dual fuzzy neighbourhood spaces i abstract a fuzzy neighbourhood space fns is determined by means of a system of fuzzy neighbourhoods of the ordinary points of that space we provide a general method for constructing bases for all fuzzy neighbourhood systems in a natural way from the fuzzy topologies of a special type of fuzzy topological spaces fts s which we call the fuzzy neighbourhood base spaces this construction is shown to be functorial of those new fts s the full subcategory of dual fuzzy neighbourhood spaces is shown to be isomorphic to the category of fns s and a fns will coincide with its dual fts iff it is topologically generated

dynamical systems disjoint from any minimal system furstenberg showed that if two topological systems x t and y s are disjoint then one of them say y s is minimal when y s is nontrivial we prove that x t must have dense recurrent points and there are countably many maximal transitive subsystems of x t such that their union is dense and each of them is disjoint from y s showing that a weakly mixing system with dense periodic points is in m the collection of all systems disjoint from any minimal system furstenberg asked the question to characterize the systems in m we show that a weakly mixing system with dense regular minimal points is in m and each system in m has dense minimal points and it is weakly mixing if it is transitive transitive systems in m and having no periodic points are constructed moreover we show that there is a distal system in m recently weiss showed that a system is weakly disjoint from all weakly mixing systems iff it is topologically ergodic we construct an example which is weakly disjoint from all topologically ergodic systems and is not weakly mixing

multiplicity results for some elliptic systems near resonance with a nonprincipal eigenvalue abstract some multiplicity results of solutions are obtained for a class of elliptic systems which are near resonance with a nonprincipal eigenvalue by the classical saddle point theorem and a local saddle point theorem in critical point theory

multiple solutions for asymptotically linear elliptic systems in this article we consider cooperative and noncooperative elliptic systems that are asymptotically linear at infinity we obtain infinitely many solutions with small energy if the potential is even if the noncooperative system is resonant both at zero and at infinity then the number of nontrivial solutions depends on the dimension of the eigenspaces between resonant values

on regularity axioms in fuzzy neighbourhood spaces abstract two notions of regularity in fuzzy neighbourhood spaces fns s have been introduced one by t m g ahsanullah and the other by the authors we prove that those two notions coincide we then characterize a regular fns as a fns whose level topologies for all 0

fuzzy nets and their application fuzzy net theory is enriched by the introduction of the fuzzy net of fuzzy sets the limsup liminf and limit of a fuzzy net of fuzzy sets are defined and their various properties are discussed alternative characterizations based on the notion of a fuzzy net are provided for several fuzzy topological concepts including open and closed fuzzy sets fuzzy continuity maps with closed fuzzy graph open fuzzy mapping etc thus the net theoretic approach is shown to be a promising tool in fuzzy topology

nonconventional ergodic averages and nilmanifolds we study the l2 convergence of two types of ergodic averages the first is the average of a product of functions evaluated at return times along arithmetic progressions such as the expressions appearing in furstenberg fs proof of szemer ledi fs theorem the second average is taken along cubes whose sizes tend to for each average we show that it is sufficient to prove the convergence for special systems the characteristic factors we build these factors in a general way independent of the type of the average to each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold from the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density

fuzzy metric neighbourhood spaces abstract hohle founded the first coherent topological study of menger s probabilistic metric spaces by endowing them with fuzzy t uniformities we continue hohle s work in the special case when the triangular norm t used is min that is when the resulting probabilistic metric uniformities are lowen fuzzy uniformities in this case we call the induced fuzzy topological spaces fuzzy pseudo metric neighbourhood spaces we introduce axioms of n regularity and n normality which are satisfied in fuzzy pseudo metric neighbourhood spaces we introduced a lowen fuzzy uniform topology on the fuzzy real line ri called the n euclidean topology by means of a fuzzy probabilistic metric on r i called the axiom of n complete regularity which characterizes the lowen fuzzy uniform topologies

