comparison results and estimates on the gradient without strict convexity in this paper we establish a comparison result for solutions to the problem mbox minimize int omega f nabla u x dx quad hbox on u u u 0 in w 1 1 0 omega

existence of equilibrium for generalized abstract fuzzy economies in this paper we introduce and study a new class of generalized abstract fuzzy economies we prove some new existence theorems of equilibrium and maximal element for generalized abstract fuzzy economies with uncountable number of agents and qualitative fuzzy games respectively

variational problems for a class of functionals on convex domains abstract let be a bounded convex open subset of r n n 2 and let j be the integral functional j u f du x u x dx where f 0 r is a lower semicontinuous function possibly nonconvex and with linear growth we prove that the functional j admits a unique minimizer in the space of w1 10 functions that depend only on the distance from the boundary of provided that the ratio between the lebesgue measure of and the n 1 dimensional hausdorff measure of is strictly less than a constant related to the growth of f at infinity

optimal paths in strong and weak disorder a unified approach we present a unified scaling theory for the optimal path connecting opposite edges of a disordered lattice of size l each bond of the lattice is assigned a cost expar where r is a uniformly distributed random variable and a is disorder strength the optimal path minimizes the sum of the costs of the bonds along the path we argue that for la where is the correlation exponent of percolation the path becomes equivalent to a directed polymer on an effective lattice consisting of blobs of size a it is self affined and characterized by the roughness exponent of directed polymers for la or on the length scales below the blob size the path behaves as an optimal path in the strong disorder limit it has a self similar fractal shape with fractal dimension dopt we derived the scaling relations for the length of the path its transversal displacement the average cost and its fluctuation we test our scaling theoretical predictions by numerical simulations on a square lattice

existence of minima for nonconvex functionals in spaces of functions depending on the distance from the boundary we prove that some nonconvex functionals admit a unique minimum in a functional space of functions which depend only on the distance from the boundary of the plane domain where they are defined the domains considered are disks and regular polygons we prove that the sequence of minima of the functional on the polygons converges to the unique minimum on the circumscribed disk as the number of sides tends to infinity our method also allows us to determine the explicit form of the minima

regularized gap functions and error bounds for generalized mixed weak vector quasivariational inequality problems in fuzzy environments abstract in this paper we establish gap functions for a new class of generalized mixed weak vector quasivariational inequality problems in fuzzy environments in short mqvipf these functions are constructed through terms of regularized gap functions using the method of nonlinear scalarization functions moreover error bounds are provided for mqvipf via regularized gap functions under suitable assumptions the main results obtained in this paper improve and extend some corresponding known results some examples are given to illustrate our results

coincidence theorems and variational inequalities for fuzzy mappings abstract in this paper some coincidence theorems and existence theorems of solutions of variational inequalities for fuzzy mappings are established

algorithm of solutions for mixed implicit quasi variational inequalities with fuzzy mappings abstract in this paper by applying the auxiliary variational principle technique some existence theorems of solutions for a class of mixed implicit quasi variational inequalities with fuzzy mappings are proved in hilbert spaces a novel and innovative iterative algorithm to compute approximate solutions is suggested and analyzed the convergence criteria is also given as special cases of these results the open problem put forward by noor is answered positively our results are new and generalize a number of known results to mixed implicity quasi variational inequalities with fuzzy mappings

on the minimum problem for a class of noncoercive nonconvex functionals we are concerned with the problem of existence of solutions to the variational problem min left int 0 r g t v t dt v in ac 0 r v r 0 right with only one fixed endpoint prescribed the map g colon 0 r times mathbb r to overline mathbb r is a normal integrand for which neither convexity nor superlinear growth conditions are assumed as an application we give an existence result for the radially symmetric variational problem min u in wuu b r int b r f pt mod x mod nabla u x a x u x dx where b r is the ball of mathbb r n centered at the origin and with radius r 0 the map f colon 0 r times 0 infty to overline mathbb r is a normal integrand and a in l 0 r again neither convexity nor superlinear growth conditions are made on f these kinds of problems with nonconvex lagrangians with respect to nabla u arise in different fields of mathematical physics such as optimal design and nonlinear elasticity

