on frechet differentiability of lipschitz maps between banach spaces a well known open question is whether every countable collection of lipschitz functions on a banach space x with separable dual has a common point of frechet differentiability we show that the answer is positive for some infinite dimensional x previously even for collections consisting of two functions this has been known for finite dimensional x only although for one function the answer is known to be affirmative in full generality our aims are achieved by introducing a new class of null sets in banach spaces called f null sets whose definition involves both the notions of category and measure and showing that the required differentiability holds almost everywhere with respect to it we even obtain existence of frechet derivatives of lipschitz functions between certain infinite dimensional banach spaces no such results have been known previously our main result states that a lipschitz map between separable banach spaces is frechet differentiable f almost everywhere provided that it is regularly gateaux differentiable f almost everywhere and the gateaux derivatives stay within a norm separable space of operators it is easy to see that lipschitz maps of x to spaces with the radon nikodym property are gateaux differentiable f almost everywhere moreover gateaux differentiability implies regular gateaux differentiability with exception of another kind of negligible sets so called a porous sets the answer to the question is therefore positive in every space in which every r porous set is f null we show that this holds for c k with k countable compact the tsirelson space and for all subspaces of co but that it fails for hilbert spaces

emerging notions of norm attainment for lipschitz maps between banach spaces abstract we classify several notions of norm attaining lipschitz maps which were introduced previously and present the relations among them in order to verify proper inclusions we also analyze some results for the sets of lipschitz maps satisfying each of these properties to be dense or not in lip 0 x y for instance we characterize a banach space y with the radon nikod m property in terms of the denseness of norm attaining lipschitz maps with values in y further we introduce a property called the local directional bishop phelps bollobas property for lipschitz compact maps which extends the one studied previously for scalar valued functions and provide some new positive results

the proximal normal formula in banach space approximation by proximal normals to the clarke generalized subdifferential for a distance function generated by a nonempty closed set and the normal cone to the set generated by the proximal normals are important tools in nonsmooth analysis we give simple general versions of such formulae in infinite dimensional banach spaces which satisfy different geometrical conditions our first class of spaces with uniformly g teaux differentiable norm includes the hilbert space case and the formulae is attained through dense subsets our second class of reflexive kadec smooth spaces is the most general for which such formulae can be obtained for all nonempty closed sets in the space our technique also allows us to establish the existence of solutions for a class of optimization problems substantially extending similar work of ekeland and lebourg

typical differentiability within an exceptionally small set abstract we verify the existence of a purely unrectifiable set in which the typical lipschitz function has a large set of differentiability points the example arises from a construction due to csornyei preiss and tiser of a universal differentiability set in which a certain lipschitz function has only a purely unrectifiable set of differentiability points

a measure zero universal differentiability set in the heisenberg group we show that the heisenberg group mathbb h n contains a measure zero set n such that every lipschitz function f mathbb h n rightarrow mathbb r is pansu differentiable at a point of n the proof adapts the construction of small universal differentiability sets in the euclidean setting we find a point of n and a horizontal direction where the directional derivative in horizontal directions is almost locally maximal then deduce pansu differentiability at such a point

differentiability of the metric projection in hilbert space a study is made of differentiability of the metric projection p onto a closed convex subset k of a hilbert space h when k has nonempty interior the gateaux or fr het smoothness of its boundary can be related with some precision to gateaux or frechet differentiability properties of p for instance combining results in 3 with earlier work of r d holmes shows that k has a c2 boundary if and only if p is cl in h k and its derivative p has a certain invertibility property at each point an example in 5 shows that if the c2 condition is relaxed even slightly then p can be nondifferentiable fr het in h k

second order optimality conditions in nonlinear programming obtained by way of epi derivatives dedicated to the memory of robin w chaney second order optimality conditions for finite dimensional smooth and nonsmooth nonlinear programming are obtained by a new method that emphasizes a close connection with geometrical approximation of the essential objective function the approximation is secured by the use of certain epi derivatives defined by epiconvergence the optimality conditions are expressed in a form that covers general interval constraints and their possible representation through penalties or an augmented lagrangian an abstract constraint involving restriction to a convex polyhedron is incorporated