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

existence and multiplicity of solutions for asymptotically linear noncooperative elliptic systems abstract using the minimax methods in critical point theory and a generalized landesman lazer type condition we establish two existence results of solutions for asymptotically linear noncooperative elliptic systems at resonance besides this we obtain two solutions in the case of near resonance

a new approach to fuzzy topological spaces and various fuzzy topological notions defined by nets n fuzzy spaces introduced by the authors are further investigated various fuzzy topological motions such as fuzzy local compactness functions with closed fuzzy graph fuzzy subcontinuity inverse fuzzy subcontinuity etc are studied in the light of n i these notions are found to be more coherent in n i than in the chang s category

pseudo closure operators in fuzzy topological spaces abstract the concept of a pseudo closure operator s of an fts x has been introduced in 6 in this paper we give an axiomatic characterization of pseudo closure operators also we give a necessary and sufficient condition on an fts x so that is the unique fuzzy topology which induces s at the end of the paper we give a characterization of fts s of structure a 6 in terms of pseudo closure operators

semi induced fuzzy topologies abstract induced fuzzy topologies were first introduced by wong 15 and weiss 14 they were studied by lowen who called them topologically generated fuzzy topologies 7 an induced fuzzy topology includes its modification topology we generalize induced fuzzy topologies to semi induced fuzzy topologies viz those which include their modification topologies the yet more general fuzzy topologies of structure c are also introduced the fully stratified fuzzy topological spaces of structure c are precisely the fuzzy neighbourhood spaces of lowen 4 in subsequent papers this relationship will be established and benefited from the two axioms of semi induction and structure c hold trivially for crisp fuzzy topologies we call such axioms fuzzy structural axioms in fuzzy topology in this paper we analyze the axioms of structure c semi induction and induction in this context we show how these axioms are linked to the three structural asioms of structures a b and d introduced by us in 11 the basic notions and results pertaining to structures a b and d are compiled in the first section the behaviour of warren s fuzzy boundary in semi induced fuzzy topological spaces fts s for short is also explained

on fuzzy almost continuous functions abstract in this paper we redefine the notions of open fuzzy set closed fuzzy set adherent point of a fuzzy set and closure of a fuzzy set all the results pertaining to these notions see velicko 17 are extended to fuzzy setting and several related characterization of fuzzy almost continuous functions are obtained which are the counterparts of the corresponding results in general topological spaces 12 some more properties of fuzzy almost continuous functions are established subsequently a characaterization of normal fuzzy topological spaces is also obtained the notion of fuzzy almost quasi compact mapping is introduced and its characterizations and relationship with fuzzy almost continuous functions are discussed

the existence of solutions for a class of semilinear elliptic systems in this paper some new existence theorems of weak solutions for a class of semilinear elliptic systems are obtained by means of the local linking theorem and the saddle point theorem

linear fuzzy neighborhood spaces some of the properties of linear fuzzy neighborhood spaces are studied a special class of these spaces is the family of all fuzzy linear spaces whose topology is given by probabilistic seminorms as in the classical case it is shown that a linear fuzzy neighborhood space is probabilistic normable iff it is hausdorff and it has an n bounded convex fuzzy neighborhood of zero

fuzzy t locality spaces abstract a lowen fuzzy topological space is said to be localizable at points if it is completely determined in some prescribed manner by a system of fuzzy neighbourhoods called here fuzzy localities of its ordinary points we introduce for each lower semicontinuous triangular norm t a new type of those spaces which we call fuzzy t locality spaces the particularity of a fuzzy t locality space is that its fuzzy interior operator is obtained from its t locality system through a definition of fuzzy containment of fuzzy sets that employs the t residuated implication operator the origins of this new theory lie in the theory of probabilistic metric spaces indeed fuzzy t locality spaces provide a fuzzy topological framework for a naturally defined notion of metric t continuity of functions between probabilistic t metric spaces in a manner described in the last two sections of this article before that we establish characterizations and aspects of good behaviour for fuzzy t locality spaces including that every t locality system has an open basis we briefly study their level topologies and we show that all topologically generated spaces are fuzzy t locality spaces we prove that the category t fls of fuzzy t locality spaces and continuous functions between them is a topological category we show that it is closed in fts under the formation of optimal lifts of sources we provide a condition on t under which t fls becomes closed in fts also under the formation of co optimal lifts of sinks

generalized possibility measures we define a generalized possibility measure gpm on a set x to be a function from the power set of x to the unit interval i 0 1 whereby the measure of a finite union of subsets of x equals the maximum of their measures we characterize gpms as the infima of downward directed families of possibility measures this helps to confer an intuitive meaning on gpms we discuss all directed families of possibility measures or equivalently of possibility distributions which generate a given gpm and we identify the largest such family we introduce extensions of gpms to real functions on ix obtained through a continuous triangular norm we provide characterizations and alternative descriptions for those extensions

