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

quantum mechanical scattering by impenetrable periodic surfaces in this paper we investigate the existence and completeness of the wave operators w s limt exp ith p exp ith0 corresponding to the quantum mechanical scattering of nonrelativistic particles by certain classes of impenetrable noncompact surfaces bounding domains r 2 which contain a half space and are contained in another half space here h0 is the usual negative distributional laplacian in h0 l2 r h is the negative dirichlet laplacian in h l2 and p is an appropriate identification operator under these conditions we prove by elementary methods that w exist as partially isometric operators whose initial sets have a transparent physical meaning suppose now that the domain r also has the periodicity property x x x l x when l ranges over a bravais lattice in r 1 where we write x r as x x with x r 1 and x r then a ranw hscatt h and b w are asymptotically complete in the sense that h hscatt h hsurf h here hscatt h and hsur

necessary conditions for constrained optimization problems with semicontinuous and continuous data we consider nonsmooth constrained optimization problems with semicontinuous and continuous data in banach space and derive necessary conditions sl without constraint qualification in terms of smooth subderivatives and normal cones these results in different versions are set in reflexive and smooth banach spaces

on the asymptotically well behaved functions and global error bound for convex polynomials in this paper we present some new and tractable sufficient conditions for convex asymptotically well behaved awb functions moreover we establish several lipschitz and holder type global error bound results for a single convex polynomial and for functions which can be expressed as maximum of finitely many nonnegative convex polynomials an advantage of our approach is that the corresponding holder exponent in our holder type global error bound results can be determined explicitly

variational analysis in nonreflexive spaces and applications to control problems with l 1 perturbations we provide a refined sensitivity analysis for finite and infinite horizon control problems where in both cases the perturbation space is l 1 our underlying technique relies on a recent sequential description of both the generalized gradient of clarke and of the approximate g subdifferential of functions defined on a smooth banach space we also show that the proximal limit formula for the generalized gradient and its l p analogue are direct consequences of these sequential formulas related characterizations of lipschitzness of a function on a smooth space are given

optimal control for a mixed flow of hamiltonian and gradient type in space of probability measures abstract in this paper we investigate an optimal control problem in the space of measures on r2 the problem is motivated by a stochastic interacting particle model which gives the 2 d navier stokes equations in their vorticity formulation as a mean field equation we prove that the associated hamiltonjacobi bellman equation in the space of probability measures is well posed in an appropriately defined viscosity solution sense

robust l_p norm decoding for bicm based secondary user systems the frequency bands used by secondary user systems such as cognitive radio and ultra wideband uwb systems are subject to various forms of non gaussian noise and interference including co channel and narrowband interference caused by the primary user and other secondary users as well as man made impulsive noise for secondary user systems employing popular bit interleaved coded modulation bicm we propose a robust lp norm metric for viterbi decoding to overcome the harmful effects of non gaussian noise and interference for the resulting system we derive an approximate upper bound on the bit error rate ber and a general analytical expression for the asymptotic ber valid for all practically relevant types of fading including rayleigh ricean nakagami m nakagami q and weibull fading and all types of noise and interference with finite moments simulation and analytical results show that in non gaussian noise and interference the proposed lp norm metric yields large performance gains over the conventional l2 norm metric and outperforms erasure decoding

global error bounds for piecewise convex polynomials in this paper by examining the recession properties of convex polynomials we provide a necessary and sufficient condition for a piecewise convex polynomial to have a holder type global error bound with an explicit holder exponent our result extends the corresponding results of li siam j control optim 33 5 1510 1529 1995 from piecewise convex quadratic functions to piecewise convex polynomials

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

a nonlinear lagrangian approach to constrained optimization problems in this paper we study nonlinear lagrangian functions for constrained optimization problems which are in general nonlinear with respect to the objective function we establish an equivalence between two types of zero duality gap properties which are described using augmented lagrangian dual functions and nonlinear lagrangian dual functions respectively furthermore we show the existence of a path of optimal solutions generated by nonlinear lagrangian problems and show its convergence toward the optimal set of the original problem we analyze the convergence of several classes of nonlinear lagrangian problems in terms of their first and second order necessary optimality conditions

global error bounds with fractional exponents using the partial order induced by a proper weakly lower semicontinuous function on a reflexive banach space x we give a sufficient condition for f to have error bounds with fractional exponents application is given to identify the set of such exponents for quadratic functions

