an elliptic parabolic free boundary problem continuity of the interface on etablit la continuite de l interface de la solution faible d un probleme elliptique parabolique lie a la theorie des ecoulements partiellement satures dans les milieux poreux on utilise une methode basee sur le principe du maximum pour les equations paraboliques en supposant l ecoulement unidimensionnel

degenerate diffusion and the stefan problem on etudie le probleme i u t u xx f x u dans 0 1 r u 0 t 0 u 1 t 0 dans r u x 0 u 0 x dans 0 1 avec certaines hypotheses sur f et u 0 on etudie le caractere des solutions quand s annule sur un intervalle

singularly perturbed ordinary differential equations abstract we show the existence of periodic solutions for a couple of ordinary differential equations depending on a parameter e when e presents a small parameter for the majorest derivative of the first equation the key idea is to connect a coincidence degree theory with a method from a previous paper by the author j differential equations 89 1991 203 223 to avoid the use of a hard implicit function theorem of nash moser

relationships between vector variational like inequality and optimization problems abstract in this paper we will establish the relationships between vector variational like inequality and optimization problems we will be able to identify the vector critical points the weakly efficient points and the solutions of the weak vector variational like inequality problem under conditions of pseudo invexity these conditions are more general those existing in the literature

on the regularity of weak solutions of a one dimensional two phase degenerate stefan problem with convection we shall be concerned with the following one dimensional two phase stefan problem let t 0 and q 0 1 x 0 t find a pair of functions u x t s t which are defined on qt and 0 t respectively and which satisfy al u uxx f u u in x t o x s l o t t l la az ub4 u xx in x t s t x l o tdt l lb u s t r 0 o t t l lc i t u s t t u s t t o t t lid u 0 t 1 06t t l le u 1 t 1 obt t l lf u x 0 u x o x l 1 k s 0 so e 0 1 l lh here so is a given number and the ar a f and u are known functions of their arguments problem 1 1 may be viewed as a model of heat conduction in a region involving phase change an example situation is an ice water mix in which circulation of the water is present then the term f u u represents the contribution of the field velocity generated by the water flow in the liquid phase a more detailed discussion on the formulation of 1 1 can be found elsewhere see e g s

generalized invex monotonicity abstract in this paper the generalized invex monotone functions are defined as an extension of monotone functions a series of sufficient and necessary conditions are also given that relate the generalized invexity of the function with the generalized invex monotonicity of its gradient function this new class of functions will be important in order to characterize the solutions of the variational like inequality problem and mathematical programming problem

solvability of semidefinite complementarity problems abstract the concept of exceptional family has been introduced to study the existence theorem for nonlinear complementarity problems and variational inequality problems we describe extensions of such concepts to complementarity problems defined over the cone of block diagonal symmetric positive semidefinite real matrices using the concept of exceptional family we propose a very general existence theorem for the semidefinite complementarity problem extensions of isac carbone s condition karamardian s condition properness and coercivity are also introduced several applications of the main results are presented and we prove that without exceptional family is a sufficient and necessary condition for the solvability of pseudomonotone semidefinite complementarity problems

a solution condition for complementarity problems with an application to spatial price equilibrium a sufficient condition is developed for the existence of solutions to complementarity problems which is applicable to all continuous functions in addition it is shown to imply a number of simpler conditions including the well known coerciveness condition finally this condition is applied to an existence problem in spatial price equilibrium where it is shown to yield not only a more general result but also a simpler method of proof

application of degree theory in stability of the complementarity problem in this paper we study the behavior of a solution of the complementarity problem when the problem is perturbed the basic result is that if the degree of a certain function is nonzero then the problem is stable we show that several well known conditions for the existence of a solution to the problem also imply stability of the problem we also show that in the nondegenerate case there are simple conditions that ensure stability

some relations between variational like inequality problems and vectorial optimization problems in banach spaces in this work we will establish some relations between variational like inequality problems and vectorial optimization problems in banach spaces under invexity hypotheses this paper extends the earlier work of ruiz garzon et al g ruiz garzon r osuna gomez a rufian lizana relationships between vector variational like inequality and optimization problems european j oper res 157 2004 113 119