the productivity classes of fuzzy topologies in a previous article fuzzy sets and systems 30 1989 175 191 we introduced classes of fuzzy topological spaces fts s and other special classes of fts s those are tntimatley related to the productivity properties of fuzzy closure and fuzzy interior operators we here provide a number of necessary and sufficient conditions on a fts to be a member of one of those classes we use those characterizations to establish that each one of our classes of fts s is closed under products and quotients in the category of fuzzy topological spaces

on product fuzzy topological spaces abstract we provide complete answers to the following three questions in fuzzy topology when is the fuzzy closure operator finitely productive on a given family of fuzzy topological spaces which need not be fully stratified when is the fuzzy interior operator finitely productive on such a family when are the coordinate projections open maps from arbitrary products of members of such a family the answer to the first question is the harder one to establish the second and the third question turn out to have same answer

fuzzy almost continuity and its pointwise characterization by dual points and fuzzy nets abstract we introduce the notions of fuzzy almost continuity and fuzzy weak continuity at a fuzzy point following the pointwise approach we investigate their fuzzy topological properties we also obtain a pointwise characterization of fuzzy almost continuous functions by dual points and fuzzy nets the main idea behind this work is to establish a local theory in the study of continuity and its weaker forms in general fuzzy topological spaces

a level topologies criterion for lowen fuzzy uniformizability throughout x will be a nonempty set we consider a subset m of x as a crisp fuzzy set and denote this again by m we denote the strong cut of a fuzzy set e ix by and its weak cut by e i

fuzzy t neighbourhood spaces part 1 t proximities this is the first one of three articles that are intended to be published consecutively in this journal together they constitute a three part work on the unified subject of fuzzy t neighbourhood spaces and their fuzzy t proximities where t stands for any continuous triangular norm these new notions generalize to arbitrary t corresponding ones due to artico hohle lowen and moresco in this part we define and study notions of fuzzy t proximity one for each t their definition subsumes that of fuzzy proximity due to artico and moresco fuzzy sets and systems 21 1987 85 as our min proximity in particular we study proximity maps the fuzzy topological space associated with a t proximity which in the second part will be proved a fuzzy t neighbourhood space t proximities as fuzzy relations in ordinary power sets t proximal neighbourhood systems and relationships to the hohle lowen fuzzy t uniformities j math anal appl 82 1981 370 manuscripta math 38 1982 289 the second part of this three part work will be concerned with the categories of fuzzy t neighbourhood spaces one for each t whereas we devote the third part to our t separation axioms in those categories

fuzzy closed and fuzzy perfect mappings abstract it is shown that a fuzzy continuous map f is fuzzy perfect iff f i z is fuzzy closed for every fuzzy topological space z

multiplicity of solutions for resonant elliptic systems we establish the existence and multiplicity of solutions for some resonant elliptic systems the results are proved by applying minimax arguments and morse theory

multiple solutions for an elliptic system near resonance abstract we consider an elliptic system with linear part depending on two parameters and a sublinear perturbation we obtain the existence of at least two solutions when the linear part is near resonance the system is associated with a strongly indefinite functional and the solutions are obtained through saddle point theorem and galerkin approximation

on separation axioms and ti fuzzy continuity abstract we introduce and generalise concepts of t continuous functions in a fuzzy setting and study them in connection with the fuzzy continuity and fuzzy separation axioms some fuzzy topological properties are also investigated under such functions

pseudo closure operators in fuzzy neighbourhood base spaces abstract fuzzy neighbourhood base spaces and dual fuzzy neighbourhood spaces are two special types of fuzzy topological spaces they were introduced by morsi as useful tools for the theory explanation and application of lowen s fuzzy neighbourhood systems the notion of the pseudoclosure operator of a fuzzy topological space was introduced by mashhour et al as a tool in establishing a necessary and sufficient condition for the productivity of fuzzy kuratowski operators pseudo closure operators carry partial information about the fuzzy closure operators of their respective spaces we establish relationships between the pseudo closure operators of fuzzy neighbourhood base spaces their fuzzy closure operators and their level topologies we provide counterexamples to disprove other nonvalid relationships we also study some categorical aspects of the operation of taking the pseudo closure operators of fuzzy topological spaces

