invex sets and preinvex functions on riemannian manifolds the concept of a geodesic invex subset of a riemannian manifold is introduced geodesic invex and preinvex functions on a geodesic invex set with respect to particular maps are defined the relation between geodesic invexity and preinvexity of functions on manifolds is studied using proximal subdifferential certain results concerning extremum points of a non smooth geodesic preinvex function on a geodesic invex set are obtained the main value inequality and the mean value theorem in invexity analysis are extended to cartan hadamard manifolds

a class of hamilton jacobi equations on banach finsler manifolds abstract the concept of subdifferentiability is studied in the context of c 1 finsler manifolds modeled on a banach space with a lipschitz c 1 bump function a class of hamilton jacobi equations defined on c 1 finsler manifolds is studied and several results related to the existence and uniqueness of viscosity solutions are obtained

radon nikodým property and thick families of geodesics abstract banach spaces without the radon nikod m property are characterized as spaces containing bilipschitz images of thick families of geodesics defined as follows a family t of geodesics joining points u and v in a metric space is called thick if there is 0 such that for every g t and for any finite collection of points r 1 r n in the image of g there is another u v geodesic g t satisfying the conditions g also passes through r 1 r n and possibly has some more common points with g on the other hand there is a finite collection of common points of g and g which contains r 1 r n and is such that the sum of maximal deviations of the geodesics between these common points is at least

dynamic programming and viscosity solutions for the optimal control of quantum spin systems abstract the purpose of this paper is to describe the application of the notion of viscosity solutions to solve the hamilton jacobi bellman hjb equation associated with an important class of optimal control problems for quantum spin systems the hjb equation that arises in the control problems of interest is a first order nonlinear partial differential equation defined on a lie group hence we employ recent extensions of the theory of viscosity solutions to riemannian manifolds in order to interpret possibly non differentiable solutions to this equation results from differential topology on the triangulation of manifolds are then used develop a finite difference approximation method for numerically computing the solution to such problems the convergence of these approximations is proven using viscosity solution methods in order to illustrate the techniques developed these methods are applied to an example problem

line search algorithms for locally lipschitz functions on riemannian manifolds this paper presents line search algorithms for finding extrema of locally lipschitz functions defined on riemannian manifolds to this end we generalize the so called wolfe conditions for nonsmooth functions on riemannian manifolds using varepsilon subgradient oriented descent directions and the wolfe conditions we propose a nonsmooth riemannian line search algorithm and establish the convergence of our algorithm to a stationary point moreover we extend the classical bfgs algorithm to nonsmooth functions on riemannian manifolds numerical experiments illustrate the effectiveness and efficiency of the proposed algorithm

a maximum principle for evolution hamilton jacobi equations on riemannian manifolds abstract we establish a maximum principle for viscosity subsolutions and supersolutions of equations of the form u t f t d x u 0 u 0 x u 0 x where u 0 m r is a bounded uniformly continuous function m is a riemannian manifold and f 0 t m r this yields uniqueness of the viscosity solutions of such hamilton jacobi equations

a characterization of the radon nikodym property abstract it is well known that every bounded below and non increasing sequence in the real line converges we give a version of this result valid in banach spaces with the radon nikodym property thus extending a former result of a prochazka

dini derivative and a characterization for lipschitz and convex functions on riemannian manifolds dini derivatives in riemannian manifold settings are studied in this paper in addition a characterization for lipschitz and convex functions defined on riemannian manifolds and sufficient optimality conditions for constraint optimization problems in terms of the dini derivative are given

the optimal control related to riemannian manifolds and the viscosity solutions to hamilton jacobi bellman equations abstract in this paper we study the optimal stochastic control problem for stochastic differential equations on riemannian manifolds the cost functional is specified by controlled backward stochastic differential equations in euclidean space under some suitable assumptions we conclude that the value function is the unique viscosity solution to the associated hamilton jacobi bellman equation which is a fully nonlinear parabolic partial differential equation on riemannian manifolds