auxiliary principle for generalized mixed variational like inequalities abstract in this paper we use the auxiliary principle technique to prove the existence of a solution of some new classes of variational inequalities which are called the generalized mixed variational like inequalities this technique is used to analyze an iterative algorithm several special cases which can be obtained from our main result are also discussed

homogenization of richardsʼ equation of van genuchten mualem model abstract this paper is devoted to the homogenization of richards equation of van genuchten mualem model which is a nonlinear degenerate parabolic differential equation it is usually used to model the motion of saturated unsaturated water flow in porous media we firstly apply the kirchhoff transformation to the equation and obtain a simpler equivalent equation with a linear oscillated diffusion term then under the real assumption for van genuchten mualem model we obtain the homogenized equation based on the two scale convergence theory some results on the first order corrector are also presented

sensitivity analysis in variational inequalities in this paper we derive conditions for the local uniqueness continuity and differentiability of a solution of parametric variational inequalities our approach is new and has a strong geometric flavor as an illustration we discuss the sensitivity of the classical complementarity problem

generalized tensor equations with leading structured tensors abstract systems of tensor equations tes have received much considerable attention in the recent literature in this paper we consider a class of generalized tensor equations gtes an important difference between gtes and tes is that gtes can be regarded as a system of non homogenous polynomial equations whereas tes is a homogenous one such a difference usually make the theoretical and algorithmic results tailored for tes not necessarily applicable to gtes to study properties of the solution set of gtes we first introduce a new class of so named z tensors which includes the set of all p tensors as a proper subset with the help of degree theory we prove that the system of gtes with a leading coefficient z tensor has at least one solution for any right hand side vector moreover we study the local error bounds under some appropriate conditions finally we employ a levenberg marquardt algorithm to find a solution to gtes and report some preliminary numerical results

analysis for a homotopy path of complementarity problems based on µ exceptional family this paper is concerned with several structural properties of a homotopy path for nonlinear continuous semi monotone complementarity problems especially several sufficient conditions are established for the existence and boundedness of this homotopy path under the p 0 property all these sufficient conditions guarantee the uniqueness and continuity of this path the concept of m exceptional family plays a key role in the analysis throughout this paper

on a degenerate diffusion equation of the form c z t ϑ zx x with application to population dynamics 51 here l u and v are strictly positive constants m 1 and i c s for 60 for s o 1 2 with c 0 the initial data z0 will always belong to w 3m q and often to c d and will always have zb 0 zo l u and z l v problem i arises in a theory developed in 173 for two interacting biological species that disperse in response to population pressure this theory is based on the assumption that the individual dispersal velocities are proportional to u u with u and v the corresponding densities if we assume that the habitat q is isolated then the resulting initial value problem has

vector variational like inequalities and non smooth vector optimization problems abstract in this paper we establish relationships between vector variational like inequality problems and non smooth vector optimization problems under non smooth invexity we identify the vector critical points the weakly efficient points and the solutions of the non smooth weak vector variational like inequality problems under non smooth pseudo invexity assumptions these conditions are more general than those existing in the literature

singular perturbed problems in ordinary differential equations abstract here we solve singularly perturbed periodic problems in ordinary differential equations when singular perturbations contain two orders higher derivatives than unperturbed equations also we apply our method to the study of singularly perturbed variational problems

d orientation sequences for continuous functions and nonlinear complementarity problems we introduce the new concept of d orientation sequence for continuous functions it is shown that if there does not exist a d orientation sequence for a continuous function then the corresponding complementarity problem cp has a solution we believe that such a result characterizes an intrinsic property of cps as the concept of exceptional family of elements the notion of d orientation sequence of a function is also a powerful tool for investigating the existence theorems of cps we use this new tool to establish among other things a new existence result for a class of p mapping cps

alternative theorems for nonlinear projection equations and applications to generalized complementarity problems let k be a closed convex subset in rn and let f g and h be three functions from rn into itself we consider the following nonlinear projection equation npe h x k g x f x x r 1 where k is the orthogonal projection operator onto the set k this equation provides a uni ed formulation of several interesting and important special cases in the case when h x g x for any x rn then 1 reduces to the equation studied by pang and yao 17 under the name of generalized normal equation which is equivalent to the following generalized variational inequality gvi g x k y g x f x 0 for all y k 2

