stability of cauchy jensen type functional equation in generalized fuzzy normed spaces in this paper we establish some stability results concerning the cauchy jensen functional equation in generalized fuzzy normed spaces the results of the present paper improve and extend some recent results

φ contractor and the solutions for nonlinear operator equations in fuzzy normed spaces in this paper we introduce the concept of contractor in fuzzy normed spaces and prove the existence and uniqueness theorems of solutions for set and single valued nonlinear operator equations with this kind of contractor as applications of these theorems we obtain some new fixed point theorems in fuzzy normed spaces

motion of a set by the curvature of its boundary abstract the study of a crystal shrinking or growing in a melt gives rise to equations relating the normal velocity of the motion to the curvature of the crystal boundary often these equations are anisotropic indicating the preferred directions of the crystal structure in the isotropic case this equation is called the curve shortening or the mean curvature flow equation and has been studied by differential geometric tools in general there are no classical solutions to these equations in this paper we develop a weak theory for the generalized mean curvature equation using the newly developed theory of viscosity solutions our approach is closely related to that of osher and sethian chen giga and goto and evans and spruck who view the boundary of the crystal as the level set of a solution to a nonlinear parabolic equation although we use their results in an essential way we give an intrinsic definition our main results are the existence of a solution large time asymptotics of this solution and its connection to the level set solution of osher and sethian chen giga and goto and evans and spruck in general there is no uniqueness even for classical solutions but we prove a uniqueness result under restrictive assumptions we also construct a class of explicit solutions which are dilations of wulff crystals

deformations of unbounded convex bodies and hypersurfaces we study the topology of the space k of complete convex hypersurfaces of r which are homeomorphic to rn 1 in particular using minkowski sums we construct a deformation retraction of k onto the grassmannian space of hyperplanes so every hypersurface in k may be flattened in a canonical way further the total curvature of each hypersurface evolves continuously and monotonically under this deformation we also show that modulo proper rotations the subspaces of k consisting of smooth strictly convex or positively curved hypersurfaces are each contractible which settles a question of h rosenberg

wkbj type approximation for finite moments perturbations of the differential equation y 0 and the analogous difference equation abstract wkbj or liouville green type approximation theorems are proved for differential equations like y g x y 0 x 1 and their discrete counterparts 2 y n g n y n 0 n v the coefficients g x g n having a finite second moment precise error bounds are obtained as in the already existing theory due to f w j olver for differential equations in other classes and in a previous work by the authors concerning certain difference equations when only the first moment is finite the same asymptotic approximation with bound still holds for recessive solutions an effort has been made to present both aspects the continuous and the discrete in a unified way basing the error analysis on a volterra stieltjes integral equation examples are given to illustrate the theory

on nonlinear equations for fuzzy mappings in probabilistic normed spaces in this paper we introduce the concept of probabilistic contractor couple which simplifies and weakens that of probabilistic g contractor given by cho et al fuzzy sets and systems 110 2000 115 122 by using this concept we study the existence and iteration convergence of solutions for set valued nonlinear operator equations and nonlinear equations of fuzzy mappings in menger pn spaces the results presented in this paper improve and extend the corresponding results of cho et al in a sense

on yau s problem of evolving one curve to another convex case abstract a revised yau s curvature difference flow is considered to deform one convex curve x 0 to another one x it is proved that this flow exists globally on time interval 0 and the evolving curve preserving its convexity and bounded area a converges to a fixed limiting curve x congruent to a a x as time tends to infinity where a is the area bounded by the target curve x

liouville green approximations for a class of linear oscillatory difference equations of the second order abstract an asymptotic approximation theorem is proved for the solutions of linear oscillatory three term recurrence equations in a certain class this result represents a discrete analogue of the well known liouville green or wkbj theorem rigorously proved by olver for second order linear differential equations as in olver s theorem precise estimates are obtained for the error terms and the double asymptotic nature of the approximation is made clear applications to certain orthogonal polynomials are also given

the stability of the additive cauchy functional equation in non archimedean fuzzy normed spaces a stability result for the additive cauchy functional equation in non archimedean fuzzy normed spaces over a field with valuation is given by using the concept of countable extension of a t norm

sublinear perturbations of the differential equation y n 0 and of the analogous difference equation abstract according to a result of a ghizzetti for any solution y t of the differential equation where y n t i 0 n 1 g i t y i t 0 t 1 1 g i x x n i 1 dx 0 i n 1 either y t 0 for t 1 or there is an integer r with 0 r n 1 such that lim t y t t r exists and 0 related results are obtained for difference and differential inequalities a special case of the former has interesting applications in the study of orthogonal polynomials