partitioned adaptive runge kutta methods for the solution of nonstiff and stiff systems for the numerical solution of initial value problems of ordinary differential equations partitioned adaptive runge kutta methods are studied these methods consist of an adaptive runge kutta methods for the treatment of a stiff system and a corresponding explicit runge kutta method for a nonstiff system first we modify the theory of butcher series for partitioned adaptive runge kutta methods we show that for any explicit runge kutta method there exists a translation invariant partitoned adaptive runge kutta method of the same order secondly we derive a special translaton invariant partitioned adaptive runge kutta method of order 3 an automatic stiffness detection and a stepsize control basing on richardson extrapolation are performed extensive tests and comparisons with the partitioned rkf4rw algorithm from rentrop 16 and the partitioned algorithm lsoda from hindmarsh 9 and petzold 15 show that the partitoned adaptive runge kutta algorithm works reliable and gives good numericals results furthermore these tests show that the automatic stiffness detection in this algorithm is effective

uniform approximation of convex function in smooth banach spaces abstract this article is devoted to smooth approximation of convex functions on banach spaces with smooth norm we prove that if x is a smooth space and f is a w lower semicontinuous lipschitzian convex function on x then there exist two w lower semicontinuous g teaux differentiable convex function sequences f n n 1 and g n n 1 such that 1 f n f n 1 f and f g n 1 g n 2 f n f and g n f uniformly on x 3 cl x x d f n x x cl x x d g n x x x

nonsmooth analysis and hamilton jacobi equations on riemannian manifolds we establish some perturbed minimization principles and we develop a theory of subdifferential calculus for functions defined on riemannian manifolds then we apply these results to show existence and uniqueness of viscosity solutions to hamilton jacobi equations defined on riemannian manifolds

differentiability and ball covering property in banach spaces abstract in this paper author proves that if x 1 and x 2 are g teaux differentiable space then x 1 and x 2 have the ball covering property if and only if x 1 x 2 p and x 1 x 2 have the ball covering property where x y p x 1 p y 2 p 1 p p 1 and x y max x y

on the structure of a free measures and applications we establish a general structure theorem for the singular part of a free radon measures where a is a linear pde operator by applying the theorem to suitably chosen differential operators a we obtain a simple proof of alberti s rank one theorem and for the first time its extensions to functions of bounded deformation bd we also prove a structure theorem for the singular part of a finite family of normal currents the latter result implies that the rademacher theorem on the differentiability of lipschitz functions can hold only for absolutely continuous measures and that every top dimensional ambrosio kirchheim metric current in rd is a federer fleming flat chain

hjb equations for the optimal control of differential equations with delays and state constraints i regularity of viscosity solutions we study a class of optimal control problems with state constraints where the state equation is a differential equation with delays this class includes some problems arising in economics in particular the so called models with time to build see p k asea and p j zak j econom dynam control 23 1999 pp 1155 1175 m bambi j econom dynam control 32 2008 pp 1015 1040 f e kydland and e c prescott econometrica 50 1982 pp 1345 1370 we embed the problem in a suitable hilbert space h and consider the associated hamilton jacobi bellman hjb equation this kind of infinite dimensional hjb equation has not been previously studied and is difficult due to the presence of state constraints and the lack of smoothing properties of the state equation our main result on the regularity of solutions to such an hjb equation seems to be entirely new more precisely we prove that the value function is continuous in a sufficiently big open set of h that it solves in the viscosity sense the associated hjb equation and that it has continuous classical derivative in the direction of the present this regularity result is the starting point to define a feedback map in the classical sense which gives rise to a candidate optimal feedback strategy

second order subgradients of convex integral functionals the purpose of this work is twofold on the one hand we study the second order behaviour of a nonsmooth convex function f defined over a reflexive banach space x we establish several equivalent characterizations of the set 2f x y known as the second order subdifferential of f at x relative to y e f3f i on the other hand we examine the case in which f if is the functional integral associated to a normal convex integrand f we extend a result of chi ngoc do from the space x lpd 1 p oo to a possible nonreflexive banach space x lp 1 p oo we also establish a formula for computing the second order subdifferential 82if x y