a note on the existence of traffic equilibria this note presents an alternative proof of the existence of a solution to the network equilibrium problem which is based on the recent work in complementarity theory by t smith 11

an alternative theorem for generalized variational inequalities and solvability of nonlinear quasi p m complementarity problems in this paper an alternative theorem and hence a sufficient solution condition is established for generalized variational inequality problems the concept of exceptional family for generalized variational inequality is introduced this concept is general enough to include as special cases the notions of exceptional family of elements and the d orientation sequence for continuous functions particularly we apply the alternative theorem for investigating the solvability of the nonlinear complementarity problems with so called quasi p m maps which are broad enough to encompass the quasi monotone maps and p maps as the special cases an existence theorem for this class of complementarity problems is established which significantly generalizes several previous existence results in the literature

an existence condition for open set nonlinear complementarity problems with an application to a closed economic system a sufficient condition for the existence of solutions to open set nonlinear complementarity problems oncp is developed this condition is applied to an existence problem in a closed general economic equilibrium system

regularity results for a one dimensional two phase degenerate stefan problem l lh 40 30 l li where a u 0 and ai 0 for i 1 2 in the classical stefan problem the specific heat e u is required to be strictly positive where a u ml u if u is in q and 11 if u is in q however in more general free boundary problems various types of degeneracy in a can occur in the electro chemical machining process the specific heat is zero in both phases resulting in elliptic differential equations but with a stefan type boundary condition see 3 for a description of this model in the study of saturated unsaturated flow in

asymptotic behaviour of solutions to parabolic elliptic variational inequalities in a real hilbert space h where j is a given interval in r 0 co j is the subdifferential of a convex function 4 on h and b is a maximal monotone operator in h the abstract setting reflects the form of a class of parabolic elliptic boundary value problems with time dependent obstacles such problems arise in particular as mathematical models of various processes in electrochemical technology and some flows with saturations in porous media time dependent obstacles may represent geometric controls moving control actions in these problems in particular such controls may be comprehended in variable boundary conditions prescribed over time dependent parts of the boundary of domain or in source terms distributed over time dependent subdomains the former refers to control of flows via variable water table in reservoirs the latter characterizes control of electrochemical process via mechanically driven motion of a toolpiece cathode the most typical control objective consists there in stabilizing the state of the process at a given spatial distribution for example constant water head and desired shape of anode this motivates our interest in the asymptotic behavior of solutions to the corresponding boundary value problems as t a to the knowledge of the authors this question has so far not been studied in literature as it has been shown in kenmochi pawlow 16 171 the class of considered time dependent problems admits a representation in the form of evolution equation e b in 16 171 the corresponding results on existence uniqueness and stability of solutions have been established in the present paper we study the asymptotic properties of solutions to e c b as t 00 provided some hypotheses are imposed on the family t e r and the limit 4 of as t a for the solution u u of e d b on r we prove that

an iterative scheme for variational inequalities in this paper we introduce and study a general iterative scheme for the numerical solution of finite dimensional variational inequalities this iterative scheme not only contains as special cases the projection linear approximation and relaxation methods but also induces new algorithms then we show that under appropriate assumptions the proposed iterative scheme converges by establishing contraction estimates involving a sequence of norms in en induced by symmetric positive definite matrices gm thus in contrast to the above mentioned methods this technique allows the possibility of adjusting the norm at each step of the algorithm this flexibility will generally yield convergence under weaker assumptions

exchange price equilibria and variational inequalities the aim of this paper is to show the relevance of the concept and the theory of variational inequalities in the study of economic equilibria

