on a class of fuzzy functional differential equations in this paper we establish the local and global existence and uniqueness results for fuzzy functional differential equations for the local existence and uniqueness we use the method of successive approximations and for global existence and uniqueness we use the contraction principle also we apply these results to fuzzy differential equations with distributed delays and fuzzy population models

iterative bregman projections for regularized transportation problems this article details a general numerical framework to approximate so lutions to linear programs related to optimal transport the general idea is to introduce an entropic regularization of the initial linear program this regularized problem corresponds to a kullback leibler bregman di vergence projection of a vector representing some initial joint distribu tion on the polytope of constraints we show that for many problems related to optimal transport the set of linear constraints can be split in an intersection of a few simple constraints for which the projections can be computed in closed form this allows us to make use of iterative bregman projections when there are only equality constraints or more generally bregman dykstra iterations when inequality constraints are in volved we illustrate the usefulness of this approach to several variational problems related to optimal transport barycenters for the optimal trans port metric tomographic reconstruction multi marginal optimal trans port and in particular its application to brenier s relaxed solutions of in compressible euler equations partial un balanced optimal transport and optimal transport with capacity constraints

density functional theory and optimal transportation with coulomb cost we present here novel insight into exchange correlation functionals in density functional theory based on the viewpoint of optimal transport we show that in the case of two electrons and in the semiclassical limit the exact exchange correlation functional reduces to a very interesting functional of novel form which depends on an optimal transport map t associated with a given density since the above limit is strongly correlated the limit functional yields insight into electron correlations we prove the existence and uniqueness of such an optimal map for any number of electrons and each and determine the map explicitly in the case when is radially symmetric

periodic boundary value problems for first order linear differential equations with uncertainty under generalized differentiability we study the existence of solutions to a class of first order linear fuzzy differential equations subject to periodic boundary conditions from the point of view of generalized differentiability the objective of this paper is to show that fuzzy differential equations under generalized differentiability can be used in the study of periodic phenomena by considering a combination of two types of derivatives with a switching point we provide sufficient conditions which guarantee that the piecewise defined solutions match adequately and illustrate through some examples the process of construction and calculation of solutions

open fuzzy cubature rule with application to nonlinear fuzzy volterra integral equations in two dimensions abstract in this paper we propose an open fuzzy cubature formula and as application we construct an iterative numerical method for solving nonlinear fuzzy volterra integral equations in two dimensions the convergence of the method is proved by providing the error estimate and it is tested on two numerical experiments

fractional calculus for interval valued functions abstract we use a generalization of the hukuhara difference for closed intervals on the real line to develop a theory of the fractional calculus for interval valued functions the properties of riemann liouville fractional integral riemann liouville fractional derivative and caputo fractional derivative for interval valued functions are investigated several examples are presented to illustrate the concepts and results

brief note on the variation of constants formula for fuzzy differential equations this note gives a theory of state transition matrices for linear systems of fuzzy differential equations this is used to give a fuzzy version of the classical variation of constants formula a simple example of a time independent control system is used to illustrate the methods while similar problems to the crisp case arise for time dependent systems in time independent cases the calculations are elementary solutions of eigenvalue eigenvector problems in particular for nonnegative or nonpositive matrices the problems at each level set can easily be solved in matlab to give the level sets of the fuzzy solution

on the existence of solutions to periodic boundary value problems for fuzzy linear differential equations in this work we provide sufficient conditions which guarantee the existence of solutions to periodic boundary value problems for first order linear fuzzy differential equations by using generalized differentiability and switching points in comparison with some previous works we consider equations whose coefficient may change its sign a finite number of times in the interval of interest we also study the existence of solutions which are crisp or real at the switching points where the diameter of the level sets changes from nonincreasing to nondecreasing character

i optimal curve for impulsive lotka volterra predator prey model abstract for the classical lotka volterra predator prey system new notion i optimal curve i is introduced this curve is disposed in the phase space of the system the curve i intersects each trajectory c of lotka volterra system at least once the points of i possess the following optimal property if m m i c0 then after a jump with magnitude i to the origin of coordinates it hits a trajectory c1 and c 1 is minimal i e c 1 is the nearest to the stable centre the minimality concerns the rest points of initial trajectory c 0 from which the impulsive jumps subtractings with magnitude i to 0 0 are realized the monotonicity continuity and linear asymptotical behaviour of i curve are proved

on differential equations with interactive fuzzy parameter via t norms abstract in this paper we study fuzzy differential equations with uncertain parameters modeled by interactive fuzzy numbers the interactivity is formalized with the help of upper semicontinuous t norms to obtain solutions to fuzzy differential equations we use two different approaches the first uses a family of differential inclusions while the second is the fuzzification of the deterministic problem using the zadeh extension principle we show that the solutions obtained by the two methods are equal interactivity among the parameters and variable of the problem are studies with respect two methods we show that a hierarchy in uncertainties of the solutions arises when we chose one of the basic t norm the minimum the t norm of the product the t norm of lukasiewicz and t norm of the drastic product to model the interaction between parameters finally to illustrate the concepts introduced in the paper we study the malthusian model with interactive parameters