minimizing non convex multiple integrals a density result we consider variational problems of the form where is a bounded open set in r n f r n r is a possibly non convex lower semicontinuous function with p growth at infinity for some 1 p u 0 is any function in w 1 p assuming that the convex envelope f of f is affine on each connected component of the set f f we prove the existence of solutions to p for every continuous function g such that i g has no strict local minima and ii every convergent sequence of extremum points of g eventually belongs to an interval where g is constant thus showing that the set of continuous functions g that yield existence to p is dense in the space of continuous functions on r

existence of vector quasi variational like inequalities for fuzzy mappings abstract the purpose of this paper is to study the existence problem of solutions for vector quasi variational like inequalities in fuzzy locally convex hausdorff topological vector spaces by using maximal element theorem and under suitable conditions some existence theorems for two types of vector quasi variational like inequalities are proved

a uniqueness result for a class of non strictly convex variational problems abstract let be a smooth domain in r 2 we prove that if g 0 0 is convex with g 0 g t whenever t 0 then there exists an unique minimizer u c 0 1 of the functional u g u d x d y among all lipschitz continuous functions that assume the same value of u on

efficiency scalability and robustness of parallel multilevel methods for nonlinear partial differential equations in this paper we compare the performance scalability and robustness of different parallel algorithms for the numerical solution of nonlinear boundary value problems arising in magnetic field computation and solid mechanics these problems are discretized by using the finite element method with triangular meshes and piecewise linear functions the nonlinearity is handled by a nested newton solver and the linear systems of algebraic equations within each newton step are solved by means of various iterative solvers namely multigrid methods and conjugate gradient methods with preconditioners based on domain decomposition multigrid or bpx techniques respectively the basis of the implementation of all solvers is a nonoverlapping domain decomposition data structure such that they are well suited for parallel machines with mimd architecture

nonconvex minimization problems for functionals defined on vector valued functions abstract we consider the minimization problem min v w 1 1 0 b n r r m b n r f v x h v x dx where b n r is the ball of r n centered at the origin and with radius r 0 f is a lower semicontinuous function and h is a convex function we give sufficient conditions for the existence and uniqueness of minimizers our technique relies on a detailed knowledge of the properties of the solutions to the convexified problem obtained using the corresponding euler lagrange inclusions

theory of minimum spanning trees i mean field theory and strongly disordered spin glass model the minimum spanning tree mst is a combinatorial optimization problem given a connected graph with a real weight cost on each edge find the spanning tree that minimizes the sum of the total cost of the occupied edges we consider the random mst in which the edge costs are quenched independent random variables there is a strongly disordered spin glass model due to newman and stein phys rev lett 72 2286 1994 which maps precisely onto the random mst we study scaling properties of random msts using a relation between kruskal s greedy algorithm for finding the mst and bond percolation we solve the random mst problem on the bethe lattice bl with appropriate wired boundary conditions and calculate the fractal dimension d 6 of the connected components viewed as a mean field theory the result implies that on a lattice in euclidean space of dimension d there are of order w d ensuremath d large connected components of the random mst inside a window of size w and that d d c d 6 is a critical dimension this differs from the value 8 suggested by newman and stein we also critique the original argument for 8 and provide an improved scaling argument that again yields d c 6 the result implies that the strongly disordered spin glass model has many ground states for dg6 and only of order one below six the results for msts also apply on the poisson weighted infinite tree which is a mean field approach to the continuum model of msts in euclidean space and is a limit of the bl in a companion paper we develop an ensuremath epsilon 6 ensuremath d expansion for the random mst on critical percolation clusters

on random variational inclusions with random fuzzy mappings and random relaxed cocoercive mappings in this paper we introduce and study the random variational inclusions with random fuzzy and random relaxed cocoercive mappings we define an iterative algorithm for finding the approximate solutions of this class of variational inclusions and establish the convergence of iterative sequences generated by proposed algorithm our results improve and generalize many known corresponding results

existence of solutions for a class of non convex minimum problems p celada1 s perrotta2 g treu3 1 dipartimento di scienze matematiche universita degli studi di trieste p le europa 1 i 34127 trieste italia e mail celada univ trieste it 2 dipartimento di matematica pura ed applicata g vitali universita degli studi di modena via campi 213 b i 41100 modena italia e mail perrotta c220 unimo it 3 dipartimento di matematica ed informatica universita degli studi di udine v le delle scienze 206 i 33100 udine italia e mail treu dimi uniud it