on first and second order conditions linebreak for error bounds we establish first order and second order sufficient conditions ensuring that a proper lower semicontinuous function f on a banach space x has an error bound we also consider similar problems with constraint namely that f is replaced by its restriction to a subset of x these results are employed to identify exactly when a quadratic function on x has an error bound

vector valued perturbed equilibrium problems the goal of this paper is to prove some general vector valued perturbed equilibrium principles and some existence results of vector equilibrium points for bifunctions satisfying a new natural notion of lower semi continuity we obtain these results by going through a new concept of approximative equilibrium point

on the fuzzy intersection rule abstract the paper provides a complete exposition of the fuzzy intersection rule in variational analysis and its applications to generalized differentiation theory new forms of the rule and further applications are presented

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

scattering by periodic surfaces if r 2 is an exterior domain containing a half space and contained in a half space we prove that the wave operators w s limt exp ith jexp ith0 are partial isometries and that the invariance principle w w holds for suitable real functions on r admissible functions here h0 is the negative distributional laplacian in l2 r 2 h is hd or hn the negative dirichlet or neumann laplacians in h l2 respectively j is an appropriate identification operator and w are defined as were w but with h0 and h replaced by h0 and h respectively suppose in addition that has a suitable periodicity property and that when h hn it has a certain mild local compactness property then we prove 1 that w are asymptotically complete in the sense that ran w hscatt h h hsurf h 2 that hscatt h hscatt h and hsurf h hsurf h for each admissible function and hence that w are asymptotically complete in a similar sense for each such

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

robust minimax detection of a weak signal in noise with a bounded variance and density value at the center of symmetry in practical communication environments it is frequently observed that the underlying noise distribution is not gaussian and may vary in a wide range from short tailed to heavy tailed forms to describe partially known noise distribution densities a distribution class characterized by the upper bounds upon a noise variance and a density dispersion in the central part is used the results on the minimax variance estimation in the huber sense are applied to the problem of asymptotically minimax detection of a weak signal the least favorable density minimizing fisher information over this class is called the weber hermite density and it has the gaussian and laplace densities as limiting cases the subsequent minimax detector has the following form i with relatively small variances it is the minimum l2 norm distance rule ii with relatively large variances it is the l1 norm distance rule iii it is a compromise between these extremes with relatively moderate variances it is shown that the proposed minimax detector is robust and close to huber s for heavy tailed distributions and more efficient than huber s for short tailed ones both in asymptotics and on finite samples

performance of bicm sc and bicm ofdm systems with diversity reception in non gaussian noise and interference in this paper we present a general mathematical framework for performance analysis of single carrier sc and orthogonal frequency division multiplexing ofdm systems employing popular bit interleaved coded modulation bicm and multiple receive antennas the proposed analysis is applicable to bicm systems impaired by general types of fading including rayleigh ricean nakagami m nakagami q and weibull fading and general types of noise and interference with finite moments such as additive white gaussian noise awgn additive correlated gaussian noise gaussian mixture noise co channel interference narrowband interference and ultra wideband interference we present an approximate upper bound for the bit error rate ber and an accurate closed form approximation for the asymptotic ber at high signal to noise ratios for viterbi decoding with the standard euclidean distance branch metric for the standard rate 1 2 convolutional code the proposed approximate upper bound and the asymptotic approximation become tight at bers of 10 6 and 10 12 respectively however if the code is punctured to higher rates e g 2 3 or 3 4 the asymptotic approximation also becomes tight at a ber of 10 6 exploiting the asymptotic ber approximation we show that the diversity gain of bicm systems only depends on the free distance of the code the type of fading and the number of receive antennas but not on the type of noise in contrast their coding gain strongly depends on the noise moments our asymptotic analysis shows that as long as the standard euclidean distance branch metric is used for viterbi decoding bicm systems optimized for awgn are also optimum for any other type of noise and interference with finite moments

solutions of the multiconfiguration equations in quantum chemistry the multiconfiguration methods are the natural generalization of the well known hartree fock theory for atoms and molecules by a variational method we prove the existence of a minimum of the energy and of infinitely many solutions of the multiconfiguration equations a finite number of them being interpreted as excited states of the molecule our results are valid when the total nuclear charge z exceeds n 1 n is the number of electrons and cover most of the methods used by chemists the saddle points are obtained with a min max principle we use a palais smale condition with morse type information and a new and simple form of the euler lagrange equations