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

fuzzy fractional functional integral and differential equations this paper is devoted to considering the existence and uniqueness results for fuzzy fractional functional integral equations employing the contraction principle moreover the approach is followed to prove the existence and uniqueness of solutions for the fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives finally the solution to fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives by a modified fractional euler method mfem is presented the method is illustrated by solving some examples

generalized differentiability of fuzzy valued functions in the present paper using novel generalizations of the hukuhara difference for fuzzy sets we introduce and study new generalized differentiability concepts for fuzzy valued functions several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them characterization and relatively simple expressions are provided for the new derivatives

the cauchy problem for continuous fuzzy differential equations abstract we prove a version of the classical peano s theorem for the initial value problem for a fuzzy differential equation in the metric space of normal fuzzy convex sets with distance given by the maximum of the hausdorff distances between level sets

initial and boundary value problems for fuzzy differential equations we prove a wintner type result for fuzzy ivps and a superlinear type result for fuzzy bvps both results rely on a generalized schauder theorem in metric spaces

global existence of solutions to fuzzy differential equations abstract the local existence and uniqueness theorems are investigated by wu et al j math anal appl 202 1996 629 644 for the cauchy problem of the fuzzy valued mappings of a real variable whose values are in the fuzzy number space e n d in this paper we point out a variety of results which assure global existence of solutions to fuzzy differential equations

almost periodic solutions of fuzzy systems we prove the existence of almost periodic solutions and asymptotically almost periodic solutions for the fuzzy functional differential equations moreover we consider uniform stable and uniformly asymptotically stable of almost periodic solutions for the fuzzy system

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

on random fuzzy functional differential equations in this paper we prove the existence and uniqueness result for random fuzzy functional differential equations for the existence and uniqueness we use the method of successive approximations some kind of boundedness of the solution is established

towards fuzzy differential calculus part 1 integration of fuzzy mappings abstract this paper deals with fuzzy set valued mappings of the real line and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line these mappings can also be viewed as fuzzy relations using zadeh s extension principle the integral of such fuzzy mappings over a crisp interval is defined provided a special analytical representation of the fuzzy mapping the practical computation of such an integral is shown to be easy practically speaking it yields the fuzzy surface of a fuzzily bounded area

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

picone s formula for linear non selfadjoint impulsive differential equations in this paper we derive a picone type formula for second order linear non selfadjoint impulsive differential equations having fixed moments of impulse actions and obtain a wirtinger type inequality a leighton type comparison theorem and a sturm picone comparison theorem for such equations moreover several oscillation criteria are also derived as applications

a new bernoulli wavelet method for accurate solutions of nonlinear fuzzy hammerstein volterra delay integral equations in this article bernoulli wavelet method has been developed to solve nonlinear fuzzy hammerstein volterra integral equations with constant delay this type of integral equation has a particular case the fuzzy variant of a mathematical model from epidemiology bernoulli wavelets have been generated by dilation and translation of bernoulli polynomials the properties of bernoulli wavelets and bernoulli polynomials are first presented the present wavelet method reduces these integral equations to a system of nonlinear algebraic equations and again these algebraic systems have been solved numerically by newton s method convergence analysis of the present method has been discussed in this article also the results obtained by present wavelet method have been compared with that of by b spline wavelet method some illustrative examples have been demonstrated to show the applicability and accuracy of the present method

the multiconfiguration equations for atoms and molecules charge quantization and existence of solutions we prove the existence of ground state solutions for the multiconfiguration self consistent field equations for atoms and molecules whenever the total nuclear charge z exceeds n 1 where n is the number of electrons moreover we show that for arbitrary values of z and n the scattering charge i e the asymptotic amount of charge lost by an energy minimizing sequence is integer quantized our analysis applies to the mc equations of arbitrary rank as special cases we recover in a new and unified way the existence theorems of zhislin zh60 for the n body schrodinger equation infinite rank mc and of lieb simon ls77 for the hartree fock equations rank n mc our approach is a direct study of an invariant orbital free formulation in n body space of the underlying variational principle proofs involve i the geometric n body localization methods for the linear schrodinger equation first introduced by enss en77 and developed in sim77 sig82 which can be adapted to become powerful tools in nonlinear many body theory as well ii weak convergence methods from the theory of nonlinear partial differential equations iii careful analysis of the structure of the one and two body density matrices of the bound and scattering fragments delivered by geometric localization which allows us to overcome the fact that the rank of the fragments is not reduced by localization