a problem of hartman and wintner approximation for discrete perturbations abstract consider the second order self adjoint difference equation cn xn an fn xn 1 0 as a perturbation of the eventually disconjugate difference equation cn zn an zn 1 0 where cn 0 asymptotic approximation for the fundamental system of solutions of the perturbed equation are expressed explicitly in terms of the coefficients and the principal or recessive solution of the unperturbed equation in particular the coefficient cn is allowed to be oscillatory and we do not assume absolute summability conditions on fn results obtained are applied to a second order poincare difference equation whose unperturbed equation has a double characteristic root nonlinear and nonhomogeneous perturbations are also considered an example is given for illustration

on a free boundary problem for the curvature flow with driving force we study a free boundary problem associated with the curvature dependent motion of planar curves in the upper half plane whose two endpoints slide along the horizontal axis with prescribed fixed contact angles our first main result concerns the classification of solutions every solution falls into one of the three categories namely area expanding area bounded and area shrinking types we then study in detail the asymptotic behavior of solutions in each category among other things we show that solutions are asymptotically self similar both in the area expanding and the area shrinking cases while solutions converge to either a stationary solution or a traveling wave in the area bounded case we also prove results on the concavity properties of solutions one of the main tools of this paper is the intersection number principle however in order to deal with solutions with free boundaries we introduce what we call the extended intersection number principle which turns out to be exceedingly useful in handling curves with moving endpoints

on set valued nonlinear equations in menger probabilistic normed spaces we introduce the concept of probabilistic f j contractor couple which allows us to consider set valued nonlinear operator equations in a large class of probabilistic normed spaces our results complete those from cho et al nonlinear equations for fuzzy mappings in probabilistic normed spaces fuzzy sets and systems 110 2000 115 122 and fang on nonlinear equations for fuzzy mappings in probabilistic normed spaces fuzzy sets and systems 131 2002 357 364

a numerical method for evaluating zeros of solutions of second order linear differential equations a numerical algorithm for computing real zeros of solutions of 2ndorder linear differential equations y q x y 0 in the oscillatory case on a half line is studied the method applies to the class q x a b x o x p with a o ber p 1 this procedure is based on a certain nonlinear 3rd order equation kummer s equation which plays a role in the theory of transformations of 2nd order differential equations into each other and was earlier introduced by f w j olver in 1950 to compute zeros of cylinder functions a rigorous asymptotic and numerical analysis is developed by combining boruvka s approach to the study of kummer s equation and olver s original idea numerical examples are presented

parabolic equations related to curve motion abstract we study curve motions based on differential equations curve motion equations are classified into two types adaptive equations which depend on the choice of coordinate systems and non adaptive equations examples from both types of equations are studied and the global existence for these equations is proved based on integral estimates

deforming a hypersurface by gauss curvature and support function abstract we study the motion of smooth strictly convex bodies in r n expanding in the direction of their normal vector field with speed depending on gauss curvature and support function

on the affine heat equation for non convex curves 1998 american mathematical society first published in journal of the american mathematical society vol 11 no 3 july 1998 published by the american mathematical society

singularities of the area preserving curve shortening flow with a free boundary condition we consider the area preserving curve shortening flow with neumann free boundary conditions outside of a convex domain or at a straight line we give a criterion on initial curves that guarantees the appearance of a singularity in finite time we prove that the singularity is of type ii furthermore if these initial curves are convex then an appropriate rescaling at the finite maximal time of existence yields a grim reaper or half a grim reaper as limit flow we construct examples of initial curves satisfying the mentioned criterion

a couple of nonlinear equations with fuzzy mappings in fuzzy normed spaces in this paper we introduce the concept of f contractor couple and study the existence of solutions for a couple of nonlinear equations with fuzzy mappings in fuzzy normed spaces we construct some new iterative algorithms for solving the couple of nonlinear equations with fuzzy mappings and prove the convergence of iterative sequences generated by the algorithms in fuzzy normed spaces as applications we obtain some new common fixed point theorems for fuzzy mappings in fuzzy normed spaces the results presented in this paper improve and generalize corresponding results of cho et al fuzzy sets and systems 110 2000 115 and fang and song fuzzy sets and systems 121 2001 267