connectedness in fuzzy topological spaces abstract we introduce four notions of connectedness for an arbitrary fuzzy set and discuss all the implications that exist between them using these notions various equivalence relations are defined on the set of fuzzy points components and quasi components are formed as the union of certain equivalence classes a fuzzy space is shown to be a dispoint union of its components in all the four cases and non overlapping union of its quasi components

multiple solutions for a class of nonlinear boundary value problems near resonance a variational approach abstract a variational approach is given to the existence of multiple solutions for some nonlinear boundary value problems near resonance at the first eigenvalue

fuzzy subcontinuity inverse fuzzy subcontinuity and a new category of fuzzy topological spaces the notions of subcontinuity and inverse subcontinuity are extended to the fuzzy setting adopting fuzzy net theoretic approach various properties of fuzzy subcontinuous and inversely fuzzy subcontinuous mappings are studied a category of fuzzy topological spaces is constructed canonical example is constructed by using fuzzy translation of fuzzy numbers introduced by rodabaugh to show that fuzzy subcontinuity does not imply kfuzzy continuity in the process n compactness is characterized and fuzzy unit interval is shown to be n compact also a direct proof is provided showing that fuzzy addition of fuzzy numbers is fuzzy continuous

various fuzzy topological notions and fuzzy almost continuity abstract we extend the notion of almost continuity in the sense of husain 1966 to the fuzzy setting we obtain a pointwise characterization of the notion by dual point and fuzzy net also we formulate conditions under which the two notions of fuzzy almost continuity the one defined in this paper and the one defined by ajmal and azad 1990 turn out to be equivalent the main idea behind this work is to show that using the localized approach developed in earlier papers the study of fuzzy almost continuity can be carried out as successfully as that of its counterpart in general topology in the process we exhibit the compatibility of various existing fuzzy topological notions such as fuzzy regularity c i connectedness i 3 4 c i local connectedness i 3 4 fuzzy t 1 axiom fuzzy t 2 axiom etc

a variational approach to subquadratic perturbations of elliptic systems abstract we consider subquadratic perturbations of semilinear elliptic systems which are in variational form a unified approach to both the cooperative and noncooperative problems is presented and a condition of nonquadraticity at infinity is introduced to treat the resonant cases through minimax techniques

existence and multiplicity results for some elliptic systems at resonance abstract the existence and multiplicity of weak solutions for some resonant elliptic systems are obtained by using ekeland s variational principle the mountain pass theorem and the saddle point theorem in critical point theory

nilsequences and a structure theorem for topological dynamical systems abstract we characterize inverse limits of nilsystems in topological dynamics via a structure theorem for topological dynamical systems that is an analog of the structure theorem for measure preserving systems we provide two applications of the structure the first is to nilsequences which have played an important role in recent developments in ergodic theory and additive combinatorics we give a characterization that detects if a given sequence is a nilsequence by only testing properties locally meaning on finite intervals the second application is the construction of the maximal nilfactor of any order in a distal minimal topological dynamical system we show that this factor can be defined via a certain generalization of the regionally proximal relation that is used to produce the maximal equicontinuous factor and corresponds to the case of order 1

a new approach to fuzzy topological spaces and fuzzy perfect mappings abstract a net theoretic approach is provided for defining a fuzzy topological space fuzzy topological spaces so defined form a category the main features and advantages of this category which is denoted by n here are discussed the notion of a perfect mapping is introduced and studied in the fuzzy setting it is found that in n a fts is n compact iff any mapping x x where x is any singleton is fuzzy perfect a fuzzy continuous mapping x y from an n compact fts x to a hausdorff fts y is fuzzy perfect in n however these results do not hold in c the chang s category in general it is found that the divergences in the study of fuzzy perfect mappings disappear if the study is carried out in n instead of c

on co optimal lifts in the category of fuzzy neighbourhood spaces the category fns of fuzzy neighbourhood spaces and their continuous functions is closed under the formation of optimal and co optimal lifts in the larger topological category fts of fuzzy topological spaces but while the fuzzy neighbourhood system and the level topologies of an optimal lift of a source in fns have simple descriptions in terms of the fuzzy neighbourhood systems and the level topologies respectively of the source spaces analogous descriptions are yet to be given for both constructs for a co optimal lift of a sink in fns we here establish such descriptions with the help of a functor introduced earlier by one of us with codomain fns and domain another full subcategory of fts the category fnbs of fuzzy neighbourhood base spaces