the abc theorem for higher dimensional function fields we generalize the abc theorems to the function field of a variety over an algebraically closed field of arbitrary characteristic which is non singular in codimension one we also obtain an upper bound for the minimal order sequence of wronskians over such function fields of positive characteristic

differentiability properties of functions that are l stable at a point abstract the class of the l stable functions is introduced in previous papers of the authors as a generalization of the class of c 1 1 functions the importance of this class is that minimization problems with l stable data admit second order optimality conditions in terms of suitable directional derivatives similar to those for ones with c 1 1 data the present paper studies differentiability properties of l stable functions in normed spaces and generalizes to infinite dimensional spaces some results of the authors

attainment and sub differentiability of the infimal convolution of a function and the square of the norm abstract let x be a banach space whose norm is simultaneously lur and gateaux frechet smooth under some assumptions it is shown that the infimal convolution of a fairly general function on x and the square of the norm is generically strongly attained and hence is gateaux frechet differentiable this contains a result of s dutta on distance functions

a dynamic programming approach to nonlinear boundary control problems of parabolic type abstract in this paper we study a hamilton jacobi equation related to the boundary control of a parabolic equation with neumann boundary conditions the state space of this problem is a hilbert space and the equation is defined classically only on a dense subset of the state space moreover the hamiltonian appearing in the equation contains fractional powers of an unbounded operator these facts render the problem difficult in this paper we give a revised definition of a viscosity solution to accommodate the unboundness of the hamiltonian we then obtain existence and uniqueness results for viscosity solutions in particular we show that under suitable assumptions the value function of the boundary control problem is the unique viscosity solution of the related hamilton jacobi equation

generic solutions for some perturbed optimization problem in non reflexive banach spaces abstract let z be a closed boundedly relatively weakly compact nonempty subset of a banach space x and j z r a lower semicontinuous function bounded from below if x 0 is a convex subset in x and x 0 has approximatively z property k then the set of all points x in x 0 z for which there exists z 0 z such that j z 0 x z 0 x and every sequence z n z satisfying lim n j z n x z n x for x contains a subsequence strongly convergent to an element of z is a dense g subset of x 0 z moreover under the assumption that x 0 is approximatively z strictly convex we show more namely that the set of all points x in x 0 z for which there exists a unique point z 0 z such that j z 0 x z 0 x and every sequence z n z satisfying lim n j z n x z n x for x converges strongly to z 0 is a dense g subset of x 0 z here x inf j z x z z z these extend s cobzas s result j math anal appl 243 2000 344 356

regular mappings and non existence of bi lipschitz embeddings for slit carpets abstract we prove that the slit carpet studied by merenkov does not admit a bi lipschitz embedding into any uniformly convex banach space in particular this includes any space r n but also spaces such as l p for p 1 this resolves question 8 in the 1997 list by heinonen and semmes

on generic second order gateaux differentiability let f x be a function on a banach space x we say that f is strictly gateaux differentiable if it is gateaux differentiable at every point of x and the mapping x h x h is continuous recall that a convex gateaux differentiable function is strictly gateaux differentiable in the case of a locally lipschitz function our definition coincides with more standard ones it requires that f be norm to weak star continuous

mean value inequalities in hilbert space we establish a new mean value theorem applicable to lower semicontinuous functions on hilbert space a novel feature of the result is its multidirectionality it compares the value of a function at a point to its values on a set we then discuss some refinements and consequences of the theorem including applications to calculus flow invariance and generalized solutions to partial differential equations rtsumt on etablit un nouveau theoreme de la valeur moyenne qui s applique aux fonctions semicontinues inferieurement sur un espace de hilbert on deduit plusieurs consequences du resultat portant par exemple sur les fonctions monotones et sur les solutions generalisees des equations aux derivees partielles

a study of rosenbrock type methods of high order this paper deals with the solution of nonlinear stiff ordinary differential equations the methods derived here are of rosenbrock type this has the advantage that they area stable or stiffly stable and nevertheless do not require the solution of nonlinear systems of equations we derive methods of orders 5 and 6 which require one evaluation of the jacobian and onelu decomposition per step we have written programs for these methods which use richardson extrapolation for the step size control and give numerical results

