fuzzy differential equation with nonlocal condition we shall prove the existence and uniqueness theorem of a solution to the nonlocal fuzzy differential equation using the contraction mapping principle

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

gauss curvature flow the fate of the rolling stones we prove firey s 1974 conjecture that convex surfaces moving by their gauss curvature become spherical as they contract to points

the lp minkowski problem and the minkowski problem in centroaffine geometry abstract the l p minkowski problem introduced by lutwak is solved for p n 1 in the smooth category the relevant monge ampere equation 0 1 is solved for all p 1 the same equation for p 1 is also studied and solved for p n 1 1 when p n 1 the equation is interpreted as a minkowski problem in centroaffine geometry a kazdan warner type obstruction for this problem is obtained

the two dimensional lp minkowski problem and nonlinear equations with negative exponents abstract in this paper we present various existence results for nonlinear differential equations related to the l p minkowski problem in the plane and the one dimensional conformal curvature problem in particular the supercritical case that is p 2 for the l p minkowski problem with sign changing data is addressed a new geometric sharp inequality is also obtained

the isoperimetric problem for pinwheel tilings in aperiodic pinwheel tilings of the plane there exist unions of tiles with ratio area perimeter 2 arbitrarily close to that of a circle such approximate circles can be constructed with arbitrary center and any sufficiently large radius the existence of such circles follows from the metric on pinwheel space being almost euclidean at large distances ifp andq are points separated by large euclidean distancer then the shortest path along tile edges fromp toq has lengthr o r

existence of convex hypersurfaces with prescribed gauss kronecker curvature let f x be a given positive function in rn 1 in this paper we consider the existence of convex closed hypersurfaces x so that its gausskronecker curvature at x x is equal to f x this problem has variational structure and the existence of stable solutions has been discussed by tso j diff geom 34 1991 389 410 using the mountain pass lemma and the gauss curvature flow we prove the existence of unstable solutions to the problem

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

on the _ minkowski problem a volume normalized formulation of the lp minkowski problem is presented this formulation has the advantage that a solution is possible for all p 1 including the degenerate case where the index p is equal to the dimension of the ambient space a new approach to the lp minkowski problem is presented which solves the volume normalized formulation for even data and all p 1

nonuniqueness of solutions to the l p minkowski problem abstract in this paper we show by example that there is no uniform estimate for the l p minkowski problem as a result we obtain the nonuniqueness of solutions to the problem

rotationally symmetric solutions to the l p minkowski problem abstract in this paper we study the l p minkowski problem for p n 1 which corresponds to the critical exponent in the blaschke santalo inequality we first obtain volume estimates for general solutions then establish a priori estimates for rotationally symmetric solutions by using a kazdan warner type obstruction finally we give sufficient conditions for the existence of rotationally symmetric solutions by a blow up analysis we also include an existence result for the l p minkowski problem which corresponds to the super critical case of the blaschke santalo inequality

a priori estimates and existence of solutions to the prescribed centroaffine curvature problem abstract in this paper we study the prescribed centroaffine curvature problem in the euclidean space r n 1 this problem is equivalent to solving a monge ampere equation on the unit sphere it corresponds to the critical case of the blaschke santalo inequality by approximation from the subcritical case and using an obstruction condition and a blow up analysis we obtain sufficient conditions for the a priori estimates and the existence of solutions up to a lagrange multiplier

on solvability of two dimensional lp minkowski problem abstract given p 0 and a positive continuous function g with g x t g x for some 0 0 the functional f u 0 t u x 2 u 2 x dx c 0 t g x u p x dx 2 p has a minimizer in the class of positive functions u c1 r for which u x t u x for all x r this minimizer is used to prove the existence of a positive periodic solution y c2 r of two dimensional lp minkowski problem y1 p x y x y x g x where p 0 2

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

a note on fuzzy volterra integral equations abstract in this paper existence theorems for certain volterra integral equations involving fuzzy set valued mappings whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in r n are obtained

new approach for studying nonlocal problems related to differential systems and partial differential equations in generalized fuzzy metric spaces abstract this study considers the solvability of nonlocal problems for fuzzy differential systems under gh differentiability using some linear transforms we convert partial differential equations into fuzzy integral systems before some similar results are obtained for fuzzy wave equations with nonlocal conditions as a consequence we give some further estimates of the convergence rates of approximation schemes to exact solutions the main tool used in our proofs is based on perov s fixed point principle which is applied to vector valued integral operators these operators appear as the sum of two integral operators where one is fredholm type and the other is of a volterra type that depends on the restriction of the domain the novel feature of this study is that we combine the matrix convergent to zero technique with calculations of fuzzy valued functions in generalized fuzzy metric spaces some examples are presented to demonstrate our theoretical results