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

analysis of a damped nonlinear multilevel method in this paper we present a new algorithm that is obtained by introducing a damping parameter in the classical nonlinear multilevel method we analyse this damped nonlinear multilevel method in particular we prove global convergence and local efficiency for a suitable class of problems

an approach for solving fuzzy implicit variational inequalities with linear membership functions in this paper we consider a class of fuzzy implicit variational inequalities with linear membership functions by using the tolerance approach we show that solving such problems can be reduced to a semi infinite programming problem a version of the method of centres with entropic regularization techniques only used a quasi newton line search using matlab software is required in our implementation we also give a numerical example to illustrate the validity of our approach

generalized vector variational like inequalities in fuzzy environment abstract the purpose of this paper is to introduce and study a new class of generalized vector variational like inequalities and generalized vector variational inequalities in fuzzy environment by using the kkm technique maximal element lemma and under suitable conditions some existence theorems of solutions for these two kinds of vector variational inequalities in fuzzy environment are proved the results presented in the paper extend and improve some recent results announced by some authors

a convergent adaptive finite element method for an optimal design problem the optimal design problem for maximal torsion stiffness of an infinite bar of given geometry and unknown distribution of two materials of prescribed amounts is one model example in topology optimisation it eventually leads to a degenerate convex minimisation problem the numerical analysis is therefore delicate for possibly multiple primal variables u but unique derivatives dw d u even fine a posteriori error estimates still suffer from the reliability efficiency gap however it motivates a simple edge based adaptive mesh refining algorithm afem that is not a priori guaranteed to refine everywhere its convergence proof is therefore based on energy estimates and some refined convexity control numerical experiments illustrate even nearly optimal convergence rates of the proposed afem

existence and regularity of minimizers of nonconvex functionals depending onuand u abstract we consider variational problems of the form min f u x g x u x dx u u 0 h 1 0 wheref r n 0 is a possibly nonconvex function with quadratic growth at infinity andg x u is lipschitz continuous and strictly increasing decreasing inu we prove the existence and local lipschitz regularity of solutions for every boundary datumu0 h1 l on the basis of the structure of the epigraph of the convex envelope off

generalized variational inequalities for fuzzy maps abstract the variational inequality problem for fuzzy maps was first studied by chang and zhu 1989 chang made further study for this kind of problem in 2 1991 this paper deals with the existence of solutions to generalized variational inequalities for fuzzy maps and then obtains some applications to fixed point and inclusion theorems the work is an extension of corresponding work of chang and zhu 1989 and shi and tan 1989

generalized vector complementarity problem in fuzzy environment abstract this paper is devoted to introduce and study a new class of generalized vector complementarity problems gvcp for short and generalized vector variational inequalities gvvi for short in fuzzy environment under suitable conditions we prove the equivalence between gvcp and gvvi in banach spaces then without any monotonicity assumption we apply kkm technique to establish an existence theorem of solutions for gvvi finally we show that the solution set of gvcp is nonempty and closed

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

random generalized nonlinear variational inclusions for random fuzzy mappings in this paper we introduce and study a new class of random set valued nonlinear generalized variational inclusions with noncompact valued random fuzzy mappings and construct a new iterative algorithm we also discuss the existence of random solutions for this class of random fuzzy variational inclusions and the convergence of random iterative sequences generated by the algorithm

variational inequalities for fuzzy mappings ii in this paper we establish the equivalence between the variational inequalities for fuzzy mappings and the wiener hopf equations for fuzzy mappings we use this equivalence to suggest a number of new iterative algorithms for solving the variational inequalities for fuzzy mappings we also study the convergence criteria of these iterative algorithms the auxiliary principle technique is used to study the existence of a solution of the variational inequality for fuzzy mappings and to suggest a novel and innovative iterative algorithm for computing the approximate solution the results proved in this paper represent an improvement of previously known results