fuzzy t neighbourhood spaces part 2 t neighbourhood systems we explore a notion of fuzzy t neighbourhood spaces for any continuous triangular norm t and we present on this notion a unified treatment our theory on one hand generalizes the theory of lowen fuzzy sets and systems 7 1982 65 from t min to arbitrary t which has been the progenitor of this work and on the other hand it is strongly related to the theory of l neighbourhoods of hohle mathematics of fuzzy sets logic topology and measure theory kluwer academic publishers dordrecht 1999 when l is restricted to the lattice 0 1 we study the t neighbourhood bases and systems of our spaces their level topologies as well as their relationship to the lowen hohle fuzzy t uniformities j math anal appl 82 1981 370 manuscripta math 38 1982 289 and to our fuzzy t proximities introduced in part 1 we show that the nearness concept underlying a fuzzy t neighbourhood space is a fuzzy relation of closeness between its ordinary subsets and ordinary points in particular our spaces are in canonical one to one correspondence with the t probabilistic topological spaces of frank j math anal appl 34 1971 67 we demonstrate that each full subcategory t fns of fts of fuzzy t neighbourhood spaces is a topological category to do that we characterize t fns within fts and we characterize continuity of functions within t fns in terms of t neighbourhood bases

the urysohn lemma for fuzzy neighbourhood spaces abstract the framework fns of lowen s fuzzy neighbourhood spaces is a special environment for the study of fuzzy topology it encompasses its own metrics uniformities and proximities among other interesting constructions this is a fresh and fast growing field of research it promises to develop quickly to a calculus which combines depth with applicability within this framework we establish the urysohn lemma for the mashhour morsi axiom of n normality the formulation employs the mashhour morsi n euclidean space as the suitable fuzzy analogue of the real line the proof makes use of the artico moresco fuzzy proximities the lowen fuzzy uniformities and the mashhour morsi axiom of n complete regularity

category n and functions with closed fuzzy graph abstract a net theoretic approach for defining a fuzzy topological space is provided the fuzzy topological spaces so obtained form a category which we denote by n in this paper it is found that the limitations exhibited by the chang category particularly with respect to projections closure of a fuzzy set compactness etc no longer exist in n the notion of closed fuzzy graph is investigated the divergences shown by this notion are immediately put to rest if the study is carried out in n instead of c the chang category

some more results on completely induced fuzzy topological spaces abstract the object of the paper is to present some new results on completely induced fuzzy topological spaces and completely semi induced spaces earlier introduced by the authors in this journal when a fuzzy topological space becomes a completely induced space is discussed examples of completely induced space are also cited

fuzzy neighborhood spaces abstract in this paper we introduce a subcategory of the category of fuzzy topological spaces and we prove basic properties of this subcategory the main idea behind these spaces is the notion of fuzzy neighborhoods which enables us to introduce a local theory of fuzzy topologies

existence and multiplicity results for some elliptic systems with discontinuous nonlinearities abstract in this paper we complete two tasks we first extend the critical point theorem obtained by li and willem s j li m willem applications of local linking to critical point theory j math anal appl 189 1995 6 32 to the nonsmooth case in which the energy functional is locally lipschitz and satisfies the weaker nonsmooth cerami condition then we study some semilinear elliptic systems with discontinuous nonlinearities and obtain some existence and multiplicity results

regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence abstract inverse limits of nilsystems in topologically dynamical systems were characterized by host kra and maass recently namely for each d n a certain generalization of the regionally proximal relation was introduced and for a distal minimal system it was shown that such a relation is an equivalence one which determines the maximal d step nilfactor one of the main results in this article is to show that the above results hold for a general minimal system a combinatorial consequence is also deduced which is the topological correspondence of the result obtained by host and kra for positive upper banach density subsets using ergodic methods

