a liouville type theorem for schrodinger operators in this paper we prove a sufficient condition in terms of the behavior of a ground state of a symmetric critical operator p 1 such that a nonzero subsolution of a symmetric nonnegative operator p 0 is a ground state particularly if p j v j for j 0 1 are two nonnegative schrodinger operators defined on omega subseteq mathbb r d such that p 1 is critical in with a ground state the function psi nleq 0 is a subsolution of the equation p 0 u 0 in and satisfies psi leq c varphi in then p 0 is critical in and psi is its ground state in particular psi is up to a multiplicative constant the unique positive supersolution of the equation p 0 u 0 in similar results hold for general symmetric operators and also on riemannian manifolds

positive unbounded and monotone solutions of the singular second painleve equation on the half line abstract in this paper we study singular boundary value problems on the half line and we prove the existence of a global monotone positive and unbounded solution the latter satisfies a neumann condition at the origin and has prescribed asymptotic behavior at infinity our approach is based on a generalization of the kneser s property continuum of the cross sections of the solutions funnel i e on the properties of the so called consequent mapping and on properties of the associated vector field on the face space two applications one on the well known second painleve type equation which is related to superconductivity theory and a second in the theory of colloids clarify our results

blow up of solutions of semilinear euler poisson darboux equations with nonlocal boundary conditions this article studies the hyperbolic initial nonlocal boundary value problem u t t k t u t u x x u 0 0 u x 0 u 0 x u t x 0 0 0 0 u a t 0 an y u y t q dy t 0 where k is a real number p and q are nonnegative constants and u 0 m and n are given functions criteria for u to blow up in finite time are given

multiple solutions of singular three point boundary value problems on 0 abstract in this paper we study the existence of single and multiple solutions of three point boundary value problems for the following nonlinear singular second order differential equations x t p x t q x t h t f t x t 0 t 0 a x 0 b x 0 k x c 0 lim t x t e r t d 0 where p b 0 a k 0 q 0 0 r 0 p p 2 4 q 2 h 0 0 is continuous and may be singular at t 0 f 0 0 is continuous and may take a negative value by applying the technique of lower and upper solutions and the theory of topological degree we obtain the conditions for the existence of at least one solution and at least three solutions respectively

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

higher order m point boundary value problems on time scales in this paper we investigate the existence of positive solutions for nonlinear even order m point boundary value problems on time scales by means of fixed point theorems

singular boundary value problems on a half line singular boundary value problems of nonlinear second order differential equations on a half line are considered the main methods used are modified upper and lower solution techniques necessary and sufficient conditions for the existence of solutions to a class of sublinear singular boundary value problems are obtained

the existence of countably many positive solutions for one dimensional p laplacian with infinitely many singularities on the half line we study the existence of countably many positive solutions of boundary value problems on the half line for differential equations of second order the fixed point index theory and a new fixed point theorem in cones are used

existence theory for nonlinear fredholm and volterra integral equations on half open intervals in this chapter we present existence theory for the nonlinear fredholm integral equation y t h t smallint 0 t k t s g s y s ds 4 1 1 and the nonlinear volterra integral equation y t h t smallint 0 t k t s g s y s ds 4 1 2 when both are defined on the compact interval 0 t naturally we first concern ourselves with existence principles for both equations

finiteness conditions co frobenius hopf algebras and quantum groups abstract some finiteness conditions for infinite dimensional coalgebras particularly right or left semiperfect coalgebras or co frobenius hopf algebras are studied as well examples of co frobenius hopf algebras are constructed via a hopf algebra structure on an ore extension of a group algebra and it is shown that bicrossproducts of co frobenius hopf algebras are themselves co frobenius

twin solutions to singular dirichlet problems the existence of two nonnegative solutions to dirichlet second order boundary value problems is established in this paper our nonlinearity may be singular at y 0 t 0 and or t 1