nonsmooth analysis and hamilton jacobi equations on riemannian manifolds we establish some perturbed minimization principles and we develop a theory of subdifferential calculus for functions defined on riemannian manifolds then we apply these results to show existence and uniqueness of viscosity solutions to hamilton jacobi equations defined on riemannian manifolds

global and fine approximation of convex functions let u subset of r d be open and convex we prove that every not necessarily lipschitz or strongly convex function f u r can be approximated by real analytic convex functions uniformly on all of u we also show that c 0 fine approximation of convex functions by smooth or real analytic convex functions on r d is possible in general if and only if d 1 nevertheless for d 2 we give a characterization of the class of convex functions on r d which can be approximated by real analytic or just smoother convex functions in the c 0 fine topology it turns out that the possibility of performing this kind of approximation is not determined by the degree of local convexity or smoothness of the given function but by its global geometrical behaviour we also show that every c 1 convex and proper function on u can be approximated by c infinity convex functions in the c 1 fine topology and we provide some applications of these results concerning prescription of sub differential boundary data to convex real analytic functions and smooth surgery of convex bodies

counting rational points on hypersurfaces for any n geq 2 let f in mathbb z x 1 ldots x n be a form of degree d geq 2 which produces a geometrically irreducible hypersurface in mathbb p n 1 this paper is concerned with the number n f b of rational points on f 0 which have height at most b for any varepsilon 0 we establish the estimate n f b o b n 2 varepsilon whenever either n leq 5 or the hypersurface is not a union of lines here the implied constant depends at most upon d n and varepsilon

invariant monotone vector fields on riemannian manifolds abstract various concepts of invariant monotone vector fields on riemannian manifolds are introduced some examples of invariant monotone vector fields are given several notions of invexities for functions on riemannian manifolds are defined and their relations with invariant monotone vector fields are studied

regularization by sup inf convolutions on riemannian manifolds an extension of lasry lions theorem to manifolds of bounded curvature abstract we show how lasry lions s result on regularization of functions defined on r n or on hilbert spaces by sup inf convolutions with squares of distances can be extended to finite or infinite dimensional riemannian manifolds m of bounded sectional curvature more specifically among other things we show that if the sectional curvature k of m satisfies k 0 k k 0 on m for some k 0 0 and if the injectivity and convexity radii of m are strictly positive then every bounded uniformly continuous function f m r can be uniformly approximated by globally c 1 1 functions defined by f sup z m inf y m f y 1 2 d z y 2 1 2 d x z 2 as 0 with 0 2 our definition of global c 1 1 smoothness is intrinsic and natural and it reduces to the usual one in flat spaces but we warn the reader that in the noncompact case this definition differs from other notions of rather local c 1 1 smoothness that have been recently used by other authors based on charts the importance of this regularization method lies rather than on the degree of smoothness obtained on the fact that the correspondence f f is explicit and preserves many significant geometrical properties that the given functions f may have such as invariance by a set of isometries infima sets of minimizers ordering local or global lipschitzness and only when one additionally assumes that k 0 local or global convexity we also give two examples showing that this result completely fails even for nonflat cartan hadamard manifolds whenever f or k are not bounded

newton algorithm for hamiltonian characterization in quantum control we propose a newton algorithm to characterize the hamiltonian of a quantum system interacting with a given laser field the algorithm is based on the assumption that the evolution operator of the system is perfectly known at a fixed time the computational scheme uses the crank nicholson approximation to explicitly determine the derivatives of the propagator with respect to the hamiltonians of the system in order to globalize this algorithm we use a continuation method that improves its convergence properties this technique is applied to a two level quantum system and to a molecular one with a double well potential the numerical tests show that accurate estimates of the unknown parameters are obtained in some cases we discuss the numerical limits of the algorithm in terms of the basin of convergence and the non uniqueness of the solution