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

numerical treatment of delay differential equations by hermite interpolation a class of numerical methods for the treatment of delay differential equations is developed these methods are based on the wellknown runge kutta fehlberg methods the retarded argument is approximated by an appropriate multipoint hermite interpolation the inherent jump discontinuities in the various derivatives of the solution are considered automatically problems with piecewise continuous right hand side and initial function are treated too real life problems are used for the numerical test and a comparison with other methods published in literature

a compact null set containing a differentiability point of every lipschitz function we prove that in a euclidean space of dimension at least two there exists a compact set of lebesgue measure zero such that any real valued lipschitz function defined on the space is differentiable at some point in the set such a set is constructed explicitly

numerical solution of retarded initial value problems local and global error and stepsize control retarded initial value problems are routinely replaced by an initial value problem of ordinary differential equations along with an appropriate interpolation scheme hence one can control the global error of the modified problem but not directly the actual global error of the original problem in this paper we give an estimate for the actual global error in terms of controllable quantities further we show that the notion of local error as inherited from the theory of ordinary differential equations must be generalized for retarded problems along with the new definition we are led to developing a reliable basis for a step selection scheme

a universal differentiability set in banach spaces with separable dual abstract we show that any non zero banach space with a separable dual contains a totally disconnected closed and bounded subset s of hausdorff dimension 1 such that every lipschitz function on the space is frechet differentiable somewhere in s

integration of multivalued operators and cyclic submonotonicity we introduce a notion of cyclic submonotonicity for multivalued operators from a banach space x to its dual we show that if the clarke subdifferential of a locally lipschitz function is strictly submonotone on an open subset u of x then it is also maximal cyclically submonotone on u and conversely that every maximal cyclically submonotone operator on u is the clarke subdifferential of a locally lipschitz function which is unique up to a constant if u is connected in finite dimensions these functions are exactly the lower c 1 functions considered by spingarn and rockafellar

an effective schmidt s subspace theorem over function fields we establish an effective version of the schmidt s subspace theorem for higher dimensional function fields of characteristic zero

zeros of analytic functions with or without multiplicities the classical mason stothers theorem deals with nontrivial polynomial solutions to the equation a b c it provides a lower bound on the number of distinct zeros of the polynomial abc in terms of deg a deg b and deg c we extend this to general analytic functions living on a reasonable bounded domain omega subset mathbb c rather than on the whole of mathbb c the estimates obtained are sharp for any and a generalization of the original result on polynomials can be recovered from them by a limiting argument

regularity for obstacle problems in infinite dimensional hilbert spaces in this paper we study fully nonlinear obstacle type problems in hilbert spaces we introduce the notion of q elliptic equation and prove existence uniqueness and regularity of viscosity solutions of q elliptic obstacle problems in particular we show that solutions of concave problems with semiconvex obstacles are in the space

well posedness of a class of perturbed optimization problems in banach spaces abstract let x be a banach space and z a nonempty subset of x let j z r be a lower semicontinuous function bounded from below and p 1 this paper is concerned with the perturbed optimization problem of finding z 0 z such that x z 0 p j z 0 inf z z x z p j z which is denoted by min j x z the notions of the j strictly convex with respect to z and of the kadec with respect to z are introduced and used in the present paper it is proved that if x is a kadec banach space with respect to z and z is a closed relatively boundedly weakly compact subset then the set of all x x for which every minimizing sequence of the problem min j x z has a converging subsequence is a dense g subset of x z 0 where z 0 is the set of all points z z such that z is a solution of the problem min j z z if additionally p 1 and x is j strictly convex with respect to z then the set of all x x for which the problem min j x z is well posed is a dense g subset of x z 0

a variational method for second order shape derivatives we consider shape functionals obtained as minima on sobolev spaces of classical integrals having smooth and convex densities under mixed dirichlet neumann boundary conditions we propose a new approach for the computation of the second order shape derivative of such functionals yielding a general existence and representation theorem in particular we consider the p torsional rigidity functional for p geq 2

