein gordon equation in the nonrelativistic limit regime we analyze rigourously error estimates and compare numerically temporal spatial resolution of various numerical methods for solving the klein gordon kg equation in the nonrelativistic limit regime involving a small parameter 0 varepsilon ll 1 which is inversely proportional to the speed of light in this regime the solution is highly oscillating in time i e there are propagating waves with wavelength of o varepsilon 2 and o 1 in time and space respectively we begin with four frequently used finite difference time domain fdtd methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size h and time step as well as the small parameter varepsilon based on the error bounds in order to compute correct solutions when 0 varepsilon ll1 the four fdtd methods share the same varepsilon scalability tau o varepsilon 3 then we propose new numerical methods by using either fourier pseudospectral or finite difference approximation for spatial derivatives combined with the gautschi type exponential integrator for temporal derivatives to discretize the kg equation the new methods are unconditionally stable and their varepsilon scalability is improved to o 1 and tau o varepsilon 2 for linear and nonlinear kg equations respectively when 0 varepsilon ll1 numerical results are reported to support our error estimates

new form 2 1 dimensional integrable lattice hierarchies with two shift operators e and eˆ abstract we present a new method to construct two 2 1 dimensional lattice hierarchies by considering a kind of spectral problems with two shift operators e and e a new 2 1 dimensional discrete soliton hierarchy and a new 3 3 lattice hierarchy are constructed by using 3 3 matrix spectral problem

a trace identity and its applications to the theory of discrete integrable systems a general theory for studying discrete integrable systems is developed based on a trace identity that the author previously proposed a scheme for generating hierarchies of discrete integrable systems is presented under broad assumptions the resulting hierarchies are shown to consist of liouville integrable hamiltonian systems

equivalence theorems of the convergence between ishikawa and mann iterations with errors for generalized strongly successively φ pseudocontractive mappings without lipschitzian assumptions abstract in this paper the equivalence of the strong convergence between the modified mann and ishikawa iterations with errors in two different schemes by xu y g xu ishikawa and mann iteration process with errors for nonlinear strongly accretive operator equations j math anal appl 224 1998 91 101 and liu l s liu ishikawa and mann iterative process with errors for nonlinear strongly accretive mappings in banach spaces j math anal appl 194 1995 114 125 respectively is proven for the generalized strongly successively pseudocontractive mappings without lipschitzian assumption our results generalize the recent results of the papers zhenyu huang f bu the equivalence between the convergence of ishikawa and mann iterations with errors for strongly successively pseudocontractive mappings without lipschitzian assumption j math anal appl 325 1 2007 586 594 b e rhoades s m soltuz the equivalence between the convergences of ishikawa and mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps j math anal appl 289 2004 266 278 b e rhoades s m soltuz the equivalence between mann ishikawa iterations and multi step iteration nonlinear anal 58 2004 219 228 by extending to the most general class of the generalized strongly successively pseudocontractive mappings and hence improve the corresponding results of all the references given in this paper by providing the equivalence of convergence between all of these iteration schemes for any initial points u 1 x 1 in uniformly smooth banach spaces

the solution by iteration of nonlinear equations in uniformly smooth banach spaces abstract letebe a uniformly smooth banach space and lett d t e ebe a strong pseudocontraction with an open domaind t ineand a fixed pointx d t we establish the strong convergence of the mann and ishikawa iterative processes with errors to the fixed point oft related results deal with the iterative solution of operator equations of the formsf txandf x tx 0 whentis a set valued strongly accretative operator our theorems include the cases in which the operatortis defined only locally explicit error estimates are also given

constructing nonlinear discrete integrable hamiltonian couplings beginning with lax pairs from special non semisimple matrix lie algebras we establish a scheme for constructing nonlinear discrete integrable couplings discrete variational identities over the associated loop algebras are used to build hamiltonian structures for the resulting integrable couplings we illustrate the application of the scheme by means of an enlarged volterra spectral problem and present an example of nonlinear discrete integrable hamiltonian couplings for the volterra lattice equations

iterative solution of nonlinear equations with strongly accretive operators abstract let e be a real banach space with a uniformly convex dual suppose t e e is a strongly accretive map with bounded range such that for each e the equation tx has a solution in e it is proved that each of the two well known fixed point iteration methods the mann and ishikawa iteration methods under suitable conditions converges strongly to a solution of the equation tx furthermore our method shows that such a solution is necessarily unique explicit error estimates are given our results resolve in the affirmative an open problem j math anal appl 151 no 2 1990 460 and generalize important known results