a new method for the existence of solutions of nonlinear differential equations many problems is analysis reduce to solving an equation au f where a is an operator on a space into another space u is the unknown and f is a given element the existence of a local solution near a trivial solution u0 of au 0 where 0 is near f is usually solved by the classical implicit function theorem as a matter of fact the picard iteration procedure is used unfortunately this procedure cannot be applied in some problems of partial differential equations these cases occur if the loss of derivatives appears 6 7 31 to get around this difficulty a more sophisticated iteration procedure is needed moser 1 has found such a method which enables one to handle a loss of derivatives by using a rapidly converging iteration scheme the nash moser technique is nicely described in the papers s lo we prove in this paper several theorems on the existence of solutions with the application to several types of specific problems the main interest is that in many cases these results can be used in situations which up to now have required the nash moser implicit function theorem yet they have classical proofs we also apply abstract results to show the existence of local invariant manifolds for singularly perturbed ordinary differential equations and we give a simple proof of a theorem of j moser for invariant tori l the plan of our paper is as follows in section 2 we present proofs of several existence theorems in an abstract setting in a scale of hilbert spaces the hypotheses are those commonly found in problems which have up to now required the application of the nash moser technique however the results have classical proofs based on the degree of mappings s we shall give in the proof of theorem 2 8 a new iteration procedure which retains supergeometric convergence properties we require obligations only for the linearization of our equation this is the main advance of this theorem section 3 of the paper presents applications of the theorems of section 2 there are demonstrated proofs of the above mentioned theorems of

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

on the existence of the weak solution and the regularity of the free boundary to a one dimensional two phase degenerate stefan problem existence and regularity of the free boundary s t are demonstrated for the weak solution u u x t and s s t of the degenerate stefan problem 1 u ut uxx 0 0 0 t t u 1 t 2 t 0 0 t t u x 0 u0 x 0 x 1 u s t t 0 0 t t t ux s t 0 t ux s t 0 t 0 t t s 0 s0 where i u 0 and i 0 0 i 1 2

disequilibrium and variational inequalities in this paper we introduce a new market disequilibrium model in a spatial economic setting which genemlizes a recent spatial disequilibrium model to the asymmetric case we derive two alternative variational inequality formulations of the market conditions in the case of price rigidities and or controls and discuss existence and uniqueness properties we then propose a decomposition algorithm which resolves the variational inequality problem into three distinct and simpler variational inequality subproblems with special structure which are then solved in sequential fashion any appropriate algorithm can then be used to solve the individual subproblems the first variational inequality subproblem however is identical to the one governing the well known spatial price e tilibritmt problem and hence a plethora of algorithms are available for its solution we conclude with computa tional experience with the decomposition algorithm on large scale market examples this work bridges the study of disequilibrium and equilibrium problems via the theory of variational inequalities kepor disequilibrium spatial equilibrium economics variational inequalities

generalized multivalued quasi variational inequalities in this paper we introduce and study a new class of variational inequalities known as generalized multivalued quasi variational inequalities this class is the most general and unifying one using essentially the projection technique we establish the equivalence between this class and the implicit wiener hopf equations this equivalence is used to suggest a number of iterative algorithms for solving the generalized multivalued quasi variational inequalities the convergence analysis of these algorithms is also studied the results proved in this paper represent a significant improvement and refinement of the previously known results in this area

mixed finite elements for the richards equation linearization procedure we consider mixed finite element discretization for a class of degenerate parabolic problems including the richards equation after regularization time discretization is achieved by an euler implicit scheme while mixed finite elements are employed for the discretization in space based on the results obtained in radu et al rana preprint 02 06 eindhoven university of technology 2002 this paper considers a simple iterative scheme to solve the emerging nonlinear elliptic problems

periodic solutions of parabolic elliptic obstacle problems this paper deals with parabolic elliptic obstacle problems the problem is formulated as follows let sz be a bounded domain in rn iv 1 and for each i e r let 9 t be a relatively compact domain in sz for a given interval z to tl cc d to t co we then define non cylindrical subsets qjz i 1 2 of ix sz by

regularity results for an elliptic parabolic free boundary problem on etudie un probleme aux limites libre elliptique parabolique a une dimension d espace on donne plusieurs resultats de regularite pour la solution faible et la frontiere libre on donne des conditions pour que la frontiere libre soit une courbe c 1

some aspects of variational inequalities abstract in this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence uniqueness computational properties various generalizations sensitivity analysis and their applications we also propose some open problems with sufficient information and references so that someone may attempt solution s in his her area of special interest we also include some new results which we have recently obtained