on a class of first order hamilton jacobi equations in metric spaces abstract we establish well posedness of a class of first order hamilton jacobi equation in geodesic metric spaces the result is then applied to solve a hamilton jacobi equation in the wasserstein space of probability measures which arises from the variational formulation of a compressible euler equation

on smooth variational principles in banach spaces abstract a new smooth variational principle for spaces admitting frechet differentiable bump functions is proved further it is shown that each proper lower semicontinuous bounded below function can be supported by a smooth function with locally holder derivative if and only if the space is superreflexive some geometrical refinements of the borwein preiss smooth variational principle using deville s techniques are obtained

partial extensions of attouch s theorem with applications to proto derivatives of subgradient mappings attouch s theorem which gives on a reflexive banach space the equivalence between the mosco epi convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients has many important applications in convex optimization in particular generalized derivatives have been defined in terms of the epi convergence or graph convergence of certain difference quotient mappings and attouch s theorem has been used to relate these various generalized derivatives these relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems we prove in a hilbert space several partial extensions of attouch s theorem to functions more general than convex these functions are called primal lower nice furthermore we use our extensions to derive a relationship between the second order epi derivatives of primal lower nice functions and the proto derivative of their associated subgradient mappings

scattering theory for high order operators in domains with infinite boundary abstract a general scattering theory is studied for high order elliptic operators in domains with infinite boundary

vector valued variational principles in the context of vector valued extensions of variational principles we are dealing with functions taking values in a banach space partially ordered by a closed convex pointed cone we introduce and study a new notion of semi continuity connected with the order and we improve the vector valued extensions of deville godefroy zizler perturbed minimization principle

convergence analysis of an augmented lagrangian method for mathematical programs with complementarity constraints abstract in this paper a mathematical program with complementarity constraints mpcc is reformulated as a nonsmooth constrained optimization problem by using the fischer burmeister function an augmented proximal lagrangian method is applied to tackle the resulting constrained optimization problem the augmented lagrangian problems are in general nonsmooth we derive first and second order optimality conditions for the augmented lagrangian problems using an approximate smooth variational principle and establish that the limit point of a sequence of points that satisfy the second order necessary optimality conditions of the augmented lagrangian problems is a strongly stationary point of the original mpcc if the limit point is feasible to mpcc and the linear independence constraint qualification for mpcc and the upper level strict complementarity condition hold at the limit point

abstract criteria for multiple solutions to nonlinear coupled equations involving magnetic schrodinger operators we consider a system of nonlinear coupled equations involving magnetic schrodinger operators and general potentials we provide the criteria for the existence of multiple solutions to these equations as special cases we get the classical results on existence of infinitely many distinct solutions within hartree and hartree fock theory of atoms and molecules subject to an external magnetic fields we also extend recent results within this theory including coulomb system with a constant magnetic field a decreasing magnetic field and a physically measurable magnetic field

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

solutions to quasi relativistic multi configurative hartree fock equations in quantum chemistry we establish the existence of infinitely many distinct solutions to the multi configurative hartree fock type equations for n electron coulomb systems with quasi relativistic kinetic energy sqrt a 2 d x n a 4 a 2 for the nth electron finitely many of the solutions are interpreted as excited states of the molecule moreover we prove the existence of a ground state the results are valid under the hypotheses that the total charge ztot of k nuclei is greater than n 1 and that ztot is smaller than a critical charge ztot the proofs are based on a new application of the lions fang ghoussoub critical point approach to nonminimal solutions on a complete analytic hilbert riemann manifold in combination with density operator techniques

necessary suboptimality and optimality conditions via variational principles the paper aims to develop some basic principles and tools of nonconvex variational analysis with applications to necessary suboptimality and optimality conditions for constrained optimization problems in infinite dimensions we establish a certain subdifferential variational principle as a new characterization of asplund spaces this result is different from conventional support forms of variational principles and appears to be convenient for applications to nonsmooth optimization based on the subdifferential variational principle we obtain new necessary conditions for suboptimal solutions in general nonsmooth optimization problems with equality inequality and set constraints in asplund spaces in this way we establish the so called sequential normal compactness properties of constraint sets that play an essential role in infinite dimensional variational analysis and its applications as a by product of our approach we derive various forms of necessary optimality conditions for nonsmooth constrained problems in infinite dimensions which extend known results in that direction