repeatedly smooting discrete scale space evolution and dominant point detection in this paper the fourier analysis is used to derive the properties of an evolving curve an arbitrary kernel repeatedly smoothing akrs evolution of a curve is then introduced it is shown that when the repeated number is large the akrs evolution of a curve is an approximately discrete implementation of the scale based evolution of this curve in the euclidean space as a special case an exponential repeatedly smoothing is proposed to implement the scale based evolution it is shown that in addition to its simple implementation and its desired approximation to a gaussian kernel an exponential function is a function that when it is selected as a repeatedly smoothing kernel the motion both magnitude and direction of a point on a curve from the i 1 th to ith instant is equal to the curvature of the ith smoothed curve at this point finally a perimeter controlled evolution method is proposed to extract dominant points it is shown experimentally that the proposed method is robust to noise object rotation and object changes in sizes

mean curvature flow of noncompact hypersurfaces with type ii curvature blow up ii abstract we continue the study initiated by the first two authors in 15 of type ii curvature blow up in mean curvature flow of complete noncompact hypersurfaces embedded in euclidean space in particular we construct mean curvature flow solutions in the rotationally symmetric class with the following precise asymptotics near the vanishing time t 1 the highest curvature concentrates at the tip of the hypersurface an umbilical point and blows up at the rate t t 1 2 in a neighbourhood of the tip the solution converges to a translating soliton known as the bowl soliton 3 near spatial infinity the hypersurface approaches a collapsing cylinder at an exponential rate

on a nonlinear parabolic equation arising from anisotropic plane curve evolution abstract we study the asymptotic behavior of a nonlinear parabolic equation arising from anisotropic plane curve evolution we show that if the solution has type one blow up then it will converge after rescaling to a self similar solution if the solution has type two blow up then its profile near the maximum point is also known in the case of anisotropic plane curve evolution if the evolving curve has type two blow up then it will converge to a translational self similar solution

symmetry reductions and exact solutions of the affine heat equation lie symmetry group method is applied to study the affine heat equation for surface its symmetry groups and corresponding optimal systems are determined and group invariant solutions associated to the symmetries are obtained and classified

blow up of solutions to a degenerate parabolic equation not in divergence form abstract we study nonglobal positive solutions to the dirichlet problem for ut up u u in bounded domains where 0 t t 1 p if either the space dimension is one or p t t 1 p u t consists of continuous functions solving w w 1 p w 1 p in one space dimension it is shown that actually t t 1 p u t w as t t where w coincides with an element of a one parameter family of functions inside each component of its positivity set furthermore we study the size of the components of w 0 with the result that this size is uniquely determined by in the case p 1 the positivity set can have the maximum possible size 2 p for certain initial data but it may also be arbitrarily close to the minimal length

quasilinear evolution equations and parabolic systems it is shown that general quasilinear parabolic systems possess unique maximal classical solutions for sufficiently smooth initial values provided the boundary conditions are time independent moreover it is shown that in the autonomous case these equations generate local semi flows on appropriate sobolev spaces our results apply in particular to the case of prescribed boundary values dirichlet boundary conditions

convex curves moving translationally in the plane abstract we show that a nontrivial homothetic self similar solution can happen only when f k k or f k k we also derive a parametric representation of a translational self similar solution a translational self similar solution may have self intersections but cannot be a simple closed curve for any f k

characterizations of metrizable topological vector spaces and their asymmetric generalizations in terms of fuzzy quasi norms we obtain characterizations of those paratopological vector spaces that are quasi metrizable locally bounded quasi metrizable locally convex and quasi normable respectively in terms of appropriate kinds of fuzzy quasi norms characterizations of distinguished kinds of metrizable topological vector spaces some of them known already are deduced as particular cases of our approach

blow up rates of solutions of initial boundary value problems for a quasi linear parabolic equation abstract we consider initial boundary value problems for a quasi linear parabolic equation k t k 2 k k with zero dirichlet boundary conditions and positive initial data it has known that each of solutions blows up at a finite time with the rate faster than t t 1 in this paper it is proved that sup k t t t 1 log log t t 1 as t t under some assumptions our strategy is based on analysis for curve shortening flows that with self crossing brought by s b angenent and j j l velazquez in addition we prove some of numerical conjectures by watterson which are keys to provide the blow up rate

