extension theorems of continuous random linear operators on random domains abstract the central purpose of this paper is to prove the following theorem let u be a complete probability space b a normed linear space over the scalar field k e 2 b a separable random domain with linear subspace values and gr e k a continuous random linear operator where gr e x b x e denotes the graph of e then there exists a continuous random linear operator b k such that x x x e and sup x x b x 1 sup x x e x 1 for each in for the case where e is not separable a result similar to the above stated theorem is also given which generalizes and improves many previous results on random generalizations of the hahn banach theorem

random strict convexity and random uniform convexity in random normed modules abstract based on the analysis of stratification structure on random normed modules we first present random strict convexity and random uniform convexity in random normed modules then we establish their respective relations to classical strict and uniform convexity in the process some known important results concerning strict convexity and uniform convexity of lebesgue bochner function spaces can be obtained as a special case of our results further we also give their important applications to the theory of random conjugate spaces as well as best approximation finally we conclude this paper with some remarks showing that the study of geometry of random normed modules will also motivate the further study of geometry of probabilistic normed spaces

two fixed point theorems in complete random normed modules and their applications to backward stochastic equations abstract let f f p be a filtered probability space with a filtration f f t t 0 t satisfying the usual conditions and t a finite time l 0 f 0 the algebra of equivalence classes of f 0 measurable real valued random variables on l p f t r d the usual function space and l f 0 p f t r d the l 0 f 0 module generated by l p f t r d the usual backward stochastic equations b s e s are studied for their terminal conditions in l p f t r d motivated by the study of continuous time conditional mean conditional convex risk measure portfolio selection this paper for the first time formulates and studies a more general class of b s e s with their terminal conditions in l f 0 p f t r d the main results of this paper are to prove two fixed point theorems in complete random normed modules which are respectively the random generalizations of banach contraction mapping principle and browder kirk fixed point theorem and give their applications to the general class of b s e s

a study of boundedness in probabilistic normed spaces abstract it was shown in lafuerza guillen rodriguez lallena and sempi 1999 8 that uniform boundedness in a serstnev pn space v named boundedness in the present setting of a subset a v with respect to the strong topology is equivalent to the fact that the probabilistic radius r a of a is an element of d here we extend the equivalence just mentioned to a larger class of pn spaces namely those pn spaces that are topological vector spaces briefly tv spaces but are not serstnev pn spaces we present a characterization of those pn spaces whether they are tv spaces or not in which the equivalence holds then a characterization of the archimedeanity of triangle functions of type t l is given this work is a partial solution to a problem of comparing the concepts of distributional boundedness d bounded in short and that of boundedness in the sense of associated strong topology

on the vectoral ekeland s variational principle and minimal points in product spaces abstract phelps 13 noticed that the scalar ekeland s variational principle evp is equivalent to the existence of a minimal point of the epigraph of the corresponding function with respect to an appropriate order attouch and riahi 1 showed that evp is equivalent to the existence of maximal points with respect to cones satisfying some additional conditions taking these into account gopfert and tammer 6 7 established a maximal point theorem in a product space the aim of this paper is to obtain several minimal point theorems in product spaces and the corresponding variants of the vectorial evp

a study of boundedness in probabilistic normed spaces probabilistic normed spaces have been redefined by alsina schweizer and sklar we give a detailed analysis of various boundedness notions for linear operators between such spaces and we study the relationship among them and also with the notion of continuity

ekeland s variational principle and its equivalent theorems in vector optimization we present a simple proof of vectorial ekeland s variational principle vectorial caristi s fixed point theorem and vectorial takahashi s nonconvex minimization theorem equivalences between the three theorems are also given

averaging operators and positive contractive projections abstract the main topic of the present paper is a generalization of two theorems about positive contractive projections on c 0 x due to seever and kelley resp to the more general structure of algebras the theorems in question are the following if a is an archimedean semiprime algebra with the stone condition then every positive contractive projection t in a satisfies the identity t a tb t ta tb for all a b a seever furthermore t is averaging if and only if the range of t is a subalgebra kelley both theorems hold in particular for an archimedean unital algebra in contrast to the original proofs of seever and kelley in the c 0 x case our proofs are purely algebraic and order theoretical