hamilton jacobi equations in infinite dimensions i uniqueness of viscosity solutions soient un espace de banach v son dual v et des solutions d une equation h x u du 0 dans etablies dans un sous ensemble de v h v r v r est continue on demontre que des solutions classiques sont determinees de facon unique par leurs valeurs limites on montre que l extension de la notion de solutions viscosite aux espaces de banach reussit dans des espaces munis de la propriete de radon nikodym

vinogradov s mean value theorem via efficient congruencing we obtain estimates for vinogradov s integral that for the rst time approach those conjectured to be the best possible several applications of these new bounds are provided in particular the conjectured asymptotic formula in waring s problem holds for sums of s kth powers of natural numbers whenever s 2k 2 2k 3

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

manin s conjecture for certain biprojective hypersurfaces using the circle method we count integer points on complete intersections in biprojective space in boxes of different side length provided the number of variables is large enough depending on the degree of the defining equations and certain loci related to the singular locus having established these asymptotics we deduce asymptotic formulas for rational points on such varieties with respect to the anticanonical height function in particular we establish a conjecture of manin for certain smooth hypersurfaces in biprojective space of sufficiently large dimension

constructive solution of a bilinear optimal control problem for a schrodinger equation abstract often considered in numerical simulations related to the control of quantum systems the so called monotonic schemes have not been so far much studied from the functional analysis point of view yet these procedures provide an efficient constructive method for solving a certain class of optimal control problems this paper aims both at extending the results already available about these algorithms in the finite dimensional case i e the time discretized case and at completing those of the continuous case this paper starts with some results about the regularity of a functional related to a wide class of models in quantum chemistry these enable us to extend an inequality due to ojasiewicz to the infinite dimensional case finally some inequalities proving the cauchy character of the monotonic sequence are obtained followed by an estimation of the rate of convergence

variational inequalities for set valued vector fields on riemannian manifolds convexity of the solution set and the proximal point algorithm we consider variational inequality problems for set valued vector fields on general riemannian manifolds the existence results of the solution convexity of the solution set and the convergence property of the proximal point algorithm for the variational inequality problems for set valued mappings on riemannian manifolds are established applications to convex optimization problems on riemannian manifolds are provided

density of rational points on a quadric bundle in mathbb p 3 times mathbb p 3 we establish an asymptotic formula for the number of rational points of bounded anticanonical height which lie on a certain zariski dense subset of the biprojective hypersurface x1y12 x4y42 0 in p3 p3 this confirms the modified manin conjecture for this variety in which the removal of a thin set of rational points is allowed

prescribing symmetric functions of the eigenvalues of the ricci tensor we study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the ricci tensor we prove an existence theorem for a wide class of symmetric functions on manifolds with positive ricci curvature provided the conformal class admits an admissible metric

a new form of the circle method and its application to quadratic forms if the coefficients r n satisfy suitable arithmetic conditions the behaviour of f will be determined by an appropriate rational approximation a q to with small values of q usually producing large values of f when lies in an interval a q a q with q small a major arc one hopes to estimate f asymptotically while if the corresponding q is large for the minor arcs one hopes that f will be small at least on average one can distinguish two different types of application of the circle method the first type following the above pattern uses one of a variety of mean value techniques hua s inequality for example to give an upper bound for the minor arc contribution results on waring s problem are generally proved this way for example the second type of application on which we shall be focusing our attention avoids the minor arcs completely and treats f asymptotically for all here one cannot take advantage of the mean value ideas used in the first type of problem the savings come instead from the availability of good estimates for complete exponential sums kloosterman s work 12 on quaternary quadratic forms and the papers of the author 5 and hooley 8 and 9 on cubic forms are of this latter type we should point out that such applications do not include any problems in which f is the generating function for a polynomial of degree 4 or more there are