a uniformly accurate ua multiscale time integrator pseudospectral method for the dirac equation in the nonrelativistic limit regime we propose and rigourously analyze a multiscale time integrator fourier pseudospectral mti fp method for the dirac equation with a dimensionless parameter varepsilon in 0 1 which is inversely proportional to the speed of light in the nonrelativistic limit regime i e 0 varepsilon ll 1 the solution exhibits highly oscillatory propagating waves with wavelength o varepsilon 2 and o 1 in time and space respectively due to the rapid temporal oscillation it is quite challenging in designing and analyzing numerical methods with uniform error bounds in varepsilon in 0 1 we present the mti fp method based on properly adopting a multiscale decomposition of the solution of the dirac equation and applying the exponential wave integrator with appropriate numerical quadratures by a careful study of the error propagation and using the energy method we establish two independent error estimates via two different mathematical approaches as h m 0 frac tau 2 varepsilon 2 and h m 0 tau 2 varepsilon 2 where h is the mesh size tau is the time step and m 0 depends on the regularity of the solution these two error bounds immediately imply that the mti fp method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o tau for all varepsilon in 0 1 and optimally with quadratic convergence rate at o tau 2 in the regimes when either varepsilon o 1 or 0 varepsilon lesssim tau numerical results are reported to demonstrate that our error estimates are optimal and sharp finally the mti fp method is applied to study numerically the convergence rates of the solution of the dirac equation to those of its limiting models when varepsilon to0

an integrable lattice hierarchy associated integrable coupling darboux transformation and conservation laws abstract a new integrable lattice hierarchy is constructed from a discrete matrix spectral problem some related properties of the new hierarchy are discussed the hamiltonian structures and liouville integrability of the new hierarchy are established by using the discrete trace identity a kind of integrable coupling for the new hierarchy is constructed through enlarging spectral problems a darboux transformation dt with two variable parameters and the infinitely many conservation laws for a typical lattice equation in the new hierarchy are constructed based on its lax representation the explicit solutions are obtained via the dt the structures for those solutions are graphically investigated all these properties might be helpful to understanding some physical phenomena

a discrete variational identity on semi direct sums of lie algebras the discrete variational identity under general bilinear forms on semi direct sums of lie algebras is established the constant involved in the variational identity is determined through the corresponding solution to the stationary discrete zero curvature equation an application of the resulting variational identity to a class of semi direct sums of lie algebras in the volterra lattice case furnishes hamiltonian structures for the associated integrable couplings of the volterra lattice hierarchy

a generalized ablowitz ladik hierarchy multi hamiltonian structure and darboux transformation starting from a discrete spectral problem we derive a hierarchy of nonlinear discrete equations which includes the ablowitz ladik hierarchy and a new hierarchy as special cases especially we investigate in detail the integrability and resolvability of the new hierarchy it is shown that the new hierarchy is integrable in liouville s sense and possesses multi hamiltonian structure a darboux transformation is established for a typical discrete system in the new hierarchy with the help of the gauge transformation of its lax pair as applications of the darboux transformation new exact solutions of the discrete system are explicitly given

a spline collocation approach for the numerical solution of a generalized nonlinear klein gordon equation in this paper a finite element collocation approach using cubic b splines is employed for the numerical solution of a generalized form of the nonlinear klein gordon equation the efficiency of the method is tested on a number of examples that represent special cases of the extended equation including the sine gordon equation the numerical results are compared with existing numerical and analytic solutions and the outcomes confirm that the scheme yields accurate and reliable results even when few nodes are used at the time levels

fourier collocation method for solving nonlinear klein gordon equation a fourier collocation scheme is proposed for solving the periodic problem of nonlinear klein gordon equation it keeps the energy discretely and is shown to be stable or generalized stable its convergence is investigated numerical results are also presented

an exponential wave integrator sine pseudospectral method for the klein gordon zakharov system an exponential wave integrator sine pseudospectral method is presented and analyzed for discretizing the klein gordon zakharov kgz system with two dimensionless parameters 0 varepsilon le 1 and 0 gamma le 1 which are inversely proportional to the plasma frequency and the speed of sound respectively the main idea in the numerical method is to apply the sine pseudospectral discretization for spatial derivatives followed by using an exponential wave integrator for temporal derivatives in phase space the method is explicit symmetric in time and it is of spectral accuracy in space and second order accuracy in time for any fixed varepsilon varepsilon 0 and gamma gamma 0 in the mathcal o 1 plasma frequency and speed of sound regime i e varepsilon mathcal o 1 and gamma mathcal o 1 we establish rigorously error estimates for the numerical method in the energy space h 1 times l 2 we also study numerically the resolution of the method in the simultaneous high plasma fr