regularity results for the minimum time function of a class of semilinear evolution equations of parabolic type under suitable controllability and smoothness assumptions the minimum time function t x of a semilinear control system is proved to be locally lipschitz continuous and semiconcave on the controllable set these properties are then applied to derive optimality conditions relating optimal trajectories to the superdifferential of t

generic frechet differentiability of convex functions dominated by a lower semicontinuous convex function abstract in this paper an extended real valued proper lower semicontinuous convex function f on a banach space is said to have the frechet differentiability property fdp if every proper lower semicontinuous convex function g with g f is frechet differentiable on a dense g subset of int dom g the interior of the effective domain of g we show that f has the fdp if and only if the w closed convex hull of the image of the subdifferential map of f has the radon nikod m property this is a generalization of the main theorem in a paper by lixin and shuzhong to appear according to this result it also gives several new criteria of asplund spaces

stability analysis of numerical methods for delay differential equations this paper deals with the stability analysis of step by step methods for the numerical solution of delay differential equations we focus on the behaviour of such methods when they are applied to the linear testproblemu t u t u t with 0 and complex a general theorem is presented which can be used to obtain complete characterizations of the stability regions of these methods

generalized subdifferentials a baire categorical approach we use baire categorical arguments to construct dramatically pathological locally lipschitz functions the origins of this approach can be traced back to banach and mazurkiewicz 1931 who independently used similar categorical arguments to show that almost every continuous real valued function defined on 0 1 is nowhere differentiable as with the results of banach and mazurkiewicz it appears that it is easier to show that almost every function possesses a certain property than to construct a single concrete example among the most striking results contained in this paper are almost every 1 lipschitz function defined on a banach space has a clarke subdifferential mapping that is identically equal to the dual ball if t 1 t 2 ldots t n is a family of maximal cyclically monotone operators defined on a banach space x then there exists a real valued locally lipschitz function g such that partial 0 g x mbox co t 1 x t 2 x ldots t n x for each x in x in a separable banach space each non empty weak compact convex subset in the dual space is identically equal to the approximate subdifferential mapping of some lipschitz function and for locally lipschitz functions defined on separable spaces the notions of strong and weak integrability coincide

essentially smooth lipschitz functions in this paper we address some of the most fundamental questions regarding the differentiability structure of locally lipschitz functions defined on separable banach spaces for example we examine the relationship between integrability d representability and strict differentiability in addition to this we show that on any separable banach space there is a significant family of locally lipschitz functions that are integrable d representable and possess desirable differentiability properties we also present some striking applications of our results to distance functions

numerical approximation of the solutions of delay differential equations on an infinite interval using piecewise constant arguments the idea of approximating the solutions of delay differential equations by equations with piecewise constant arguments epca has been suggested by gyijri l who proved convergence of the method for linear and nonlinear delay equations on compact intervals and under certain conditions also on the half line the approximating equations with piecewise constant argument can in turn be solved by use of difference equations the latter then also provide an approximation of the original delay equation the theory of epca was initiated and studied by cooke and wiener in 2 and 3 in this paper we begin to investigate the question of whether in this method of approximation the asymptotic dynamics of the delay differential equations are preserved a number of authors have dealt with the problem of showing that discretization of ordinary differential equations does not significantly alter the basic qualitative features readers may refer to the papers of kloeden and lorenz 4 and beyn 5 for some of this work for delay differential equations questions of this sort have been studied by cryer 6 barwell 7 zennaro 8 and later authors the paper of strehmel et al 9 contains an up to date list of references in our method of approximation the delay equation is first replaced by an epca and then by a difference equation and our objective is to relate the qualitative dynamics of these three equations we obtain results for autonomous and nonautonomous equations with one or many delays for autonomous equations the resulting difference algorithm is just a simple euler scheme but for nonautonomous equations it includes other possibilities in order to make the method and the nature of results as clear as possible we begin with a particular test equation k t pz t 7 p 0 7 0