on the kurchatov method for solving equations under weak conditions we present a new convergence analysis for the kurchatov method in order to solve nonlinear equations in a banach space setting in the semilocal convergence case the sufficient convergence conditions are weaker than in earlier studies such as argyros 2005 2007 ezquerro et al 2013 and kurchatov 1971 this way we extend the applicability of this method moreover in the local convergence case our radius of convergence is larger leading to a wider choice of initial guesses and fewer iterations to achieve a desired error tolerance numerical examples are also presented

adm pade technique for the nonlinear lattice equations adm pade technique is a combination of adomian decomposition method adm and pade approximants we solve two nonlinear lattice equations using the technique which gives the approximate solution with higher accuracy and faster convergence rate than using adm alone bell shaped solitary solution of belov chaltikian bc lattice and kink shaped solitary solution of the nonlinear self dual network equations sdnes are presented comparisons are made between approximate solutions and exact solutions to illustrate the validity and the great potential of the technique

some new results on the blaszak marciniak lattice backlund transformation and nonlinear superposition formula the so called blaszak marciniak lattice is considered by the dependent variable transformation the blaszak marciniak lattice is transformed into a quart linear form by introducing an auxiliary variable we further transform it into the bilinear form a corresponding backlund transformation is obtained furthermore a nonlinear superposition formula is proved rigorously as an application of the obtained results soliton solutions are derived

on almost surely bounded semigroups of random linear operators menger proposed transferring the probabilistic notions of quantum mechanics to the underlying geometry following menger s idea the notion of random metric spaces is a random generalization of that of metric spaces and also plays an important role in the study of random operator equations the main difficulty of this article is to work out a skill to give a property peculiar to a special semigroup of random operators which is not involved in the classical case subsequently some random operators equations are studied in particular we discuss the schrodinger type random equation which considerably generalizes the corresponding result in the sense of skorohod

complete monotone quasiconcave duality we introduce a notion of complete monotone quasiconcave duality motivated by some economic applications we show that this duality holds for important classes of quasiconcave functions

conditionally evenly convex sets and evenly quasi convex maps abstract evenly convex sets in a topological vector space are defined as the intersection of a family of open half spaces we introduce a generalization of this concept in the conditional framework and provide a generalized version of the bipolar theorem this notion is then applied to obtain the dual representation of conditionally evenly quasi convex maps which turns out to be a fundamental ingredient in the study of quasi convex dynamic risk measures

weak topologies for modules over rings of bounded random variables abstract in order to establish a functional analytic basis for representation theorems for conditional and multi period risk measures we study locally convex modules over the ring l g their topology is determined by seminorms as expected central mathematical tools of the analysis are hahn banach type and separation theorems which however have to be treated more carefully in the module case once a dual module is introduced one can establish a module version of the bipolar theorem we also prove the krein smulian as well as the alaoglu bourbaki theorem for modules for banach modules their reflexivity is characterized by a compactness criterium in a very weak topology

separation and duality in locally l 0 convex modules motivated by financial applications we study convex analysis for modules over the ordered ring l0 of random variables we establish a module analogue of locally convex vector spaces namely locally l0 convex modules in this context we prove hyperplane separation theorems we investigate continuity subdifferentiability and dual representations of fenchel moreau type for l0 convex functions from l0 modules into l0 several examples and applications are given

farkas lemma in random locally convex modules and minkowski weyl type results in l0 f rn abstract in this paper we investigate the structure of a finitely generated l 0 positively homogeneous and l 0 convex cone in a random locally convex module first we establish a farkas type characterization for such a kind of cone then we can further give some minkowski weyl type results for the specific case when the random locally convex module is the space l 0 f r n of equivalence classes of r n valued random variables