characteristic inequalities of uniformly convex and uniformly smooth banach spaces abstract let x be a real banach space with dual x and moduli of convexity and smoothness x e and x respectively for 1 p denotes the duality mapping from x into 2 x with gauge function t p 1 and j p denotes an arbitrary selection for j p let a r 0 0 t is strictly increasing and there exists c 0 such that t c x t 2 and f r 0 0 t is convex nondecreasing and there exists k 0 such that k x it is proved that x is uniformly convex if and only if there is a a such that x y p x p p j p x y x y x y x and x is uniformly smooth if and only if there is a f such that x y p x p p j p x y x y x y x where for given function f f x y is defined by f x y p 0 1 x ty x p t f t y x ty x dt these inequalities which have various applications can be regarded as general banach space versions of the well known polarization identity occurring in hilbert spaces

convergence theorems for multivalued φ hemicontractive operators and φ strongly accretive operators abstract suppose e is a uniformly smooth banach space and t e 2e is a multivalued hemicontractive operator with bounded range suppose an bn cn and an bn cn are real sequences in 0 1 satisfying the following conditions i a n b n c n a n b n c n 1 for all n n ii lim n b n lim n b n lim n c n 0 iii n 1 b n iv c n o b n for arbitrary xi u1 v1 e define the sequence xn n 1 by xn 1 anxn bn n cnun n tyn n n yn a nxn b n n c nvn n txn n n where un n 1 vn n 1 are arbitrary bounded sequences in e then xn n 1 converges strongly to the unique fixed point of t related results deal with the iterative solutions of nonlinear multivalued accretive operator equation f tx

a necessary and sufficient condition for convergence of steepest descent approximation to accretive operator equations abstract a necessary and sufficient condition is established which ensures the strong convergence of the steepest descent approximation to a solution of equations involving quasi accretive operators defined on a uniformly smooth banach space

stability and convergence of trigonometric integrator pseudospectral discretization for n coupled nonlinear klein gordon equations abstract this work serves as an improvement on a recent paper dong 2013 9 in which the n coupled nonlinear klein gordon equations were solved numerically by a fully explicit trigonometric integrator fourier pseudospectral tifp method this tifp method is second order accurate in time and spectral order accurate in space however in the previous work there was an absence of rigorous stability and convergence analysis moreover numerical studies in this work suggest that this tifp method suffers from a stability condition o h and h refer to time step and space mesh size to relax such a restriction while keeping the convergence properties and explicitness we propose two modified tifp methods motivated by the mollified impulse and gautschi type integrators for oscillatory odes for the modifications considered here linear stability and rigorous error estimates in the energy space are carried out which are the main achievements gained in this work meanwhile numerical results are also presented ideas of this work also suggest a general framework for proposing and analyzing efficient numerical methods for coupled wave type equations

a new hierarchy of integrable differential difference equations and darboux transformation a new discrete isospectral problem and the corresponding hierarchy of nonlinear differential difference equations are proposed it is shown that the hierarchy of differential difference equations possesses the hamiltonian structures a darboux transformation for the discrete spectral problem is found as an application two soliton solutions for the first system of differential difference equations in the hierarchy are given

steepest descent approximations for accretive operator equations a necessary and sufficient condition is established for the strong convergence of the steepest descent approximation to a solution of equations involving quasi accretive operators defined on a uniformly smooth banach space miramare trieste march 1993