frechet differentiability of regular locally lipschitzian functions abstract this paper considers frechet differentiability almost everywhere in the sense of category of regular locally lipschitzian real valued functions defined on open subsets of a banach space it is first shown that for separable banach spaces clarke s generalized gradient of such a function is a minimal convex and compact valued upper semicontinuous multifunction using a theorem of christensen and kenderov it is then shown that for separable asplund spaces such a function is frechet differentiable on a dense g subset of its domain

the stability of difference formulas for delay differential equations a new simple geometric technique is presented for analyzing the stability of difference formulas for the model delay differential equation y t py t qy t 8 where p and q are complex constants and the delay 8 is a positive constant the technique is based on the argument principle and directly relates the region of absolute stability for ordinary differential equations corresponding to the py t term with the region corresponding to the delay term qy t 8 a sufficient condition for stability is that these regions be disjoint the technique is used to show that for each a stable a a stable or stiffly stable linear multistep formula for ordinary differential equations there is a correspond ing linear multistep formula for delay differential equations with analogous stability properties the analogy does not extend however to a stable one step formulas

avoiding σ porous sets in hilbert spaces abstract we give a constructive proof that any porous subset of a hilbert space has lebesgue measure zero on typical c 1 curves further we discover that this result does not extend to all forms of porosity we find that even power p porous sets may meet many c 1 curves in positive measure

a note on the abc conjecture in this note we have generalized the abc conjecture for integers and extended mason s theorem to polynomials in m by dint of nevanlinna s value distribution theory 2002 wiley periodicals inc

on the number of polynomial solutions of bernoulli and abel polynomial differential equations abstract in this paper we determine the maximum number of polynomial solutions of bernoulli differential equations and of some integrable polynomial abel differential equations as far as we know the tools used to prove our results have not been utilized before for studying this type of questions we show that the addressed problems can be reduced to know the number of polynomial solutions of a related polynomial equation of arbitrary degree then we approach to these equations either applying several tools developed to study extended fermat problems for polynomial equations or reducing the question to the computation of the genus of some associated planar algebraic curves

stability analysis of linearly implicit one step interpolation methods for stiff retarded differential equations the aim of this paper is to analyze the stability properties of the modified problem and of linearly implicit runge kutta interpolation methods for stiff retarded differential equations with respect to the model equation y t py qy t tau quad t geqq 0 y t varphi t quad t leqq 0 where p and q are co mplex constants and tau 0 is a constant d necessary and sufficient conditions for the stability of the modified problem i e the retarded term y t tau of the model equation is replaced by an interpolation polynomial and of linearly implicit one step interpolation methods are presented these conditions allow us to compute the boundary locus of the stability region it is shown that any modified problem and any linearly implicit runge kutta lagrange method with appropriate stability function are p beta stable and that linearly implicit runge kutta lagrange methods possess better stability properties than linearly implicit runge kutta hermite methods furthermore

proximal normal cone analysis on smooth banach spaces and applications unlike the frechet penot clarke and other order one notions of normal cones the notion of proximal normal cone describes variational behavior of order two and most of results on proximal normal cone are established in the hilbert space framework a possible reason is that in a general banach space such as in the classical banach spaces l p and l p with 1 leq p 2 the proximal normal cone of a smooth subset a at every point of a may consist of the zero element only nevertheless on the other hand we show in this paper that for a fairly big class of smooth banach spaces including the classical banach spaces l p and l p with 2 leq p infty the consideration of proximal normal cones provides a lot of useful information especially for leading to satisfactory treatment for constrained optimization problems some of our results are new even in the hilbert space case

a stothers mason theorem with a difference radical differential calculus is not a unique way to observe polynomial equations such as a b c we propose a way of applying difference calculus to estimate multiplicities of the roots of the polynomials a b and c satisfying the equation above then a difference abc theorem for polynomials is proved using a new notion of a radical of a polynomial results for example on the non existence of polynomial solutions to difference fermat and difference super fermat functional equations are given as applications we also introduce a truncated second main theorem for differences and use it to consider these functional equations with non polynomial entire solutions equations with polynomial or non polynomial solutions are observed to see the sharpness of results obtained