the algebraic structure of finitely generated l0 f k modules and the helly theorem in random normed modules abstract let k be the scalar field of real numbers or complex numbers and l 0 f k the algebra of equivalence classes of k valued random variables defined on a probability space f p in this paper we first characterize the algebraic structure of finitely generated l 0 f k modules and then combining the recently developed separation theorem in random locally convex modules we prove the helly theorem in random normed modules with the countable concatenation property under the framework of random conjugate spaces at the same time a simple counterexample shows that it is necessary to require the countable concatenation property

a characterization for an l μ k topological module to admit enough canonical module homomorphisms let k be the scalar field of all real or complex numbers let a be a finite measure space and let l k be the algebra of the equivalence classes of all k valued measurable functions defined on a l k is a topological algebra over k when endowed with the topology of convergence locally in measure topological modules over this topological algebra l k briefly l k topological modules are an extensive class of topological modules which arise naturally in the course of the study of the theory of probabilistic normed spaces the purpose of this paper is to show that an arbitrary regular l k topological module admits enough canonical module homomorphisms if and only if all of its quasi free submodules of finite rank are complemented in the sense of topological modules

a kantorovich type analysis for a fast iterative method for solving nonlinear equations abstract we revisit a fast iterative method studied by us in i k argyros on a two point newton like method of convergent order two int j comput math 88 2 2005 219 234 to approximate solutions of nonlinear operator equations the method uses only divided differences of order one and two function evaluations per step this time we use a simpler kantorovich type analysis to establish the quadratic convergence of the method in the local as well as the semilocal case moreover we show that in some cases our method compares favorably and can be used in cases where other methods using similar information cannot s amat s busquier v f candela a class of quasi newton generalized steffensen s methods on banach spaces j comput appl math 149 2 2002 397 406 d chen on the convergence of a class of generalized steffensen s iterative procedures and error analysis int j comput math 31 1989 195 203 numerical examples are provided to justify the theoretical results

characterizations of evenly convex sets and evenly quasiconvex functions the aim of this paper is to present a geometric characterization of even convexity in separable banach spaces which is not expressed in terms of dual functionals or separation theorems as an application an analytic equivalent definition for the class of evenly quasiconvex functions is derived

a new integrable symplectic map associated with lattice soliton equations a method is developed that extends the nonlinearization technique to the hierarchy of lattice soliton equations associated with a discrete 3 3 matrix spectral problem a new integrable symplectic map and its involutive system of conserved integrals are obtained by the nonlinearization of spatial parts and the time parts of lax pairs and their adjoint lax pairs of the hierarchy moreover the solutions of the typical system of lattice equations in the hierarchy are reduced to the solutions of a system of ordinary differential equations and a simple iterative process of the symplectic map

optimization of convex risk functions we consider optimization problems involving convex risk functions by employing techniques of convex analysis and optimization theory in vector spaces of measurable functions we develop new representation theorems for risk models and optimality and duality theory for problems with convex risk functions

farkas type results with conjugate functions we present some new farkas type results for inequality systems involving a finite as well as an infinite number of convex constraints for this we use two kinds of conjugate dual problems namely an extended fenchel type dual problem and the recently introduced fenchel lagrange dual problem for the latter which is a combination of the classical fenchel and lagrange duals the strong duality is established

probabilistic norms and convergence of random variables abstract we prove that the probabilistic norms of suitable probabilistic normed spaces induce convergence in probability l p convergence and almost sure convergence

on convergence of a new secant like method for solving nonlinear equations in this paper we prove that the order of a new secant like method presented recently with the so called order of 2 618 is only 2 414 some mistakes in the derivation leading to such a conclusion are pointed out meanwhile under the assumption that the second derivative of the involved function is bounded the convergence radius of the secant like method is given and error estimates matching its convergence order are also provided by using a generalized fibonacci sequence

best possible bounds for the distribution of a sum a problem of kolmogorov recently in answer to a question of kolmogorov g d makarov obtained best possible bounds for the distribution function of the sumx y of two random variables x andy whose individual distribution functions fx andfy are fixed we show that these bounds follow directly from an inequality which has been known for some time the techniques we employ which are based on copulas and their properties yield an insightful proof of the fact that these bounds are best possible settle the question of equality and are computationally manageable furthermore they extend to binary operations other than addition and to higher dimensions