bicrossproducts of multiplier hopf algebras abstract in this paper we generalize majid s bicrossproduct construction we start with a pair a b of two regular multiplier hopf algebras we assume that b is a right a module algebra and that a is a left b comodule coalgebra the right action of a on b gives rise to the smash product a b the left coaction of b on a gives a possible coproduct on a b we discuss in detail the necessary compatibility conditions between the action and the coaction for to be a proper coproduct on a b the result is again a regular multiplier hopf algebra majid s construction is obtained when we have hopf algebras we also look at the dual case constructed from a pair c d of regular multiplier hopf algebras where now c is a left d module algebra while d is a right c comodule coalgebra we show that indeed these two constructions are dual to each other in the sense that a natural pairing of a with c and of b with d yields a duality between a b and the smash product c d we show that the bicrossproduct of an algebraic quantum group is again an algebraic quantum group i e a regular multiplier hopf algebra with integrals the algebra case is also considered some special cases are treated and they are related with other constructions available in the literature the basic example coming from a not necessarily finite group g with two subgroups h and k such that g k h and h k e where e is the identity of g is used to illustrate our theory more examples will be considered in forthcoming papers on the subject

principal eigenvalues and sturm comparison via picone s identity we characterize the principal eigenvalues and eigenfunctions in rn and present comparison results for higher dimensional p laplacian our main tool is picone s identity in this way we extend several recent results on spectral and comparison properties for differential equations

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

criticality theory of half linear equations with the p a laplacian abstract we study positive solutions of half linear second order elliptic equations of the form q a v u div u a p 2 a x u v x u p 2 u 0 in where 1 p is a domain in r n n 2 v l loc a a i j l loc r n 2 is a symmetric and locally uniformly positive definite matrix in and a 2 a x i j 1 n a i j x i j x 1 n r n we extend criticality theory which has been established for linear operators and for half linear operators involving the p laplacian to the operator q a v we prove liouville type theorems and study the behavior of positive solutions of the equation q a v u 0 near an isolated singularity and near infinity in and obtain some perturbations results

the drinfel d double of multiplier hopf algebras abstract let a b be a pairing of two regular multiplier hopf algebras a and b the drinfel d double associated to this pairing is constructed by using appropriate representations of a and b on the same vector space b a we realize the drinfel d double denoted by d as an algebra of operators on the vector space b a in the case that a b is a multiplier hopf algebra pairing we prove that d is again a multiplier hopf algebra if a and b carry positive integrals we prove that d also has a positive integral this proof is not given before

calculus of variations for a boundary value problem of differential system on the half line boundary value problem of second order differential equations on the half line is investigated in this paper the method mainly depends on the calculus of variations to the classical functionals sufficient conditions are obtained for the existence of the solutions

the existence of countably many positive solutions for nonlinear singular m point boundary value problems on the half line in this paper by introducing a new operator improving and generating a p laplace operator for some p 1 we study the existence of countably many positive solutions for nonlinear boundary value problems on the half line f u a t f u t 0 0r is the increasing homeomorphism and positive homomorphism and f 0 0 we show the sufficient conditions for the existence of countably many positive solutions by using the fixed point index theory and a new fixed point theorem in cones

twisted tensor product of multiplier hopf algebras abstract we combine two not necessarily commutative or co commutative multiplier hopf algebras to a new multiplier hopf algebra we use the construction of a twisted tensor product for algebras as introduced by a van daele we then proceed to find sufficient conditions on the twist map for this twisted tensor product to be a multiplier hopf algebra with the natural comultiplication for usual hopf algebras we find that majid s double crossed product by a matched pair of hopf algebras is exactly the twisted tensor product hopf algebra according to an appropriate twist map starting from two dually paired multiplier hopf algebras we construct the drinfel d double multiplier hopf algebra in the framework of twisted tensor products

twisted tensor coproduct of multiplier hopf algebras we put a non trivial comultiplication on the natural tensor product algebra of two multiplier hopf algebras by means of a cotwisting map as a special case we characterize the dual of the drinfel d double of an algebraic quantum group because any finite dimensional hopf algebra is an algebraic quantum group our characterization applies to the dual of the drinfel d double of a finite dimensional hopf algebra then it coincides with a result of lu