smoothness properties for the optimal mixture of two isotropic materials the compliance and eigenvalue problems the present paper is devoted to obtaining some smoothness results for the solution of two classical control problems relative to the optimal mixture of two isotropic materials in the first one the goal is to maximize the energy in the second one we want to minimize the first eigenvalue of the corresponding elliptic operator at least for the first problem it is well known that it does not have a solution in general thus we deal with a relaxed formulation one of the applications of our results is in fact the nonexistence of a solution for the unrelaxed problem in this sense we improve a classical nonexistence result by murat and tartar for the maximization of the energy which was obtained assuming the solution smooth we also get a counterexample to the existence of a solution for the eigenvalue problem which to our knowledge was an open problem

on the convergence of finite element approximations of a relaxed variational problem the accuracy of finite element approximations to a convex but not strictly convex variational problem is considered convergence is proved for a finite element approximation of a particular vector field related to the solution in a special one dimensional case 0 h convergence is shown for a piecewise linear approximation of the derivative h denotes the size of each element domain numerical results are also presented for this one dimensional case

on radially symmetric minima of nonconvex functionals abstract we study a minimization problem in the space w1 10 br where br is the ball of radius r with center at the origin the functional considered is not necessarily convex under suitable assumptions we prove the existence of a radially symmetric decreasing solution by strengthening the assumptions we obtain uniqueness results finally we study under which assumptions and in which sense the solutions found solve the corresponding euler equation the proofs are very direct and simple they only make use of the functions t n introduced by the author arch rational mech anal 1999

iterative algorithm for generalized quasi variational like inclusions with fuzzy mappings in banach spaces in this paper we introduce and study a class of generalized quasi variational like inclusions with fuzzy mappings in real banach spaces and show its equivalence with a class of relations using this equivalence we suggest and analyze an iterative algorithm for finding the approximate solution of this class of inclusions further we prove the existence of solution of this class of inclusions and discuss the convergence criteria of iterative sequences generated by the iterative algorithm

a sharp upper bound for the torsional rigidity of rods by means of web functions using web functions we approximate the dirichlet integral which represents the torsional rigidity of a cylindrical rod with planar convex cross section to this end we use a suitably defined piercing function which enables us to obtain bounds for both the approximate and the exact value of the torsional rigidity when varies we show that the ratio between these two values is always larger than we prove that this is a sharp lower bound and that it is not attained several extremal cases are also analyzed and studied by numerical methods

an optimal design problem for a two phase isolating material in the wall of a cavity abstract we are interested in the optimal distribution of two isolating materials in the wall of a cavity in order to get the best isolation when the amount of the best isolating material is limited we assume that the wall is of width e 0 small the problem can be formulated as the minimization of the first eigenvalue of a certain diffusion operator where the diffusion constant is of order one inside the cavity and of order e in the wall as it is usual in optimal design the problem has not solution in general and therefore it is necessary to work with a relaxed formulation passing to the limit when e tends to zero we get an asymptotic model where the variable control is now in a robin boundary condition instead of the diffusion operator we also present some numerical simulations showing the behavior of the solutions

existence uniqueness and qualitative properties of minima to radially symmetric non coercive non convex variational problems we are concerned with the problem of existence uniqueness and qualitative properties of solutions to the radially symmetric variational problem min u in wuu b r int b r pq f pt mod x mod nabla u x h x u x dx where b r is the ball of mathbb r n centered at the origin and with radius r 0 the map f colon 0 r times 0 infty to overline mathbb r is a normal integrand and h colon 0 r times r to r is a convex function of the second variable this kind of problems with non convex lagrangians with respect to nabla u arise in various fields of applied sciences such as optimal design and nonlinear elasticity

existence results of a perturbed variational inequality with a fuzzy mapping abstract this paper is devoted to the existence of a perturbed variational inequality with a fuzzy mapping in a finite dimensional space we introduce two perturbation methods for a fuzzy mapping in one method it is perturbed by a nonlinear mapping and in the other it is perturbed by a vector under suitable coercivity conditions we show that the perturbed variational inequality with a fuzzy mapping has a solution when perturbed by both of the methods mentioned above the main results presented in this paper generalize and improve some known results