risk preferences and their robust representation to address the plurality of interpretations of the subjective notion of risk we describe it by means of a risk order and concentrate on the context invariant features of diversification and monotonicity our main results are uniquely characterized robust representations of lower semicontinuous risk orders on vector spaces and convex sets this representation covers most instruments related to risk and allows for a differentiated interpretation depending on the underlying context that is illustrated in different settings for random variables risk perception can be interpreted as model risk and we compute among others the robust representation of the economic index of riskiness for lotteries risk perception can be viewed as distributional risk and we study the value at risk for consumption patterns which excerpt an intertemporality dimension in risk perception we provide an interpretation in terms of discounting risk and discuss some examples

equilibrium pricing in incomplete markets under translation invariant preferences we propose a general discrete time framework for deriving equilibrium prices of financial securities it allows for heterogeneous agents unspanned random endowments and convex trading constraints we give a dual characterization of equilibria and provide general results on their existence and uniqueness in the special case where all agents have preferences of the same type and in equilibrium all random endowments are replicable by trading in the financial market we show that a one fund theorem holds and give an explicit expression for the equilibrium pricing kernel

ergodic characterizations of reflexivity of banach spaces abstract let x be a banach space with a basis we prove the following characterizations i x is finite dimensional if and only if every power bounded operator is uniformly ergodic ii x is reflexive if and only if every power bounded operator is mean ergodic iii x is quasi reflexive of order one if and only if for every power bounded operator t t or t is mean ergodic

ekelandʼs variational principle for an l 0 valued function on a complete random metric space abstract motivated by the recent work on conditional risk measures this paper studies the ekeland s variational principle for a proper lower semicontinuous and lower bounded l 0 valued function where l 0 is the set of equivalence classes of extended real valued random variables on a probability space first we prove a general form of ekeland s variational principle for such a function defined on a complete random metric space then we give a more precise form of ekeland s variational principle for such a local function on a complete random normed module finally as applications we establish the bishop phelps theorem in a complete random normed module under the framework of random conjugate spaces

conditional lp spaces and the duality of modules over f algebras abstract motivated by dynamic asset pricing we extend the dual pairs theory of dieudonne 1942 and mackey 1945 to pairs of modules over a dedekind complete f algebra with multiplicative unit the main tools are a hahn banach theorem for modules of this kind a topology on the f algebra that has the special feature of coinciding with the norm topology when the algebra is a banach algebra and with the strong order topology of filipovic kupper and vogelpoth 2009 when the algebra of all random variables on a probability space g p is considered as a leading example we study in some detail the duality of conditional l p spaces

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

r matrix approach to lattice integrable systems an r matrix formalism is applied to the construction of the integrable lattice systems and their bi hamiltonian structure miura like gauge transformations between the hierarchies are also investigated in the end the ladder of linear maps between generated hierarchies is established and described

new solitary wave solutions for two nonlinear evolution equations abstract in this paper multiple exp function method is employed to investigate exact multiple wave solutions for 2 1 dimensional potential kadomtsev petviashvili equation and 3 1 dimensional jimbo miwa equation not only already known multiple wave solutions are recovered but also several new or more general multiple wave solutions are obtained

the bishop phelps theorem in complete random normed modules endowed with the e λ topology abstract in this paper we adopt a new approach so that we can prove that the bishop phelps theorem in complete random normed modules still holds under the e topology which solves an open problem posed in t x guo y j yang ekeland s variational principle for an l 0 valued function on a complete random metric space j math anal appl 389 2012 1 14

nonlinear superposition formulae for the differential difference analogue of the kdv equation and two dimensional toda equation in this paper nonlinear superposition formulae of the differential difference analogue of the kdv equation and two dimensional toda equation are proved rigorously some particular solutions of the differential difference analogue of the kdv equation are given as an illustrative application of the obtained result