impact of wimax interference on mb ofdm uwb systems analysis and mitigation this paper presents an analysis of the performance of the multiband orthogonal frequency division multiplexing mb ofdm system for ultra wideband uwb communication in the presence of interference from ieee 802 16 wimax systems operating in the 3 5 ghz band we present an exact analysis of the uncoded bit error rate ber of the mb ofdm system based on laplace transform techniques we also present a simple gaussian approximation for the wimax interference and establish its relative accuracy and usefulness by means of comparison with the exact analysis and simulations such a gaussian approximation is especially useful for simplified performance analysis as well as for the design of interference mitigation techniques motivated by the gaussian approximation we propose a simple two stage interference mitigation technique for coded mb ofdm transmissions consisting of interference spectrum estimation during silent periods followed by appropriate bit metric weighting during viterbi decoding we compare parametric and non parametric spectrum estimation techniques for various scenarios of interest in the presence of wimax interference the two stage interference mitigation provides substantial gains in performance in return for modest increases in complexity and without requiring any modifications to the mb ofdm transmitter or signal structure

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

comments on bit interleaved coded modulation caire taricco and biglieri presented a detailed analysis of bit interleaved coded modulation bicm a simple and popular technique used to improve system performance especially in the context of fading channels they derived an upper bound to the probability of error called the expurgated bound in this correspondence the proof of the expurgated bound is shown to be flawed a new upper bound is also derived it is not known whether the original expurgated bound is valid for the important special case of square qam with gray labeling but the new bound is very close to and slightly tighter than the original bound for a numerical example

borwein preiss variational principle revisited abstract in this article we refine and slightly strengthen the metric space version of the borwein preiss variational principle due to li and shi 2000 12 clarify the assumptions and conclusions of their theorem 1 as well as theorem 2 5 2 in borwein and zhu 2005 4 and streamline the proofs our main result theorem 3 is formulated in the metric space setting when reduced to banach spaces corollary 9 it extends and strengthens the smooth variational principle established in borwein and preiss 1987 3 along several directions

global łojasiewicz type inequality for non degenerate polynomial maps abstract let f f 1 f p r n r p be a polynomial map this paper studies the existence of the following global ojasiewicz type inequality f x f x c d x f 1 0 for all x r n for some constants c 0 0 and 0 we show that the above inequality holds if one of the following conditions is satisfied i f is convenient and khovanskii non degenerate at infinity ii f is convenient and non degenerate at infinity iii f is mikhailov gindikin non degenerate further in cases ii and iii the exponents and can be determined explicitly

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

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

the hypertangent cone in borwein and strojwas 5 we have observed that in baire metrizable spaces the hypertangent cone plays an important role in the theory of tangent cones and generalized subgradients because its properties relate to lipschitz behaviour of sets and functions here more accurate results for banach spaces are presented they follow from a formula which may be viewed as the discrete version of the treiman inclusion 18 we prove this in section 1 our discrete formula has many applications it implies generalizations of the bishop phelps theorem on the density of support points 11 and the dense clarke subdifferentiability theorem of mclinden 14 in section 2 we present a new characterization of the clarke derivative and new conditions characterizing directionally lipschitzian functions on a banach space in section 3 we use our discrete formula to obtain mean value theorems for lower semicontinuous functions on banach spaces in the vein of penot 16 also we prove a strengthening of lebourg s 13 mean value theorem for reflexive spaces in section 4 we prove a general inversion theorem in particular we use it to furnish conditions for the clarke tangent cone of the intersection of sets to contain the intersection of the tangent cones for these sets this generalizes the finite dimensional version of ioffe ll and an epi lipschitzian version of rockafellar 16 in a banach space the basic facts about tangent cones and generalized subgradients that we use in this paper are discussed in clarke l0

sufficient conditions for error bounds for a lower semicontinuous l s c inequality system on a banach space it is shown that error bounds hold provided every element in an abstract subdifferential of the constraint function at each point outside the solution set is norm bounded away from zero a sufficient condition for a global error bound to exist is also given for an l s c inequality system on a real normed linear space it turns out that a global error bound closely relates to metric regularity which is useful for presenting sufficient conditions for an l s c system to be regular at sets under the generalized slater condition a continuous convex system on rn is proved to be metrically regular at bounded sets