exact solutions to the transportation problem on the line given distributions w of production and v of consumption on the line the monge kantorovich problem is to decide which producer should supply each consumer in order to minimize the total transportation costs here cost will be assumed to be a strictly concave function of the distance which translates into an economy of scale for longer trips and may encourage cross hauling the resulting solutions display a hierarchical structure that reflects a striking separation into local and global scales also found in the real world moreover this structure can be exploited to reduce the infinite dimensional linear problem to a convex minimization in m variables where 2 m 2 counts the number of times that w v changes sign a combinatorial algorithm is then derived which yields exact solutions by optimizing a certain finite sequence of convex separable network flows

stability and periodicity in fuzzy differential equations formulations of fuzzy differential equations des in terms of the hukuhara derivative do not reproduce the rich and varied behavior of crisp des another interpretation in terms of differential inclusions expressed level setwise overcomes much of this deficiency and opens up for profitable investigation such properties as stability attraction periodicity and the like this is especially important for investigating continuous systems which are uncertain or incompletely specified this paper studies studies lyapunov stability of fuzzy des and the periodicity of the fuzzy solution set for both the time dependent and autonomous cases

on the convergence of fuzzy sets three kind of convergences are defined on a space of fuzzy sets relationships between these convergences are studied some comments on the fixed point property are represented

some results on boundary value problems for fuzzy differential equations with functional dependence abstract we calculate the exact solution for a class of boundary value problems for first order linear fuzzy differential equations with impulses under hukuhara differentiability by finding green s function some conclusions can be derived for impulsive fuzzy differential equations including a term with functional dependence for which existence and uniqueness results are proved

perron s theorem for linear impulsive differential equations with distributed delay in this paper it is shown that under a perron condition trivial solution of linear impulsive differential equation with distributed delay is uniformly asymptotically stable

periodic boundary value problems for impulsive fuzzy differential equations the peculiarity of the hukuhara derivative makes it impossible to find periodic solutions for fuzzy differential equations with the exception of very restrictive situations in this work we consider a boundary value problem associated with an impulsive fuzzy differential equation and approximate the extremal solutions in a fuzzy functional interval using the monotone method fuzzy comparison results are useful in our procedure and the expression of the solution for some impulsive periodic linear differential problems is also provided

membership induced on manifolds by vector fields and flows abstract in many engineering and scientific applications the need arises to perform differential calculations on manifolds if such calculations involve imprecise data methods for computing with such data must be determined definitions and results are given for memberships on manifolds induced by differentiation along vector fields and by flows of vector fields with uncertain parameters examples are presented which illustrate the concepts

fuzzy delay differential equations under generalized differentiability abstract we interpret a fuzzy delay differential equation using the concept of generalized differentiability in this setting we prove the existence of two fuzzy solutions each one corresponding to a different type of differentiability uniqueness is understood in the sense that the solutions considered do not have switching points the applicability of the theoretical results is illustrated with some real world examples

on the structure of solutions for fuzzy initial value problem in this paper by means of banach fixed point theorem we establish the existence and uniqueness of solutions for set differential equations the continuous dependence of solutions on initial values and the structural stability of solutions on the basis of these results we define level sets of solutions to fuzzy differential equations by the solutions of set differential equations and then we obtain the corresponding results for the large solutions of the fuzzy initial value problem fivp finally discuss the relationship between small solutions and large solutions of fivp

multi marginal optimal transport on riemannian manifolds we study a multi marginal optimal transportation problem on a riemannian manifold with cost function given by the average distance squared from multiple points to their barycenter under a standard regularity condition on the first marginal we prove that the optimal measure is unique and concentrated on the graph of a function over the first variable thus inducing a monge solution this result generalizes mccann s polar factorization theorem on manifolds from two to several marginals in the same sense that a well known result of gangbo and swiech generalizes brenier s polar factorization theorem on bbb r n

impulsive functional differential inclusions and fuzzy population models in this paper we shall establish some existence results for the impulsive functional differential inclusion and the fuzzy impulsive functional differential equation with some conditions and study the properties of the solution set and the attainable set finally the results will be used to fuzzy population models