a backlund transformation and nonlinear superposition formula for the belov chaltikian lattice the so called belov chaltikian lattice is considered by the dependent variable transformation the belov chaltikian lattice is transformed into a trilinear form by introducing an auxiliary variable we further transform it into the bilinear form a corresponding backlund transformation for it is obtained furthermore a nonlinear superposition formula is proved rigorously as an application of the obtained results soliton solutions are derived

backlund transformation and its superposition principle of a blaszak marciniak four field lattice a four field lattice furnished by blaszak and marciniak is transformed into bilinear form upon introducing two auxiliary independent variables a backlund transformation in bilinear form is found for the lattice and the corresponding nonlinear superposition formula is rigorously established as a consequence soliton solutions to the lattice are derived

stability in locally l0 convex modules and a conditional version of james compactness theorem abstract locally l 0 convex modules were introduced in filipovic et al 2009 10 as the analytic basis for the study of conditional risk measures later the algebra of conditional sets was introduced in drapeau et al 2016 8 in this paper we study locally l 0 convex modules and find exactly which subclass of locally l 0 convex modules can be identified with the class of locally convex vector spaces within the context of conditional set theory second we provide a version of the classical james theorem of characterization of weak compactness for conditional banach spaces finally we state a conditional version of the fatou and lebesgue properties for conditional convex risk measures and as application of the developed theory we establish a version of the so called jouini schachermayer touzi theorem for robust representation of conditional convex risk measures defined on a l type module

vector valued variational principles in the context of vector valued extensions of variational principles we are dealing with functions taking values in a banach space partially ordered by a closed convex pointed cone we introduce and study a new notion of semi continuity connected with the order and we improve the vector valued extensions of deville godefroy zizler perturbed minimization principle

algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations an algebraic structure related to discrete zero curvature equations is established it is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries the key of the theory is to generate nonisospectral flows t l l 0 from the discrete spectral problem associated with a given system of discrete evolution equations three examples are given

integrable discretizations of the 2 1 dimensional sinh gordon equation in this paper we propose two semi discrete equations and one fully discrete equation and study them by hirota s bilinear method these equations have continuum limits into a system which admits the 2 1 dimensional generalization of the sinh gordon equation as a result two integrable semi discrete versions and one fully discrete version for the sinh gordon equation are found backlund transformations nonlinear superposition formulae determinant solution and lax pairs for these discrete versions are presented

a basic strict separation theorem in random locally convex modules abstract the central idea of this paper is to make full use of the recently developed theory of random conjugate spaces to establish a basic strict separation theorem that is universally suitable in an arbitrary random locally convex module a series of interesting corollaries of the basic theorem are also included

hirota bilinear approach to a new integrable differential difference system a new integrable differential difference system is proposed by the dependent variable transformation the system is transformed into multilinear form by introducing an auxiliary variable we further transform it into the bilinear form a corresponding backlund transformation for it is obtained furthermore a nonlinear superposition formula is presented as an application of the obtained results soliton solutions to the system are derived

on quasi convex duality the familiar fenchel moreau rockafellar duality scheme deduced from a conjugation is shown to be applicable to quasi convex problems here quasi affine functions take the place of affine functions the links with other quasi convex dualities are examined

the drop theorem the petal theorem and ekeland s variational principle on montre que le principe variationnel d ekeland ou plus precisement une forme legerement modifiee est une consequence d un resultat geometrique connu sous le terme de theoreme de la goutte

relations between some basic results derived from two kinds of topologies for a random locally convex module the purpose of this paper is to exhibit the relations between some basic results derived from the two kinds of topologies namely the e topology and the stronger locally l0 convex topology for a random locally convex module first we give an extremely simple proof of the known hahn banach extension theorem for l0 linear functions as well as its continuous variant then we give the relations between the hyperplane separation theorems in d filipovic m kupper n vogelpoth separation and duality in locally l0 convex modules j funct anal 256 2009 3996 4029 and a basic strict separation theorem in t x guo h x xiao x x chen a basic strict separation theorem in random locally convex modules nonlinear anal 71 2009 3794 3804 in the process we also obtain a very useful fact that a random locally convex module with the countable concatenation property must have the same completeness under the two topologies as applications of the fact we prove that most of the previously established principal results of random conjugate spaces of random normed modules under the e topology are still valid under the locally l0 convex topology which considerably enriches financial applications of random normed modules