viscosity solutions of hamilton jacobi equations in infinite dimensions iv hamiltonians with unbounded linear terms in this work we show how to interpret s and e in the viscosity sense when the solutions may be expected to be weakly sequentially continuous in particular we introduce notions which allow us to handle the terms and for which we can obtain existence and uniqueness results in a variety of circumstances

hamilton jacobi equations in infinite dimensions iii on etudie des equations de la forme f x u du 0 dans ou est un sous ensemble ouvert d un espace de banach v la fonction inconnue u r est continue du x est la derivee de frechet de u en x et f est une fonction non lineaire qui est une application continue de r v dans r on introduit la notion de solution viscosite stricte afin d eliminer l hypothese que v a la propriete de radon nikodym

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

counting rational points on biquadratic hypersurfaces abstract an asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain zariski open subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables the proof uses the hardy littlewood circle method

counting rational points on algebraic varieties for any n 2 let z pn be a geometrically integral algebraic variety of degree d this article is concerned with the number nz b of q rational points on z which have height at most b for any e 0 we establish the estimate nz b od e n bdimz e provided that d e as indicated the implied constant depends at most on d e and n

on the representation of integers by quadratic forms let n 4 and let q 2 z x1 xn be a non singular quadratic form when q is indefinite we provide new upper bounds for the least non trivial integral solution to the equation q 0 and when q is positive definite we provide improved upper bounds for the greatest positive integer k for which the equation q k is insoluble in integers despite being soluble modulo every prime power

infinite games banach space geometry and the eikonal equation we study a class of infinite games which turns out to be closely related to banach space geometry using one of these games we construct a bounded differentiable almost everywhere solution of the eikonal equation u 1 onr d w ithd 2

viscosity solutions to second order partial differential equations on riemannian manifolds we prove comparison uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations f x u du d 2 u 0d ef ined on afi nite dimensional riemannian manifold m finest results with hypothesis that require the function f to be degenerate elliptic that is nonincreasing in the second order derivative variable and uniformly continuous with respect to the variable x are obtained under the assumption that m has nonnegative sectional curvature while if one additionally requires f to depend on d 2 u in a uniformly continuous manner then comparison results are established with no restrictive assumptions on curvature

the failure of rolle s theorem in infinite dimensional banach spaces we prove the following new characterization of cp lipschitz smoothness in banach spaces an infinite dimensional banach space x has a cp smooth lipschitz bump function if and only if it has another cp smooth lipschitz bump function f such that its derivative does not vanish at any point in the interior of the support of f that is f does not satisfy rolle s theorem moreover the support of this bump can be assumed to be a smooth starlike body the twisted tube method we use in the proof is interesting in itself as it provides other useful characterizations of cp smoothness related to the existence of a certain kind of deleting diffeomorphisms as well as to the failure of brouwer s fixed point theorem even for smooth self mappings of starlike bodies in all infinite dimensional spaces

rolle s theorem and negligibility of points in infinite dimensional banach spaces abstract in this note we prove that if a differentiable function oscillates between e and e on the boundary of the unit ball then there exists a point in the interior of the ball in which the differential of the function has norm equal or less than e this kind of approximate rolle s theorem is interesting because an exact rolle s theorem does not hold in many infinite dimensional banach spaces a characterization of those spaces in which rolle s theorem does not hold is given within a large class of banach spaces this question is closely related to the existence of c 1 diffeomorphisms between a banach space x and x 0 which are the identity out of a ball and we prove that such diffeomorphisms exist for every c 1 smooth banach space which can be linearly injected into a banach space whose dual norm is locally uniformly rotund lur