boundary value problems for impulsive integro differential equations on unbounded domains in a banach space in this paper the fixed point theory is used to investigate the existence and uniqueness of solutions of two point boundary value problems bvps for second order non linear impulsive integro differential equations on infinite intervals in a banach space

the analytic structure of algebraic quantum groups abstract in adv math 140 1998 323 366 van daele introduced the notion of an algebraic quantum group we proved in internat j math 8 8 1997 1067 1139 that such algebraic quantum groups give rise to reduced c algebraic quantum groups in the sense of j kustermans s vaes ann sci ecole norm sup 4 33 2000 837 934 after introducing the universal c algebraic quantum group associated to an algebraic one we will pull down the analytic structure of this c algebraic quantum group to the algebraic quantum group the multiplier algebra of the dual quantum group will be realized as a space of linear functionals on the original algebra we also identify the analytic structure of the dual quantum group in terms of the analytic structure of the original algebraic quantum group

heteroclinic connections for fully non linear non autonomous second order differential equations we investigate the solvability of the following strongly non linear non autonomous boundary value problem p a x t x t f t x t x t a e t r x x with given constants where a x is a generic continuous positive function and f is a caratheodory non linear function we show that the solvability of p is strictly connected to a sharp relation between the behaviors of f t x as x 0 and f x x as t such a relation is optimal for a wide class of problems for which we prove that p is not solvable when it does not hold

positive solutions for second order m point boundary value problems by using the schauder fixed point theorem and analysis method we establish the existence of solutions for the m point boundary value problem u t a t f u 0 u 0 0 u 1 i 1m 2kiu i b where b ki 0 i 1 2 m 2 0 b

unbounded solutions of the singular boundary value problems for second order differential equations on the half line this paper presents some results on the existence of multiple unbounded solutions for the boundary value problems for second order differential equations on half line and on the uniqueness of unbounded solution

a sharper condition for the solvability of a three point second order boundary value problem letf 0 1 r2 rbe a function satisfying caratheodory s conditions ande t l1 0 1 let 0 1 r 1 1 be given this paper is concerned with the problem of existence of a solution for the three point boundary value problem formula omitted this problem was studied earlier by gupta ntouyas and tsamatos when 1 and when 1 with 1 in the general case this problem was studied by gupta as a multi point boundary value problem in this paper sharper existence conditions are obtained for the solvability of the above boundary value problem in the general case

solvability of an m point boundary value problem for second order ordinary differential equations let 0 1 r2 r be a function satisfying caratheodory s conditions and e t l1 0 1 let i 0 1 ai r all of the ai s having the same sign i 1 2 m 2 0 1 2 m 2 1 be given this paper is concerned with the problem of existence of a solution for the m point boundary value problem x t t x t x t e t t 0 1 e formula conditions for the existence of a solution for the above boundary value problem are given using the leray schauder continuation theorem

a class of second order impulsive integro differential equations on unbounded domain in a banach space by establishing a comparison result the author obtains the existence of maximal and minimal solutions of the initial value problem ivp for a class of second order impulsive integro differential equations in a banach space

on positive solutions of boundary value problems on the half line we study the existence of positive solutions of boundary value problems on the half line for differential equations of second order the krasnoselskii fixed point theorem on cone compression and expansion is used

solutions of impulsive boundary value problems on the half line abstract this paper presents some existence results for a class of impulsive boundary value problems with nonhomogeneous boundary data on the half line by applying fixed point theorems on cones we obtain the existence of positive bounded and unbounded solutions our methods are integral equations approximate solutions a priori bounds and modified arzela techniques