the james theorem in complete random normed modules the purpose of this paper is to generalize the classical james theorem characterizing the reflexivity of banach spaces to one characterizing the random reflexivity of complete random normed modules

the e support function of an e convex set and conjugacy for e convex functions a subset of a locally convex space is called e convex if it is the intersection of a family of open halfspaces an extended real valued function on such a space is called e convex if its epigraph is e convex in this paper we introduce a suitable support function for e convex sets as well as a conjugation scheme for e convex functions

bse s bsde s and fixed point problems in this paper we introduce a class of backward stochastic equations bses that extend classical bsdes and include many interesting examples of generalized bsdes as well as semimartingale backward equations we show that a bse can be translated into a fixed point problem in a space of random vectors this makes it possible to employ general fixed point arguments to establish the existence of a solution for instance banach s contraction mapping theorem can be used to derive general existence and uniqueness results for equations with lipschitz coefficients whereas schauder type fixed point arguments can be applied to non lipschitz equations the approach works equally well for multidimensional as for one dimensional equations and leads to results in several interesting cases such as equations with path dependent coefficients anticipating equations mckean vlasov type equations and equations with coefficients of superlinear growth

the mean ergodic theorem on random reflexive random normed modules in this paper we prove a mean ergodic theorem for an almost surely power bounded random linear operator on a random reflexive random normed module which general izes and improves several known important results

a unified approach to time consistency of dynamic risk measures and dynamic performance measures in discrete time in this paper we provide a flexible framework allowing for a unified study of time consistency of risk measures and performance measures also known as acceptability indices the proposed framework not only integrates existing forms of time consistency but also provides a comprehensive toolbox for analysis and synthesis of the concept of time consistency in decision making in particular it allows for in depth comparative analysis of most of the existing types of time consistency a feat that has not been possible before and which is done in the companion paper by the authors in our approach the time consistency is studied for a large class of maps that are postulated to satisfy only two properties monotonicity and locality we call these maps lm measures the time consistency is defined in terms of an update rule the form of the update rule introduced here is novel and is perfectly suited for developing the unifying framework that is worked out in this paper as an illustration of the applicabi

subdifferential representations of risk measures measures of risk appear in two categories risk capital measures serve to determine the necessary amount of risk capital in order to avoid ruin if the outcomes of an economic activity are uncertain and their negative values may be interpreted as acceptability measures safety measures pure risk measures risk deviation measures are natural generalizations of the standard deviation while pure risk measures are typically convex acceptability measures are typically concave in both cases the convexity concavity implies under mild conditions the existence of subgradients supergradients the present paper investigates the relation between the subgradient supergradient representation and the properties of the corresponding risk measures in particular we show how monotonicity properties are reflected by the subgradient representation once the subgradient supergradient representation has been established it is extremely easy to derive these monotonicity properties we give a list of examples

the algebra of conditional sets and the concepts of conditional topology and compactness abstract the concepts of a conditional set a conditional inclusion relation and a conditional cartesian product are introduced the resulting conditional set theory is sufficiently rich in order to construct a conditional topology a conditional real and functional analysis indicating the possibility of a mathematical discourse based on conditional sets it is proved that the conditional power set is a complete boolean algebra and a conditional version of the axiom of choice the ultrafilter lemma tychonoff s theorem the borel lebesgue theorem the hahn banach theorem the banach alaoglu theorem and the krein smulian theorem are shown

hilbert a modules abstract we consider real pre hilbert modules h on archimedean f algebras a with unit e we provide conditions on a and h such that a riesz representation theorem for bounded continuous a linear operators holds