degree theory for s mappings in non reflexive banach spaces abstract let e be a real banach space e the dual space of e e the dual space of e and e an open bounded subset and let t 2 e be a weak compact mapping of class s see definition 1 2 we will construct a degree theory for such a mapping

holomorphic kernels and commuting operators necessary and sufficient conditions in terms of operator polynomials are obtained for an m tuple t tl t l of commuting bounded linear operators on a separable hilbert space x to extend to an m tuple s s1 sm of operators on some hilbert space x where each s is realized as a representation of the adjoint of a multiplication operator on the tensor product of a special type of functional hilbert spaces a1so necessary and sufficient conditions in terms of operator polynomials are obtained for t to have a commuting normal extension o introduction in this paper some results in 1 and 2 for a single bounded linear operator t on a separable hilbert space x are generalized to m commuting operators on x in 1 agler introduces a special class of functional hilbert spaces and describes conditions under which an operator t on x extends to m where m denotes the multiplication operator on x and m denotes the countable direct sum of m with itself special cases of spaces x of which the classical hardy space is the prototype are considered in 1 2 the relevant conditions for the kind of extension of t referred to above are expressed in terms of the positivity of certain operator polynomials involving t and t these conditions are closely related to the reproducing kernel associated with the space s in 2 necessary and sufficient conditions are also given for a contraction t to be subnormal these conditions are really the requirement that a certain sequence of polynomials in t and t be positive it is natural to seek generalizations of these results to m commuting operators t1 tm on x an appropriate model for this generalization is obtained by constructi ng a finite tensor product of spaces s and exploiting the well known fact that the reproducing kernel of such a tensor product is the product of the reproducing kernels of the illdividual spaces suitable modifications of the reproducing kernels which render them holomorphic on the unit polydisc can be used to describe analogous extension results for t t1 tm the question of t having a commuting normal extension n n1 nm turns out to have a direct link with the multi dimensional hausdorff moment problem from the theory of probability received by the editors march 21 1986 1980 mathematics subject classification primary 47a20 secondary 47b20 47b38

an indefinite type equation involving two p laplacians abstract we study a problem involving the operators w and its p homogeneous version p v the existence of three non negative solutions is obtained under indefinite type conditions on these operators to this aim we proceed by minimization of the energy functional on open parts of the nehari manifold

a multiplicity theorem for problems with the p laplacian abstract we consider nonlinear elliptic equations driven by the p laplacian differential operator using degree theoretic arguments based on the degree map for operators of type s we prove the existence of two nontrivial smooth solutions one of which is of constant sign

degree of mapping for nonlinear mappings of monotone type abstract a classical degree function is constructed for pseudomonotone mappings from a reflexive banach space to its dual using galerkin approximations this generalizes the leray schauder degree when the banach space is a hilbert space and yields a flexible analytical tool for the study of nonlinear elliptic problems of higher order in divergence form

solutions of asymptotically linear operator equations via morse theory abstract in this paper we use the classical morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations as special cases we include multiplicity results for semilinear elliptic bvp s as well as existence and multiplicity of periodic solutions of semilinear wave equation and of hamiltonian systems these results simplify and complement recent work of amann zehnder and castro lazer author

critical groups at zero and multiple solutions for a quasilinear elliptic equation abstract in this paper by morse theory we will compute the critical groups at zero for a functional i w 0 1 p r defined by setting i u 1 p u p d x 1 2 u 2 d x f x u d x where p 2 is a bounded domain in r n f x u 0 u f x t d t and we assume that f is resonant at zero for the spectrum of in w 0 1 2 as an application of these critical groups estimates some multiplicity results are also given

nonresonance conditions for generalised ϕ laplacian problems with jumping nonlinearities abstract we consider the boundary value problem 0 1 x u x u x f x u x u x a e x 0 1 0 2 c 00 u 0 c 01 u 0 c 10 u 1 c 11 u 1 where c j 0 c j 1 0 for each j 0 1 and f 0 1 r 2 r are caratheodory functions with suitable additional properties the differential operator generated by the left hand side of 0 1 together with the boundary conditions 0 2 is a generalisation of the usual p laplacian and also of the so called laplacian which corresponds to x s t t with an odd increasing homeomorphism for the p laplacian problem and more particularly the semilinear case p 2 nonresonance conditions which ensure the solvability of the problem 0 1 0 2 have been obtained in terms of either eigenvalues for non jumping f or the fucik spectrum or half eigenvalues for jumping f of the p laplacian in this paper under suitable growth conditions on and f we extend these conditions to the general problem 0 1 0 2

averaging theory for discontinuous piecewise differential systems we develop the averaging theory of first and second order for studying the periodic solutions of discontinuous piecewise differential systems in arbitrary dimension and with an arbitrary number of systems with the minimal conditions of differentiability we also provide two applications