the sine gordon equation in the finite line a rather complete numerical study of the perturbed and nonperturbed sine gordon equation in one dimensional cartesian coordinates and finite domains is presented the numerical study is based on five three point linearly implicit finite difference methods of different spatial accuracy for the unperturbed sge the accuracy of these five methods is found to be very sensitive to the implicitness parameter less sensitive to the spatial order of accuracy and almost no sensitive to the time step for second order accurate temporal discretizations the largest errors of the five methods were found to occur at the front of the soliton solution of the unperturbed sge but the techniques were very accurate for long time computations of the unperturbed sge in both infinite domains and in finite domains upon many collisions of the kinks and antikinks with the boundaries despite the fact that they do not preserve a discrete energy it is shown that the largest values of the kinetic and total energies and the smallest values of the strain and potential energies occur when solitons and antisolitons strike on the boundaries subject to homogeneous neumann conditions and the four energies remain constant between collisions with the boundaries for the malomed s perturbation one of the fronts of the soliton soliton doublet and soliton antisoliton doublet is trapped at one boundary and the other one undergoes a change in amplitude and its front describes a curved trajectory the soliton antisoliton breather oscillates in a damped manner and slowly drifts towards one boundary the soliton antisoliton soliton soliton doublet and soliton antisoliton doublet of the unperturbed sge are robust under kivshar and malomed s perturbations whereas the soliton antisoliton breather loses its integrity becomes a kink antikink pair and then yields a chaotic behaviour in both space and time small amplifications proportional to the first order temporal derivative of the amplitude do not alter substantially the initial dynamics of the soliton antisoliton soliton soliton doublet and soliton antisoliton doublet but their amplitudes exhibit oscillatory behaviour of increasing amplitude at large times however small amplifications cause first a transition from the soliton antisoliton breather to the kink antikink solution a linearized stability analysis of the spatially homogeneous solutions of the unperturbed sge is performed and analytical solutions are obtained by means of the method of separation of variables for both homogeneous dirichlet and neumann boundary conditions and the results show that the solution corresponding to a zero amplitude is oscillatory in both space and time whereas that corresponding to an amplitude equal to p may exhibit exponential or linear growth depending on the length of the domain if either of these growths is present it is shown that the evolution of small initial perturbations may result in homoclinic crossings and chaotic behaviour in both space and time in the absence of perturbations in the sge and the existence of an attractor when perturbations of the malomed s type are present in either case the solution may exhibit certain spatial symmetries or antisymmetries at large times finally the results presented in this paper indicate that the accurate computation of homoclinic orbits of the sge is not only controlled by the spatial accuracy of the numerical method but also by its energy preserving characteristics and perhaps by the number of grid points used in the numerical technique

iterative solution of nonlinear equations of the φ strongly accretive type abstract letq 1 and letebe a realq uniformly smooth banach space lett e ebe a lipschitz strongly accretive operator and suppose the equationtx f f e has a solution it is proved that under suitable conditions on the real sequences n n 0 and n n 0 the iteration process formula converges strongly to the unique solution of the equationtx f a related result deals with the approximation of fixed points of hemicontractive operators a class of operators which is much more general than the important class of strongly pseudocontractive operators

a hierarchy of integrable nonlinear lattice equations and its two integrable coupling systems abstract starting from a discrete spectral problem a hierarchy of integrable lattice soliton equations is derived it is shown that the hierarchy is integrable in the liouville sense a few subalgebras of the lie algebra are constructed from which the corresponding loop algebras can be presented it follows that the enlarging algebra systems are introduced moreover we construct the two integrable coupling systems with the help of the enlarging algebra systems

error estimates in the energy space for a gautschi type integrator spectral discretization for the coupled nonlinear klein gordon equations we propose and analyze an efficient and accurate numerical method for solving the coupled nonlinear klein gordon equations the method is based on the application of a gautschi type exponential integrator in time combined with sine spectral discretization in space the main results achieved in this paper are the rigorous error estimates in the energy space h 1 h 1 for the proposed scheme numerical tests are reported and agree with the error estimates quite well

semidirect sums of lie algebras and discrete integrable couplings a relation between semidirect sums of lie algebras and integrable couplings of lattice equations is established and a practicable way to construct integrable couplings is further proposed an application of the resulting general theory to the generalized toda spectral problem yields two classes of integrable couplings for the generalized toda hierarchy of lattice equations the construction of integrable couplings using semidirect sums of lie algebras provides a good source of information on complete classification of integrable lattice equations

arbitrarily high order time stepping schemes based on the operator spectrum theory for high dimensional nonlinear klein gordon equations abstract in this paper we explore arbitrarily high order lagrange collocation type time stepping schemes for effectively solving high dimensional nonlinear klein gordon equations with different boundary conditions we begin with one dimensional periodic boundary problems and first formulate an abstract ordinary differential equation ode on a suitable infinity dimensional function space based on the operator spectrum theory we then introduce an operator variation of constants formula which is essential for the derivation of our arbitrarily high order lagrange collocation type time stepping schemes for the nonlinear abstract ode the nonlinear stability and convergence are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi definite matrix under some suitable smoothness assumptions with regard to the two dimensional dirichlet or neumann boundary problems our new time stepping schemes coupled with discrete fast sine cosine transformation can be applied to simulate the two dimensional nonlinear klein gordon equations effectively all essential features of the methodology are present in one dimensional and two dimensional cases although the schemes to be analysed lend themselves with equal to higher dimensional case the numerical simulation is implemented and the numerical results clearly demonstrate the advantage and effectiveness of our new schemes in comparison with the existing numerical methods for solving nonlinear klein gordon equations in the literature