on the equivalence of some basic principles in variational analysis the primary goal of this paper is to study relationships between certain basic principles of variational analysis and its applications to nonsmooth calculus and optimization considering a broad class of banach spaces admitting smooth renorms with respect to some bornology we establish an equivalence between useful versions of a smooth variational principle for lower semicontinuous functions an extremal principle for nonconvex sets and an enhanced fuzzy sum rule formulated in terms of viscosity normals and subgradients with controlled ranks further refinements of the equivalence result are obtained in the case of a frechet differentiable norm based on the new enhanced sum rule we provide a simplified proof for the refined sequential description of approximate normals and subgradients in smooth spaces

convergence of a continuous approach for zero one programming problems abstract in this paper a new continuous formulation for the zero one programming problem is presented followed by an investigation of the algorithm for it this paper first reformulates the zero one programming problem as an equivalent mathematical programs with complementarity constraints then as a smooth ordinary nonlinear programming problem with the help of the fischer burmeister function after that the augmented lagrangian method is introduced to solve the resulting continuous problem with optimality conditions for the non smooth augmented lagrangian problem derived on the basis of approximate smooth variational principle and with convergence properties established to our benefit the sequence of solutions generated converges to feasible solutions of the original problem which provides a necessary basis for the convergence results

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

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

error bounds for lower semicontinuous functions in normed spaces without the convexity or analyticity assumption we study error bounds for an inequality system defined by a general lower semicontinuous function and establish sufficient necessary conditions on the existence of error bounds in infinite dimensional normed spaces some characterizations for a convex inequality system to possess an error bound in a reflexive banach space are also given as applications in dealing with the hoffman error bound result in normed spaces we give a computable lipschitz bound constant which is better than previous lipschitz bound constants in some examples we also consider error bounds for quadratic functions on rn

quantum mechanical scattering in exterior domains with impenetrable periodic boundaries and short range potentials we study the scattering of a nonrelativistic particle in an exterior domain open connected subset r 2 containing a half space and contained in another half space and having an impenetrable periodic boundary impenetrable means that generalized homogeneous dirichlet conditions are imposed on we prove the existence and completeness of the wave operators w limt exp ith1 p exp ith0 corresponding to the scattering of a nonrelativistic particle in by the combined effect of the boundary and a short range potential present in here h0 is the negative distributional laplacian in the hilbert space h0 l2 r h1 d v d being the dirichlet laplacian in the hilbert space h l2 v an operator of multiplication in h by a bounded measurable function v x on having the periodicity of the boundary and p h0 h an identification operator the operators w model the quantum mechanical scattering of low energy atoms by crystal surfaces with v modeling the interaction between the

morse type information on palais smale sequences obtained by min max principles in the context of the min max approach to critical point theory but without the usual compactness assumptions a la palais smale and the nondegeneracy conditions a la fredholm we construct almost critical points with two sided estimates on their approximate morse indices which are arbitrarily close to certain a priori given dual sets this additional topological and analytical information about almost critical sequences can sometimes be crucial in the proof of their convergence and therefore in solving the corresponding variational problem 1994 john wiley sons inc

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

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

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

nonsmooth analysis on smooth manifolds we study infinitesimal properties of nonsmooth nondifferentiable functions on smooth manifolds the eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function a subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems nonclassical problems of calculus of variations and generalized solutions of first order partial differential equations on manifolds we also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions

morse sard type results in sub riemannian geometry let m g be a sub riemannian manifold and x0 m assuming that chow s condition holds and that m endowed with the sub riemannian distance is complete we prove that there exists a dense subset n1 of m such that for every point x of n1 there is a unique minimizing path steering x0 to x this trajectory admitting a normal extremal lift if the distribution is everywhere of corank one we prove the existence of a subset n2 of m of full lebesgue measure such that for every point x of n2 there exists a minimizing path steering x0 to x which admits a normal extremal lift is nonsingular and the point x is not conjugate to x0 in particular the image of the sub riemannian exponential mapping is dense in m and in the case of corank one is of full lebesgue measure in m