isometries for the legendre fenchel transform it is shown that on the space of lower semicontinuous convex functions defined on r the conjugation map the legendre fenchel transform is an isometry with respect to some metrics consistent with the epi topology we also obtain isometries for the infinite dimensional case hilbert space and reflexive banach space but this time they correspond to topologies finer than the mosco epi topology

calculus rules on the approximate second order directional derivative of a convex function given a real valued convex function f the approximate second order directional derivative f varepsilon x 0 d d of f at x 0 in the direction d is an object which is defined whenever the parameter varepsilon is chosen strictly positive the aim of this work is to derive expressions of the approximate second order directional derivative of a function f which has been constructed from other functions f i whose properties are better known we address ourselves to the problem of calculating f varepsilon having the f i eta eta 0 at our disposal calculus rules are given for the main functional operations preserving convexity composition with an affine mapping sum of functions image of a function under a linear mapping maximum of functions

reconstruction of the clarke subdifferential by the lasry lions regularizations abstract we prove that the clarke subdifferential of a locally lipschitz function with a growth condition defined on a hilbert space can be represented by the derivatives of its lasry lions regularizations we complete a result of j benoist showing a similar representation for the clarke subdifferential of a lipschitz function in an arbitrary banach space by the clarke subdifferentials of its lasry lions regularizations

directional upper derivatives and the chain rule formula for locally lipschitz functions on banach spaces motivated by an attempt to find a general chain rule formula for differentiating the composition f g of lipschitz functions f and g that would be as close as possible to the standard formula f g x f g x g x we show that this formula holds without any artificial assumptions provided derivatives are replaced by complete derivative assignments the idea behind these assignments is that the derivative of f at y is understood as defined only in the direction of a suitable tangent space u f y and so it exists at every point but these tangent spaces are chosen in such a way that for any g they contain the range of g x for almost every x showing the existence of such assignments leads us to detailed study of derived sets and the ways in which they describe pointwise behavior of lipschitz functions

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

approximately convex functions and approximately monotonic operators abstract we present characterizations of some generalized convexity properties of functions with the help of a general subdifferential we stress the case of lower semicontinuous functions we also study the important case of marginal functions and we provide representation results

semiregularity and generalized subdifferentials with applications to optimization the michel penot subdifferential of a locally lipschitzian function is the principal part of the clarke subdifferential it coincides with the g derivative at differentiable points a locally lipschitzian function can be determined by its michel penot subdifferential uniquely up to an additive constant though this cannot be done by its clarke subdifferential if the set of abnormal points is not negligible a set valued operator is the michel penot subdifferential of a locally lipschitzian function if and only if it is a seminormal operator satisfying a cyclical condition various calculus rules hold for the michel penot subdifferential equalities hold for these rules at a point under semiregularity which is weaker than regularity for a locally lipschitzian function in a separable banach space semiregularity holds everywhere except for a haar zero set applications in optimization are discussed

calculus rules on a new set valued second order derivative for convex functions etant donnee une fonction convexe f qui a ete construite a partir d autres fonctions f i dont les differentielles secondes sont aisees a determiner on cherche a exprimer les differentielles secondes de f en fonction des 2 f i

generic gâteaux differentiability of convex functions on small sets the theory of generic differentiability of convex functions on banach spaces is by now a well explored part of infinite dimensional geometry all the attempts to solve this kind of problem have in common as a working hypothesis one special feature of the finite dimensional case namely convex functions are always considered to be defined on convex sets with nonempty interior but typically a convex set in a banach space does not have interior points even when it is not contained in a closed hyperplane so this raises the problem of expounding a theory of differentiability of convex functions defined on small sets i e sets without interior points in our paper n2 we have made an attempt to solve this problem discussing questions of generic frechet differentiability of convex functions on small sets in the present paper we deal with problems of generic gateauxdifferentiability in this context

coarse differentiation of quasi isometries i spaces not quasi isometric to cayley graphs in this paper we prove that certain spaces are not quasi isometric to cayley graphs of nitely generated groups in particular we answer a question of woess and prove a conjecture of diestel and leader by showing that certain homogeneous graphs are not quasi isometric to a cayley graph of a nitely generated group this paper is the rst in a sequence of papers proving results announced in our 2007 article quasi isometries and rigidity of solvable groups in particular this paper contains many steps in the proofs of quasi isometric rigidity of lattices in sol and of the quasi isometry classication of lamplighter groups the proofs of those results are completed in coarse dierentiation of quasi isometries ii rigidity for lattices in sol and lamplighter groups the method used here is based on the idea of coarse dierentiation introduced in our 2007 article