isolated singularities of positive solutions of p laplacian type equations in r d we study the behavior of positive solutions of p laplacian type elliptic equations of the form q u p u v u p 2u 0in near an isolated singular point where 1 1 we obtain removable singularity theorems for positive solutions near in particular using a new three spheres theorems for certain solutions of the above equation near we prove that if v belongs to a certain kato class near and p d respectively p d then any positive solution u of the equation q u 0 in a punctured neighborhood of 0 respectively is in fact continuous at under further assumptions we find the asymptotic behavior of u near

an algebraic framework for group duality abstract a hopf algebra is a pair a whereais an associative algebra with identity and a homomorphism formatoa asatisfying certain conditions if we drop the assumption thatahas an identity and if we allow to have values in the so called multiplier algebram a a we get a natural extension of the notion of a hopf algebra we call this a multiplier hopf algebra the motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group also for these multiplier hopf algebras there is a natural notion of left and right invariance for linear functionals called integrals in hopf algebra theory we show that if such invariant functionals exist they are unique up to a scalar and faithful for a regular multiplier hopf algebra a i e with invertible antipode with invariant functionals we construct in a canonical way the dual it is again a regular multiplier hopf algebra with invariant functionals it is also shown that the dual of is canonically isomorphic with the original multiplier hopf algebra a it is possible to generalize many aspects of abstract harmonic analysis here one can define the fourier transform one can prove plancherel s formula because any finite dimensional hopf algebra is a regular multiplier hopf algebra and has invariant functionals our duality theorem applies to all finite dimensional hopf algebras then it coincides with the usual duality for such hopf algebras but our category of multiplier hopf algebras also includes in a certain way the discrete quantum groups and the compact quantum groups our duality includes the duality between discrete quantum groups and compact quantum groups in particular it includes the duality between compact abelian groups and discrete abelian groups one of the nice features of our theory is that we have an extension of this duality to the non abelian case but within one category this is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type we prove that also these are dual to each other we treat an example that is sufficiently general to illustrate most of the different features of our theory it is also possible to construct the quantum double of drinfel d within this category this provides a still wider class of examples so we obtain many more than just the compact and discrete quantum within this setting

positive solutions for multi point boundary value problem on the half line this paper presents a variety of existence results for nonlinear multi point boundary value problem on the half line in particular we consider x t px t qx t f t x t 0 t 0 x 0 x 0 i 1nkix i a limt x t ert b r 0 p p2 4q2 existence results are established using fixed point theorems on cone

multiple solutions of second order three point singular boundary value problems on the half line in this paper the existence of multiple solutions of second order three point singular boundary value problems on the half line is discussed by using the technique of upper and lower solutions and the theory of fixed point index in a cone moreover it generalizes and includes some known results

decay estimates for wolff potentials in rn and gradient dependent quasilinear elliptic equations abstract we prove decay estimates of positive weak solutions u d 1 p r n c 1 r n and their gradients u of the p laplacian problem 1 p n u d 1 p r n p u a x where the measurable function a satisfies the decay estimate 0 0 a x 1 1 x n x r n the positive borel measure on r n generated by the right hand side function a and its associated wolff potential is known to be the main tool for pointwise estimates of u x 0 and u x therefore one of the main goal of this paper is to prove decay estimates for corresponding wolff potentials in r n the obtained decay estimates in conjunction with a sub supersolution method which is developed here for quasilinear elliptic equations are then applied to prove the existence of positive solutions for classes of gradient dependent quasilinear equations in the form u d 1 p r n p u a x f u u

on positive solutions of boundary value problems for second order functional differential equations on infinite intervals we study the existence of positive solutions for the following boundary value problem on infinite interval for second order functional differential equations x t px t qx t f t xt 0 t 0 x t x t t t 0 limt x t 0 and x t px t qx t f t xt x t 0 t 0 x t x t t t 0 limt x t 0 where p 0 2 2 0 and q 0 the fixed point theorem on cone is used

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h

positive solutions for singular semipositone boundary value problems on infinite intervals by using the fixed point theory on a cone with a special norm and space we discuss the existence of positive solutions for a class of semipositone boundary value problems on infinite intervals the work improves many known results including singular and non singular cases