nonstationary filtration in partially saturated porous media from the mathematical formulation of a one dimensional flow through a partially saturated porous medium we arrive at a nonlinear free boundary problem the boundary being between the saturated and the unsaturated regions in the medium in particular we obtain an equation which is parabolic in the unsaturated part of the domain and elliptic in the saturated part

holderian continuity of solutions of a dynamic walrasian price equilibrium problem with application this paper considers a qualitative analysis of the solution of a pure exchange general economic equilibrium problem regulated by walras law in the dynamic parametric case the equilibrium conditions that describe this pure exchange economic model are expressed in terms of a time dependent parametric variational inequality for which a holderian type stability of the solutions have been proved for this model we have also focused our attention on a numerical application with the parameter n m l where m represents the variation of the average annual inflation rate and l represents the variation of the vat share both considered in italy in the year 2008 for this numerical example it has been proved the existence of the solutions of the evolutionary parametric variational inequality that describes the equilibrium conditions by means of the direct method and by using matlab computation after a linear interpolation the curves of equilibria have been visualized

hyperbolic phenomena in a strongly degenerate parabolic equation we consider the equation u t u u x x where 0 and where is a strictly increasing function with lim s we solve the associated cauchy problem for an increasing initial function and discuss to what extent the solution behaves qualitatively like solutions of the first order conservation law u t u x equations of this type arise for example in the theory of phase transitions where the corresponding free energy functional has a linear growth rate with respect to the gradient

exceptional family and solvability of copositive complementarity problems abstract in this paper by extending the concept of exceptional family to complementarity problems over the cone of symmetric copositive real matrices we propose an existence theorem of a solution to the copositive complementarity problem extensions of isac carbone s condition karamardian s condition weakly properness and coercivity are also introduced several applications of these results are presented and we prove that without exceptional family is a sufficient and necessary condition for the solvability of pseudomonotone copositive complementarity problems

exceptional families and finite dimensional variational inequalities over polyhedral convex sets a new concept of exceptional family for nonlinear variational inequalities over polyhedral sets is introduced in this paper it generalizes the concepts for complementarity problem introduced by smith and isac we applied the new analytical tool to the study of existence problem for variational inequalities it is shown that our existence condition is weaker than most of the sufficient conditions which have been known

what is the best causal scale space for three dimensional images we study the unique affine invariant morphological scale space in three dimensions we discuss its properties and show that it improves the dynamic shape model we explain the algorithms and display the first numerical experiments

uniqueness of the bounded solution to a strongly degenerate parabolic problem abstract this paper concerns the uniqueness of the bounded solution to a strongly degenerate parabolic problem the equation considered may have two kinds of strong degeneracies and there is no restriction on the relation between the two degeneracies by using holmgren s approach we prove that the bounded solution of the associated initial boundary value problem is unique under some essentially necessary condition on the growth of the convection

a unified description of iterative algorithms for traffic equilibria the purpose of this paper is to provide a unified description of iterative algorithms for the solution of traffic equilibrium problems we demonstrate that a large number of well known solution techniques can be described in a unified manner through the concept of partial linearization and establish close relationships with other algorithmic classes for nonlinear programming and variational inequalities in the case of nonseparable travel costs the class of partial linearization algorithms are shown to yield new results in the theory of finite dimensional variational inequalities the possibility of applying truncated algorithms within the framework is also discussed

regularity of the interfaces with sign changes of solutions of the one dimensional porous medium equation abstract in previous papers we considered the cauchy problem for the one dimensional evolution p laplacian equation for nonzero bounded and nonnegative initial data having compact support and showed that after a finite time the set of spatial critical points of the nonnegative solution u u x t in u 0 consists of one point the spatial maximum point of u and the curve of the spatial maximum points is continuous with respect to the time variable since the spatial derivative x u satisfies the porous medium equation with sign changes the curve of the spatial maximum points is regarded as an interface with sign changes of x u on the other hand in a paper by m bertsch and d hilhorst 1991 appl anal 41 111 130 the interfaces where the solutions change their sign were studied in detail for the initial boundary value problems of the generalized porous medium equation over two dimensional cylinders but the monotonicity of the initial data is assumed there as is noted in section 4 of our earlier work 1996 j math anal appl 203 78 103 the monotonicity of x u t in some neighborhood of the spatial maximum point of u t cannot be assumed and therefore if this monotonicity for some large t 0 is proved then by the method of bertsch and hilhorst cited above one may get more precise regularity properties of the curve of the spatial maximum points the purpose of the present paper is twofold one is to remove some monotonicity assumption for initial data in bertsch and hilhorst s theorem concerning the regularity of the interfaces with sign changes of solutions of the one dimensional generalized porous medium equation by comparing the solution with appropriate symmetric nonnegative solutions we shall get the monotonicity of the solution near the interface after a finite time the other is as a by product of the method to get c 1 regularity of the curves of the spatial maximum points of nonnegative solutions of the cauchy problem for the evolution p laplacian equation for sufficiently large t