motion by curvature of planar curves with end points moving freely on a line this paper deals with the motion by curvature of planar curves having end points moving freely along a line with fixed contact angles to this line we first prove the existence and uniqueness of self similar shrinking solution then we show that the curve shrinks to a point in a self similar manner if initially the curve is a graph

curve shortening and the topology of closed geodesics on surfaces we study flat knot types of geodesics on compact surfaces m2 for every flat knot type and any riemannian metric g we introduce a conley index associated with the curve shortening flow on the space of immersed curves on m2 we conclude existence of closed geodesics with prescribed flat knot types provided the associated conley index is nontrivial

classical solutions for a class of fully nonlinear degenerate parabolic equations this paper is concerned with the existence and comparison principle of classical solutions for a class of fully nonlinear degenerate parabolic equations

stability of additive mappings in non archimedean fuzzy normed spaces in this paper we introduce a notion of a non archimedean fuzzy norm and study the stability of the cauchy equation in the context of non archimedean fuzzy spaces in the spirit of hyers ulam rassias gavruta as a corollary the stability of the jensen equation is established we indeed present an interdisciplinary relation between the theory of fuzzy spaces the theory of non archimedean spaces and the theory of functional equations

the variational problem for a certain space time functional defined on planar closed curves abstract we consider a variational problem for a certain space time functional defined on planar closed curves the functional is related to the functional appeared in bellettini and mugnai 2008 4 the variational problem is stated as follows let 0 and 1 denote planar closed curves and t be a positive constant minimize the space time functional over family of planar closed curves which change from 0 at time t 0 into 1 at time t t concerning the variational problem we prove the existence of minimizer in a radially symmetric class and determine all the minimizers for each initial final data moreover we show that there exists a unique non radially symmetric critical point in a neighborhood of a certain radially symmetric minimizer

evolution of plane curves driven by a nonlinear function of curvature and anisotropy in this paper we study evolution of plane curves satisfying a geometric equation v beta k nu where v is the normal velocity and k and nu are the curvature and tangential angle of a plane curve gamma we follow the direct approach and we analyze the so called intrinsic heat equation governing the motion of plane curves obeying such a geometric equation the intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional alpha we show how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities from an analytical point of view we present some new results on short time existence of a regular family of evolving curves in the degenerate case when beta k nu gamma nu k m 0 m le 2 and the governing system of equations includes a nontrivial tangential velocity functional

symmetries and invariant solutions for the geometric heat flows we study lie symmetries and invariant solutions of the geometric heat flows the basic similarity reductions for the ghe are performed reduced equations and exact solutions associated with the symmetries are obtained group invariant solutions and reductions for the affine case are also discussed in a special case

evolving a convex closed curve to another one via a length preserving linear flow abstract motivated by a recent curvature flow introduced by professor s t yau s t yau private communication on his curvature difference flow 2007 we use a simple curvature flow to evolve a convex closed curve to another one under the assumption that both curves have the same length we show that under the evolution the length is preserved and if the curvature is bounded above during the evolution then an initial convex closed curve can be evolved to another given one

on loglog blow up in higher order reaction diffusion and wave equations a formal geometric approach abstract in 1934 petrovskii s boundary regularity study of the heat equation gave the first appearance of the ln ln t t blow up factor in pde theory we discuss the origin of analogous non self similar blow up in higher order reaction diffusion parabolic or wave hyperbolic equations of the form u t u 2 u x x x x u or u t t u 2 u x x x x u in l l 0 t with zero dirichlet boundary conditions at x l where l l 0 2 we present formal arguments that the standard similarity blow up rate 1 t t acquires an extra universal ln ln t t factor the explanation is based on a geometric matching with the so called logarithmic travelling waves as group invariant solutions of the pdes we also discuss connections with log log blow up factors occurring in earlier studies of plasma physics parabolic equations and the nonlinear critical schrodinger equation

riccati techniques and approximation for a second order poincare difference equation approximation results are obtained for second order self adjoint difference equations and applied to a second order poincare difference equation whose unperturbed equation has a double characteristic root

curve shortening straightening flow for non closed planar curves with infinite length abstract we consider a motion of non closed planar curves with infinite length the motion is governed by a steepest descent flow for the geometric functional which consists of the sum of the length functional and the total squared curvature we call the flow shortening straightening flow in this paper first we prove a long time existence result for the shortening straightening flow for non closed planar curves with infinite length then we show that the solution converges to a stationary solution as time goes to infinity moreover we give a classification of the stationary solution