fuzzy t neighbourhood spaces part 3 t separation axioms we address the problem of identifying a useful set of mutually compatible separation axioms for each category t fns of t neighbourhood spaces it should contain one of t complete regularity that characterizes t uniformizability as indeed we achieve here this seems to necessitate the parameterization of most separation axioms by the triangular norm t with the exception of the two axioms t0 and r0 which have categorical definitions valid in any topological category we introduce t separation axioms in terms of t and fuzzy closure operators only making them meaningful for the whole of fts we then characterize them within t fns in terms of t fuzzy closures of crisp fuzzy subsets and t neighbourhoods we study our t axioms and establish some entailments between them our t complete regularity commits us to investigate a t real line which is a t neighbourhood space induced by a certain probabilistic t metric due to hohle fuzzy sets and systems 1 1978 311

higher order regionally proximal equivalence relations for general minimal group actions abstract we introduce higher order regionally proximal relations suitable for an arbitrary acting group for minimal abelian group actions these relations coincide with the ones introduced by host kra and maass our main result is that these relations are equivalence relations whenever the action is minimal this was known for abelian actions by a result of shao and ye we also show that these relations lift through extensions between minimal systems answering a question by tao given a minimal system we prove that the regionally proximal equivalence relation of order d corresponds to the maximal dynamical antolin camarena szegedy nilspace factor of order at most d in particular the regionally proximal equivalence relation of order one corresponds to the maximal abelian group factor finally by using a result of gutman manners and varju under some restrictions on the acting group it follows that the regionally proximal equivalence relation of order d corresponds to the maximal pronilfactor of order at most d a factor which is an inverse limit of nilsystems of order at most d

completely induced fuzzy topological spaces abstract the object of this paper is to introduce and to study the concepts of completely induced fuzzy topological spaces and completely lower semi continuous functions completely lower semi continuous functions turn out to be the natural tool for studying the completely induced fuzzy topological spaces

connectedness and local connectedness in fuzzy topological spaces and heyting algebra valued sets abstract in this paper we introduce four notions of local connectedness in fuzzy topological spaces the question of these notions to be good extensions is discussed and their preservation under fuzzy quotients fuzzy sums and fuzzy products is investigated moreover we introduce the notion of quasi local connectedness which is a peculiarity of the fuzzy setting

the structure theory of nilspaces iii inverse limit representations and topological dynamics abstract this paper forms the third part of a series by the authors 12 11 concerning the structure theory of nilspaces a nilspace is a compact space x together with closed collections of cubes c n x x 2 n n 1 2 satisfying some natural axioms our goal is to extend the structure theory of nilspaces obtained by antolin camarena and szegedy and to provide new proofs our main result is that under the technical assumption that c n x is a connected space for all n then x is isomorphic in a strong sense to an inverse limit of nilmanifolds this is a direct and slight generalization of the main result of antolin camarena and szegedy we also apply our methods to obtain structure theorems in the setting of topological dynamics specifically if h is a group subject to very mild topological assumptions and h x is a minimal dynamical system then we give a simple characterization of the maximal pronilfactor of x this generalizes the case h z which is a theorem of host kra and maass although even in that case we give a significantly different proof

fuzzy topological spaces we start by discussing a class of special fuzzy topological spaces that is the product induced spaces s first we show that every fuzzy topological space is topologically isomorphic with a certain topological space and then proceed to prove every open fuzzy set is defined by some lower semicontinuous function taking this as the background we introduce the concept of dual points 6 and thus establish a kind of neighborhood structure of fuzzy points such that the q neighborhood 4 which is one of the important notions in fuzzy topology and the neighborhood arc integrated in this structure this neighborhood structure will be the core of our developing the theory of fuzzy topological spaces we introduce the concept of strong quasi discoincident 1 i and so give a group of fuzzy separation properties which is a most natural generalization of the usual separation properties next we introduce a kind of fuzzy metrics and use this metrics directly to discuss the fuzzy metric space by means of fuzzy points wc define kinds of uniformities discuss their fundamental properties and extend weil s theorem on usual topology to fuzzy topological spaces and hence obtain their separation character naturally these fuzzy uniformities can still be characterized by a family of fuzzy metrics finally we discuss the problem of fuzzy mctrization on fuzzy topological spaces and obtain a fuzzy metrization theorem which contains the nagata smirnov theorem as a special example

multiple solutions for neumann and periodic problems with singular ϕ laplacian abstract we use the critical point theory for convex lower semicontinuous perturbations of c 1 functionals to establish existence of multiple radial solutions for some one parameter neumann problems involving the operator v div v 1 v 2 similar results for periodic problems are also provided