on retarded fuzzy functional differential equations and nonabsolute fuzzy integrals abstract we prove a generalized convergence theorem for fuzzy henstock integrals based on the sequence of fuzzy number valued functions which are weak generalized uniformly absolute continuous u a c g to widen the applications of this convergence theorem we provide some existence theorems for the generalized solution for retarded fuzzy functional differential equations and the dependence of the solutions on a parameter under the assumption of the strong generalized derivative of the solution our results generalize existence results in a kaleva integral setting to a fuzzy henstock integral setting

numerical solutions of fuzzy differential equations by an efficient runge kutta method with generalized differentiability in this paper an extended fourth order runge kutta method is studied to approximate the solutions of first order fuzzy differential equations using a generalized characterization theorem in this method new parameters are utilized in order to enhance the order of accuracy of the solutions using evaluations of both f and f instead of using the evaluations of f only the proposed extended runge kutta method and its error analysis which guarantees pointwise convergence are given in detail furthermore the accuracy and efficiency of the proposed method are demonstrated in a series of numerical experiments

smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the hohenberg kohn functional we prove rigorously that the exact n electron hohenberg kohn density functional converges in the strongly interacting limit to the strictly correlated electrons sce functional and that the absolute value squared of the associated constrained search wavefunction tends weakly in the sense of probability measures to a minimizer of the multi marginal optimal transport problem with coulomb cost associated to the sce functional this extends our previous work for n 2 cotar etal in commun pure appl math 66 548 599 2013 the correct limit problem has been derived in the physics literature by seidl phys rev a 60 4387 4395 1999 and seidl gorigiorgi and savin phys rev a 75 042511 1 12 2007 in these papers the lack of a rigorous proofwas pointed out we also give amathematical counterexample to this type of result by replacing the constraint of given one body density an infinite dimensional quadratic expression in the wavefunction by an infinite dimensional quadratic expression in the wavefunction and its gradient connections with the lawrentiev phenomenon in the calculus of variations are indicated

numerical solutions of the nonlinear fuzzy hammerstein volterra delay integral equations in this paper an iterative numerical method that solves nonlinear fuzzy hammerstein volterra integral equations with constant delay is developed by using the error estimates a practical stopping criterion of the algorithm is obtained the convergence and the numerical stability of the method is proved and its accuracy is illustrated on three numerical examples the study of this integral equation is important because it has as a particular case the fuzzy variant of a mathematical model from epidemiology

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

optimal transportation with infinitely many marginals abstract we formulate and study an optimal transportation problem with infinitely many marginals this is a natural extension of the multi marginal problem studied by gangbo and wi ch 1998 15 we prove results on the existence uniqueness and characterization of the optimizer which are natural extensions of the results in gangbo and wi ch 1998 15 the proof relies on a relationship between this problem and the problem of finding barycenters in the wasserstein space a connection first observed for finitely many marginals by agueh and carlier 2011 1

towards fuzzy differential calculus part 2 integration on fuzzy intervals abstract the problem of integrating mappings over fuzzy domains has already received some attention in the literature however there is not a unique approach but several definitions have been proposed each carrying its own meaning this paper tries to emphasize the relationships and the differences between the various points of view

existence and uniqueness of a fuzzy solution for the nonlinear fuzzy neutral functional differential equation this work is concerned with a class of nonlinear fuzzy neutral functional differential equations specifically existence and uniqueness of fuzzy solutions for the nonlinear fuzzy neutral functional differential equation where a is a fuzzy coefficient and f and g are continuous functions are established via banach fixed point analysis approach and using the fuzzy number whose values are normal convex upper semicontinuous and compactly supported interval in en

data driven optimal transport the problem of optimal transport between two distributions x and y is extended to situations where the distributions are only known through a finite number of samples xi and yj a weak formulation is proposed based on the dual of the kantorovich formulation with two main modifications replacing the expected values in the objective function by their empirical means over the xi and yj and restricting the dual variables u x and v y to a suitable set of test functions adapted to the local availability of sample points a procedure is proposed and tested for the numerical solution of this problem based on a fluidlike flow in phase space where the sample points play the role of active lagrangian markers 2016 wiley periodicals inc

existence of the solutions of fuzzy differential equations with parameters in this paper we study the existence of the solutions of fuzzy differential equations with parameters by the topological degree method

towards fuzzy differential calculus part 3 differentiation this paper deals successively with differentiation of ordinary functions at a fuzzy point and with differentiation of fuzzy valued functions at a nonfuzzy point in both cases the concept of differentiation is extended and the problem of the linearity is addressed in the second case practical methods of computation are presented in practical situations the available knowledge about mappings may be fuzzy it is the motivation of this work