nonlinear elliptic parabolic problems we introduce a notion of viscosity solutions for a general class of elliptic parabolic phase transition problems these include the richards equation which is a classical model in filtration theory existence and uniqueness results are proved via the comparison principle in particular we show existence and stability properties of maximal and minimal viscosity solutions for a general class of initial data these results are new even in the linear case where we also show that viscosity solutions coincide with the regular weak solutions introduced in alt and luckhaus math z 183 311 341 1983

lagrangean variables in infinite dimensional spaces for a dynamic economic equilibrium problem abstract this paper is focused on the study of a dynamic competitive equilibrium by using lagrangean multipliers this mathematical formulation allows us the improve the walrasian model by considering the common possibility of an uncharged delayed payment in a given time for example by using a credit card firstly the economic equilibrium problem is reformulated as an evolutionary variational problem then the lagrangean theory in infinite dimensional spaces is applied thanks to the application of this theory we obtain the existence of lagrangean multipliers which allows us to give a computational procedure for the equilibrium solutions

bounded weak solutions of an elliptic parabolic neumann problem abstract in this paper we establish existence and uniqueness for boundedweak solutions of an elliptic parabolic neumann problem we also describethe asymptotic behavior as t oo 1 introduction let c r r be a continuous function which is strictlyincreasing on r and identically equal to a constant on r see figure 1 fora bounded domain fi c rn with smooth boundary dfl we consider the neumann problem 1 1 f c u t au inqt fix 0 t 1 2 i du du f on du x 0 t 3 c u x 0 vo x xeu here u u x t a denotes the laplacian in r and 3 du the outward normal derivative on du for given functions and i n we discuss existence uniqueness and asymptotic behavior of weak solutions of problem i equation 1 1 is elliptic if u 0 and parabolic if u 0 because of this change of type we cannot a prioriexpect solutions to be classical the physical background of 1 1 lies in the theory of partially saturated flowsin porous media in that context u stands for the hydrostatic potential and c u for the moisture content or saturation the part of fi where u is negative is calledthe unsaturated region and that where u is positive the saturated region the set

exceptional family and solvability of the second order cone complementarity problems abstract in this paper we introduce a concept of exceptional family for second order cone complementarity problems denoted by soccp which is the particular case of the concept of exceptional family of elements introduced by isac and carbone 1999 25 for the nonlinear complementarity problems and the exceptional family for second order cone complementarity problems has some pretty unique properties furthermore we propose a new sufficient existence condition of a solution to soccp and give a particular example to show that the new condition is not stronger than known isac carbone s condition

characterizations of weakly sharp solutions for a variational inequality with a pseudomonotone mapping abstract for a variational inequality problem with a pseudomonotone mapping f we characterize the weak sharpness of its solution set c without using gap functions when the mapping f is also constant on c the characterizations of weak sharpness of c become more succinct as an example a pseudomonotone mapping on c is shown to be constant on c consequently the weak sharpness of c can further be described by a g teaux differentiable function which itself characterizes the pseudomonotonicity of f on c it turns out that several existing relevant results with differentiable gap functions can be obtained from ours

strong solutions to the richards equation in the unsaturated zone abstract we study the unsaturated case of the richards equation in three space dimensions with dirichlet boundary data we first establish an a priori l estimate with its help by means of a fixed point argument we prove global in time existence of a unique weak solution in sobolev spaces finally we are able to improve the regularity of this weak solution in order to gain a strong one