the density of rational points on curves and surfaces let c be an irreducible projective curve of degree d in mathbb p 3 defined over overline mathbb q it is shown that c has o varepsilon d b 2 d varepsilon rational points of height at most b for any varepsilon 0 uniformly for all curves c this result extends an estimate of bombieri and pila duke math j 59 1989 337 357 to projective curves for a projective surface s in mathbb p 3 of degree d ge 3 it is shown that there are o varepsilon d b 2 varepsilon rational points of height at most b of which at most o varepsilon d b 52 27 varepsilon do not lie on a rational line in s for non singular surfaces one may reduce the exponent to 4 3 16 9d for d 4 or 5 or max 1 3 sqrt d 2 d 1 for d ge 6 even for the surface x 1 d x 2 d x 3 d x 4 d this last result improves on the previous best known as a further application it is shown that almost all integers represented by an irreducible binary form f x y in mathbb z x y have essentially only one such representation this extends a result of hooley j reine angew math 226 1967 30 87 which concerned cubic forms only the results are not restricted to projective surfaces and as an application of other results in the paper it is shown that x 1 x 2 x 3 in mathbb n 3 x 1 d x 2 d x 3 d n ll varepsilon d n theta d varepsilon with theta frac 2 sqrt d frac 2 d 1 when d ge 8 this provides the first non trivial bound for the number of representations as a sum of three d th powers

hamilton jacobi equations in infinite dimensions ii existence of viscosity solutions on demontre des resultats d existence generaux pour 2 problemes 1 le probleme de cauchy u t h x t u du 0 dans v 0 t u x 0 x dans v et 2 le probleme stationnaire u h x u du 0 dans v ou h et sont donnees

simultaneous integer values of pairs of quadratic forms we prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent almost all pairs of integers that satisfy the necessary local conditions provided that the forms satisfy a suitable nonsingularity condition in particular such forms simultaneously attain prime values if the obvious local conditions hold the proof uses the circle method and in particular pioneers a two dimensional version of a kloosterman refinement

pseudo jacobian and global inversion of nonsmooth mappings on riemannian manifolds abstract we introduce a notion of pseudo jacobian for continuous nonsmooth mappings between riemannian manifolds then we discuss the global inversion of continuous mappings between riemannian manifolds which may be non locally lipschitz for this purpose we obtain a version of the hadamard integral condition for invertibility by using pseudo jacobian

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

viscosity solutions of hamilton jacobi equations in infinite dimensions v unbounded linear terms and b continuous solutions we continue the systematic study of first order hamilton jacobi equations in infinite dimensions begun in parts i iv of this series 121 our approach relies on as before an appropriate interpretation of the notion of viscosity solutions as introduced in 11 lo in the interim the strong development of viscosity solution theory in many directions has continued unabated in addition to the many references given in earlier papers in this series which we will not repeat here see 12 we would like to point out that a complete theory of second order fully nonlinear and possibly degenerate elliptic equations in finite dimensions has been made possible by the use of viscosity solutions s 9 18 20 22 19 32 331 for example and there is now a substantial body of work concerning second order equations in infinite dimensions 25 271 however in this work we come back to the particular situation studied in 12 part iv namely we come back to hamilton jacobi equations in

local convergence of the proximal point method for a special class of nonconvex functions on hadamard manifolds abstract local convergence analysis of the proximal point method for a special class of nonconvex functions on hadamard manifold is presented in this paper the well definedness of the sequence generated by the proximal point method is guaranteed moreover it is proved that each cluster point of this sequence satisfies the necessary optimality conditions and under additional assumptions its convergence for a minimizer is obtained

smooth approximation of lipschitz functions on riemannian manifolds abstract we show that for every lipschitz function f defined on a separable riemannian manifold m possibly of infinite dimension for every continuous e m 0 and for every positive number r 0 there exists a c smooth lipschitz function g m r such that f p g p e p for every p m and lip g lip f r consequently every separable riemannian manifold is uniformly bumpable we also present some applications of this result such as a general version for separable riemannian manifolds of deville godefroy zizler s smooth variational principle

