a numerical verification method for nonlinear functional equations based on infinite dimensional newton like iteration this paper describes a numerical verification of solutions for infinite dimensional functional equations based on residual forms and newton like iteration the method is based upon a verification method previously developed by the authors several computer assisted proofs for differential equations including nonlinear partial differential equations are presented

dominant and subdominant positive solutions to generalized dickman equation abstract the paper considers a generalized dickman equation t x t i 1 s a i x t i for t where s n ai 0 i 0 i 1 s and i 1 s a i 1 it is proved that there are two mutually disjoint sets of positive decreasing solutions such that for every two solutions from different sets the limit of their ratio for t equals 0 or the asymptotic behavior of such solutions is derived and a structure formula utilizing such solutions and describing all the solutions of a given equation is discussed in addition a criterion is proved giving sufficient conditions for initial functions to generate solutions falling into the first or the second set illustrative examples are given some open problems are suggested to be solved

explicit h2 estimates and pointwise bounds for solutions of second order elliptic boundary value problems abstract it is well known that the h 2 norm and the c 0 norm of a function u h 2 where r n is a bounded domain n 3 can be estimated in terms of a given uniformly elliptic second order differential operator l and some boundary operator b applied to u if certain regularity assumptions are satisfied if these bounds shall be used for numerical purposes the constants occurring in the estimates must be known explicitly the main goal of the present article is the computation of such explicit constants for simplicity of presentation we restrict ourselves to the case where l u u c x u as an application we prove an existence and inclusion result for nonlinear boundary value problems

generalized lyapunov schmidt reduction for parametrized equations at near singular points abstract the lyapunov schmidt reduction is generalized such that it uniformly handles bothperfect and imperfect singularities the results presented need neither very precise information about the location of the near singularities nor a precise knowledge of near null spaces

orbital stability investigations for travelling waves in a nonlinearly supported beam abstract we consider the fourth order wave equation t t x x x x f 0 x t r r with a nonlinearity f vanishing at 0 travelling waves x t u x c t satisfy the ode u c 2 u f u 0 on r and for the case f u e u 1 the existence of at least 36 solitary travelling waves was proved in 4 by computer assisted means we investigate the orbital stability of these solutions via computation of their morse indices and using results from 11 and 12 in order to achieve our results we make use of both analytical and computer assisted techniques

on computational proofs of the existence of solutions to nonlinear parabolic problems abstract this paper is an extension of the preceding study nakao this journal 1991 in which we described a numerical verification method of the solution for one space dimensional parabolic problems to the several space dimensional case here numerical verification means the automatic proof of the existence of solutions to the problems by some numerical techniques on a computer we reformulate the verification condition for nonlinear parabolic initial boundary value problems using the fixed point problem of a compact operator on certain function spaces as in the preceding study based upon a simple c 0 finite element approximation and its constructive a priori error estimates a numerical verification procedure is presented with some numerical examples

numerical verification of solutions for obstacle problems using a newton like method abstract this paper is a continuation of the preceding study 1 in which we described a method which automatically proves the existence of solutions for variational inequalities by computer we newly formulate a verification method using a newton like method this approach enables us to remove the restriction in the previous paper to the retraction property of the operator in a neighborhood of the solution we show some numerical examples which confirm that the method is really applicable to problems which have no retraction property

numerical inclusion methods of solutions for variational inequalities we consider a numerical method that enables us to verify the existence of solutions for variational inequalities this method is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations using the finite element approximations and explicit a priori error estimates we present an effective verification procedure that through numerical computation generates a set which includes the exact solution copyright 2002 john wiley sons ltd

numerical verifications of solutions for elliptic equations in nonconvex polygonal domains in this paper methods for numerical verifications of solutions for elliptic equations in nonconvex polygonal domains are studied in order to verify solutions using computer it is necessary to determine some constants which appear in a priori error estimations we propose some methods for determination of these constants in numerical examples calculating these constants for anl shaped domain we verify the solution of a nonlinear elliptic equation

computer assisted enclosure methods for elliptic differential equations abstract we report on methods for computing enclosures of solutions of second order nonlinear elliptic boundary value problems simultaneously proving the existence of a solution in the enclosing set the old fashioned monotonicity methods are well suited for this task but only for a restricted class of problems therefore we propose a new approach which is based on a suitable fixed point formulation of the problem and uses as an essential ingredient norm bounds for the inverse of the linearization of the given problem at some approximate solution which is computed numerically these norm bounds are obtained via eigenvalue enclosures we also give a brief description of an alternative method proposed by m t nakao

constructive a priori error estimates for a full discrete approximation of the heat equation in this paper we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation our method is based on the finite element galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space the present estimates play an essential role in the numerical verification method of exact solutions for the nonlinear parabolic equations this implies that by utilizing the present results we could get the guaranteed a posteriori error estimates for various kinds of nonlinear evolutional problems our results can also be considered as an explicit optimal estimate with the limited regularity of solutions which should be unknown up to the present

solving nonlinear parabolic problems with result verification part i one space dimensional case abstract we propose some numerical methods for the automatic proof of existence of weak solutions for parabolic initial boundary value problems with one space dimension it also means that one can obtain a posteriori error bounds for the approximate solutions of the problems based upon schauder s fixed point theorem a verification condition is formulated and by the use of finite element approximation and its error estimates for a simple parabolic problem we present a numerical verification algorithm of exact solutions in a computer some numerical examples which are verified by the method are illustrated

numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element the verifications of solutions to weakly nonlinear elliptic equations by the method described e g by nakao 1988 1989 etc are sometimes hardly accomplished when the right hand sides of the equations are very large to overcome such difficulties a residual iteration technique with approximate solution was introduced by nakao 1993 in the present paper we propose an a posteriori method for the residual iteration and show that a remarkable improvement in efficiency and in accuracy of the verification can be obtained when we use a higher order finite element

global smooth solution curves using rigorous branch following in this paper we present a new method for rigorously computing smooth branches of zeros of nonlinear operators f l 1 b 1 l 2 b 2 where b 1 and b 2 are banach spaces the method is first introduced for parameter continuation and then generalized to pseudo arclength continuation examples in the context of ordinary partial and delay differential equations are given

on very accurate enclosure of the optimal constant in the a priori error estimates for h02 projection we present constructive a priori error estimates for h 0 2 projection into a space of polynomials on a one dimensional interval here constructive indicates that we can obtain the error bounds in which all constants are explicitly given or are represented in a numerically computable form using the properties of legendre polynomials we consider a method by which to determine these constants to be as small as possible using the proposed technique the optimal constant could be enclosed in a very narrow interval with result verification furthermore constructive error estimates for finite element h 0 2 projection in one dimension are presented these types of estimates will play an important role in the numerical verification of solutions for nonlinear fourth order elliptic problems as well as in the guaranteed a posteriori error analysis for the finite element method or the spectral method e g hashimoto et al 2006 2 nakao et al 2008 3 watanabe et al 2009 11

numerical verification of solutions for elasto plastic torsion problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for the elasto plastic torsion problems governed by the variational inequality based upon the finite element approximations and the explicit a priori error estimates for a simple problem we present an effective verification procedure that through numerical computation generates a set which includes the exact solution this paper is an extension of the previous paper 1 in which we mainly dealt with the obstacle problems but some special techniques are utilized to verify the solutions for nondifferentiable nonlinear equations concerned with the present problem a numerical example is illustrated

numerical verification of solutions for signorini problems using newton like method we proposed a numerical method to verify the existence of solutions for a simplified signorini problem comput math appl 2000 40 1003 1013 using sequential iteration method we numerically constructed a set containing solutions that satisfies the hypothesis of schauder s fixed point theorem in a certain sobolev space it is difficult to apply this method to the problem of which associated operator is not retractive in a neighborhood of the solution in this paper in order to overcome such a difficulty we describe an alternative approach to this problem numerical examples are presented copyright 2007 john wiley sons ltd

a computer assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam abstract for a nonlinear beam equation with exponential nonlinearity we prove existence of at least 36 travelling wave solutions for the specific wave speed c 1 3 this complements the result in smets van den berg homoclinic solutions for swift hohenberg and suspension bridge type equations j differential equations 184 2002 78 96 stating that for almost all c 0 2 there exists at least one solution our proof makes heavy use of computer assistance starting from numerical approximations we use a fixed point argument to prove existence of solutions close to the computed approximations

mckay centralizer algebras for a finite subgroup g of the special unitary group su2 we study the centralizer algebra zk g endg v k of g acting on the k fold tensor product of its defining representation v c2 the mckay corre spondence relates the representation theory of these groups to an associated affine dynkin diagram and we use this connection to study the structure and representation theory of zk g via the combinatorics of the dynkin diagram when g equals the binary tetrahedral octahedral or icosahedral group we exhibit remarkable connections between zk g and the martin jones set partition algebras

numerical verification of solutions for nonlinear elliptic problems using anl residual method abstract we consider a numerical enclosure method with guaranteedl error bounds for the solution of nonlinear elliptic problems of second order by using an a posteriori error estimate for the approximate solution of the problem with a higher orderc0 finite element it is shown that we can obtain the guaranteedl error bounds with high accuracy a particular emphasis is that our method needs no assumption of the existence of the solution of the original nonlinear equation but it follows as the result of computation itself a numerical example that confirms the effectiveness of the method is presented

a numerical verification method for the existence of weak solutions for nonlinear boundary value problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for nonlinear two point boundary value problems bvp we formulate the problem as a fixed point of a newton like operator and present a verification algorithm by computer based on sadovskii s fixed point theorem in order to compute the newton type iterative sequences in infinite dimensional space two concepts the rounding and the rounding error are introduced also some numerical examples are illustrated we emphasize that since the present method uses the weak formulation of bvp and the finite element approximations instead of the existing validation techniques it is aslo applicable in principle to the bvp for partial differential equations as it stands

numerical verification of solutions for variational inequalities of the second kind abstract the purpose of this paper is to present an approach to the numerical proof of existence of solutions for the problem of the flow of a viscous plastic fluid in a pipe using the finite element approximations and the explicit a priori error estimates for the problem of the flow of a viscous plastic fluid in a pipe we present an effective verification procedure that through numerical computation generates a set which includes the exact solution further a numerical example is presented

a posteriori and constructive a priori error bounds for finite element solutions of the stokes equations we describe a method to estimate the guaranteed error bounds of the finite element solutions for the stokes problem in mathematically rigorous sense we show that an a posteriori error can be computed by using the numerical estimates of a constant related to the so called inf sup condition for the continuous problem also a method to derive the constructive a priori error bounds are considered some numerical examples which confirm us the expected rate of convergence are presented