a differentiable structure for metric measure spaces abstract the main result of this paper is the provision of conditions under which a metric measure space admits a differentiable structure this differentiable structure gives rise to a finite dimensional l cotangent bundle over the given metric measure space and then to a sobolev space h 1 p over the given metric measure space the latter which is reflexive for p 1 this extends results of cheeger geom funct anal 9 1999 3 428 to a wider collection of metric measure spaces

second order sufficient conditions for error bounds in banach spaces recently huang and ng presented second order sufficient conditions for error bounds of continuous and ga teaux differentiable functions in banach spaces wu and ye dropped the assumption of huang and ng on ga teaux differentiability but required the space to be a hilbert space we carry on this research in two directions first we extend wu and ye s result to some non hilbert spaces second same as huang and ng we work on banach spaces but provide different second order sufficient conditions that may allow the function to be non ga teaux differentiable

differentiability of lipschitz functions in lebesgue null sets we show that if n 1 then there exists a lebesgue null set in mathbb r n containing a point of differentiability of each lipschitz function f mathbb r n rightarrow mathbb r n 1 in combination with the work of others this completes the investigation of when the classical rademacher theorem admits a converse avoidance of sigma porous sets arising as irregular points of lipschitz functions plays a key role in the proof

a generalization of ekeland s ϵ variational principle and its borwein preiss smooth variant abstract we give a generalization of ekeland s variational principle and of its borwein preiss smooth variant replacing the distance and the norm by a gauge type lower semi continuous function as an application of this generalization we show that if on a banach space x there exists a lipschitz smooth bump function then every continuous convex function on an open subset u of x is densely differentiable in u this generalizes the borwein preiss theorem on the differentiability of convex functions

asplund sets differentiability and subdifferentiability of functions in banach spaces abstract we show that asplund sets are effective tools to study differentiability of lipschitz functions and e subdifferentiability of lower semicontinuous functions on general banach spaces if a locally lipschitz function defined on an asplund generated space x t y has a minimal clarke subdifferential mapping then it is t b y uniformly strictly differentiable on a dense g subset of x examples are given of locally lipschitz functions that are t b y uniformly strictly differentiable everywhere but nowhere frechet differentiable

smooth approximation of convex functions in banach spaces abstract this paper shows that every extended real valued lower semi continuous proper respectively lipschitzian convex function defined on an asplund space can be represented as the point wise limit respectively uniform limit on every bounded set of a sequence of lipschitzian convex functions which are locally affine hence c at all points of a dense open subset and shows an analogous for w lower semi continuous proper respectively lipschitzian convex functions defined on dual spaces whose pre duals have the radon nikodym property

hilbert generated spaces we classify several classes of the subspaces of banach spaces x for which there is a bounded linear operator from a hilbert space onto a dense subset in x dually we provide optimal affine homeomorphisms from weak star dual unit balls onto weakly compact sets in hilbert spaces or in c0 spaces in their weak topology the existence of such embeddings is characterized by the existence of certain uniformly g teaux smooth norms

generalized hessian properties of regularized nonsmooth functions we take up the question of second order expansions for a class of functions of importance in optimization namely moreau envelope regularizations of nonsmooth functions f it is shown that when f is prox regular which includes convex functions and the extended real valued functions representing problems of nonlinear programming the many second order properties that can be formulated around the existence and stability of expansions of the envelopes of f or of their gradient mappings are linked by surprisingly extensive lists of equivalences with each other and with generalized differentiation properties of f itself this clarifies the circumstances conducive to developing computational methods based on envelope functions such as second order approximations in nonsmooth optimization and variants of the proximal point algorithm the results establish that generalized second order expansions of moreau envelopes at least can be counted on in most situations of interest in finite dimensional optimization