fuzzy differential equations we will concentrate on the second order linear constant coefficient ordinary differential equation for x in interval i i can be 0 t for t 0 or i can be 0 the initial conditions are y 0 0 y 0 1 we assume g is continuous on i

random fuzzy fractional integral equations theoretical foundations abstract this paper presents mathematical foundations for studies of random fuzzy fractional integral equations which involve a fuzzy integral of fractional order we consider two different kinds of such equations their solutions have different geometrical properties the equations of the first kind possess solutions with trajectories of nondecreasing diameter of their consecutive values on the other hand the solutions to equations of the second kind have trajectories with nonincreasing diameter of their consecutive values firstly the existence and uniqueness of solutions is investigated this is showed by using a method of successive approximations an estimation of error of n th approximation is given also a boundedness of the solution is indicated to show well posedness of the considered theory we prove that solutions depend continuously on the data of the equations some concrete examples of random fuzzy fractional integral equations are solved explicitly

a possibilistic argument for irreversibility abstract our aim is to explicate the idea of irreversibility by a mathematical representation of irreversible processes occurring in possibilistic dynamical systems with this in mind solutions of the cauchy problem in the space of fuzzy points are examined to meet the correspondence argument solutions of the cauchy problem in the space of concentrated points are examined too

the initial value problem for interval valued second order differential equations under generalized h differentiability in this paper the interval valued second order differential equations under generalized hukuhara differentiability isdes are introduced under suitable conditions we obtain the existence and uniqueness results of solutions to isdes to prove this assertion we use idea of contraction principle and successive approximations furthermore we use the method based on properties of linear transformations ltm to find the explicit solution of initial value problem for linear second order differential equation with interval valued forcing function and with interval initial values iivp we apply the linear transformations method to two example problems including a vibrating mass and an electrical circuit finally the method based on the analysis of a solution to real valued second order differential equation is investigated to solve interval valued second order differential equations with interval valued coefficients interval valued forcing function and interval initial values some examples are presented to illustrate applicability of the proposed method

fractional discrete time diffusion equation with uncertainty applications of fuzzy discrete fractional calculus abstract this study provides some basics of fuzzy discrete fractional calculus as well as applications to fuzzy fractional discrete time equations with theories of r cut set fuzzy caputo and riemann liouville fractional differences are defined on a isolated time scale discrete leibniz integral law is given by use of w monotonicity conditions furthermore equivalent fractional sum equations are established fuzzy discrete mittag leffler functions are obtained by the picard approximation finally fractional discrete time diffusion equations with uncertainty is investigated and exact solutions are expressed in form of two kinds of fuzzy discrete mittag leffler functions this paper suggests a discrete time tool for modeling discrete fractional systems with uncertainty

next order asymptotic expansion for n marginal optimal transport with coulomb and riesz costs abstract motivated by a problem arising from density functional theory we provide the sharp next order asymptotics for a class of multimarginal optimal transport problems with cost given by singular long range pairwise interaction potentials more precisely we consider an n marginal optimal transport problem with n equal marginals supported on r d and with cost of the form i j x i x j s in this setting we determine the second order term in the n asymptotic expansion of the minimum energy for the long range interactions corresponding to all exponents 0 s d we also prove a small oscillations property for this second order energy term our results can be extended to a larger class of models than power law type radial costs such as non rotationally invariant costs the key ingredient and main novelty in our proofs is a robust extension and simplification of the fefferman gregg decomposition 20 26 extended here to our class of kernels and which provides a unified method valid across our full range of exponents our first result generalizes a recent work of lewin lieb and seiringer 36 who dealt with the second order term for the coulomb case s 1 d 3

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

hukuhara derivative of the fuzzy expected value in this paper we establish sufficient conditions to guarantee the exchange of the fuzzy expected value for a fuzzy random variable depending on certain parameter and its hukuhara derivative with respect to the parameter main results are obtained for two different sets of general conditions one of them requiring measurability conditions with respect to the product space the fuzzy random variable is defined from and the second one just requiring conditions over the projections of the fuzzy random variables

existence and uniqueness of solutions to cauchy problem of fuzzy differential equations we studied the cauchy problem of fuzzy dierential equations on the basis of introducing the concept of dierentiation which is a generalization of the denition of h dierentiability due to puri and ralescu 1983 for fuzzy set valued mappings of real variables whose values are in the fuzzy number space e n d the existence and uniqueness theorem is obtained for the cauchy problem x 0 f t x x t 0 x0 when f satises the generalized lipschitz condition c 2000

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159