on existence and uniqueness of solutions of hamilton jacobi equations abstract the theory of scalar first order nonlinear partial differential equations has been enjoying a rapid development in the last few years this development occurred because the authors established uniqueness criteria for generalized solutions called viscosity solutions which correctly identify the solutions sought in areas of application including control theory differential games and the calculus of variations the concept of viscosity solutions is relatively easy to work with and many formally heuristic or difficult proofs have been made rigorous or simple using this concept a feedback process has begun and the experience recently gained in working with viscosity solution has suggested new existence and uniqueness results the current paper continues this interaction by establishing new existence ane uniqueness results in a natural generality suggested by earlier proofs it is also felt that the presentation of the comparison results which imply uniqueness continuous dependence and are used to estimate moduli of continuity has something to offer over earlier presentations in special cases author

on some problems on smooth approximation and smooth extension of lipschitz functions on banach finsler manifolds abstract let us consider a riemannian manifold m either separable or non separable we prove that for every e 0 every lipschitz function f m r can be uniformly approximated by a lipschitz c 1 smooth function g with lip g lip f e as a consequence every riemannian manifold is uniformly bumpable these results extend to the non separable setting those given in 1 for separable riemannian manifolds the results are presented in the context of c l finsler manifolds modeled on banach spaces sufficient conditions are given on the finsler manifold m and the banach space x where m is modeled so that every lipschitz function f m r can be uniformly approximated by a lipschitz c k smooth function g with lip g c lip f for some c depending only on x some applications of these results are also given as well as a characterization on the separable case of the class of c l finsler manifolds satisfying the above property of approximation finally we give sufficient conditions on the c 1 finsler manifold m and x to ensure the existence of lipschitz and c 1 smooth extensions of every real valued function f defined on a submanifold n of m provided f is c 1 smooth on n and lipschitz with the metric induced by m

monotonic time discretized schemes in quantum control most of the numerical simulations in quantum bilinear control have used monotonically convergent algorithms of krotov introduced by tannor et al 15 of zhu rabitz 16 or their unified formulation in 17 however the properties of the discrete version of these procedures have not been yet tackled with we present in this paper a stable time and space discretization which preserves the monotonic properties of the monotonic algorithms numerical results show that the newly derived algorithms are stable and enable various experimentations

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

proximal point method for minimizing quasiconvex locally lipschitz functions on hadamard manifolds abstract in this paper we propose an extension of the proximal point method to solve minimization problems with quasiconvex locally lipschitz objective functions on hadamard manifolds to reach this goal we use the concept of clarke subdifferential on hadamard manifolds and assuming that the function is bounded from below we prove the global convergence of the sequence generated by the method to a critical point of the function

multigrade efficient congruencing and vinogradov s mean value theorem we develop a multigrade enhancement of the efficient congru encing method to estimate vinogradov s integral of degree k for moments of order 2s thereby obtaining near optimal estimates for 5 k 2 1 543k 2 the asymptotic formula is also established for sums of 28 fifth powers

uniform approximation of continuous mappings by smooth mappings with no critical points on hilbert manifolds we prove that every continuous mapping from a separable infinite dimensional hilbert space x into r m can be uniformly approximated by c infinity smooth mappings with no critical points this kind of result can be regarded as a sort of strong approximate version of the morse sard theorem some consequences of the main theorem are as follows every two disjoint closed subsets of x can be separated by a one codimensional smooth manifold that is a level set of a smooth function with no critical points in particular every closed set in x can be uniformly approximated by open sets whose boundaries are c infinity smooth one codimensional submanifolds of x finally since every hilbert manifold is diffeomorphic to an open subset of the hilbert space all of these results still hold if one replaces the hilbert space x with any smooth manifold m modeled on x

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

equilibrium problems in hadamard manifolds abstract an equilibrium theory is developed in hadamard manifolds the existence of equilibrium points for a bifunction is proved under suitable conditions and applications to variational inequality fixed point and nash equilibrium problems are provided the convergence of picard iteration for firmly nonexpansive mappings along with the definition of resolvents for bifunctions in this setting is used to devise an algorithm to approximate equilibrium points