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

variational inequalities and fixed point theorems for pm maps abstract we study mapstfor whichj tis pseudo monotone we call suchta pm map this includes compact maps in suitable spaces and pseudo contractive maps in hilbert spaces we study variational inequalities in a situation which was previously done only whentis compact we show that our variational inequalities are well suited to treating existence of fixed points for generalized inward in particular weakly inward pm maps in hilbert spaces our new results on existence of fixed points generalizes many earlier results obtained by using other methods an application to homogeneous integral equations is provided

a variational inequality theory in reflexive smooth banach spaces and applications to p laplacian elliptic inequalities abstract we develop a theory of variational inequality for demicontinuous s contractive maps in reflexive smooth banach spaces variational inequality theorems are proved and applied to study existence of nonzero positive weak solutions for p laplacian elliptic inequalities and a population model of one species arising in mathematical biology

multiplicity of solutions for a class of the quasilinear elliptic equations at resonance abstract in this paper by morse theory we obtain the existence and multiplicity for a class of the quasilinear elliptic equations at resonance

on the degree theory for general mappings of monotone type degree theory has been developed as a tool for checking the solution existence of nonlinear equations in his classic paper published in 1983 browder developed a degree theory for mappings of monotone type f t w heref is a mapping of class s from a bounded open set in a reflexive banach space x into its dual x a ndt is a maximal monotone mapping from x into x this breakthrough paved the way for many applications of degree theoretic techniques to several large classes of nonlinear partial differential equations in this paper we continue to develop the results of browder on the degree theory for mappings of monotone type f t by enlarging the class of maximal monotone mappings and pseudo monotone homotopies we obtain some new results of the degree theory for such mappings 2007 elsevier inc all rights reserved

analysis of higher order mode effects in an expansion chamber using modal theory and equivalent electrical circuits abstract modal theory for propagation in an acoustic waveguide with variable cross section in particular in an expansion chamber can lead to several different numerical or approximate analytical solutions the advantages of these are compared and one of their number is shown to be a generalization of the plane piston inlet outlet approximation made by ih and lee the results are expressed in the form of two kinds of equivalent electrical circuits connecting the plane modes in the inlet and outlet in the first one the higher modes are described by two series inductances and one shunt inductance in the second one there is no shunt inductance and series inductances are replaced by complex series impedances the existence of resistive elements is discussed in terms of energy transport by higher modes especially at low frequencies when tunneling occurs approximate analytical numerical and experimental results for the elements of circuits are given with a study of coupling between the two step discontinuities

strongly quasibounded maximal monotone perturbations for the berkovits mustonen topological degree theory abstract let x be a real reflexive banach space with dual x let l x d l x be densely defined linear and maximal monotone let t x d t 2 x with 0 d t and 0 t 0 be strongly quasibounded and maximal monotone and c x d c x bounded demicontinuous and of type s w r t d l a new topological degree theory has been developed for the sum l t c this degree theory is an extension of the berkovits mustonen theory for t 0 and an improvement of the work of addou and mermri for t x 2 x bounded unbounded maximal monotone operators with 0 d t are strongly quasibounded and may be used with the new degree theory

on the semi monotone operator theory and applications abstract in this paper eis a real banach space a e e e is a semi monotone operator we first study the variational inequality problem findu k such that a u u v u 0 wherek e is a closed convex subset then the operator equationa u u p and we also construct a degree theory for demi continuous semi monotone operators in reflexive banach spaces

analytic models for commuting operator tuples on bounded symmetric domains for a domain in c d and a hilbert space h of analytic functions on which satisfies certain conditions we characterize the commuting d tuples t t 1 t d of operators on a separable hilbert space h such that t is unitarily equivalent to the restriction of m to an invariant subspace where m is the operator d tuple z i on the hilbert space tensor product h h for the unit disc and h the hardy space h 2 this reduces to a well known theorem of sz nagy and foias for h a reproducing kernel hilbert space on c c d such that the reciprocal 1 k x y of its reproducing kernel is a polynomial in x and y this is a recent result of ambrozie muller and the second author in this paper we extend the latter result by treating spaces h for which 1 k ceases to be a polynomial or even has a pole namely the standard weighted bergman spaces or rather their analytic continuation h h on a cartan domain corresponding to the parameter v in the continuous wallach set and reproducing kernel hilbert spaces h for which 1 k is a rational function further we treat also the more general problem when the operator m is replaced by m w w being a certain generalization of a unitary operator tuple for the case of the spaces h on cartan domains our results are based on an analysis of the homogeneous multiplication operators on which seems to be of an independent interest

nonlinear second order differential inclusions with a series of convex functions abstract in this paper we study a class of second order differential inclusions attached to which the convex potential varies depending on the time and the boundary conditions are described by the subdifferentials of convex functions applying the theory of lower semicontinuous convex and proper functionals and fixed points we give some results about the representation of the subdifferential operators of some convex functionals and the existence of solutions for the boundary value problems

on the sum of two subnormal kernels abstract we show by means of a class of examples that if k 1 and k 2 are two positive definite kernels on the unit disc such that the multiplication by the coordinate function on the corresponding reproducing kernel hilbert space is subnormal then the multiplication operator on the hilbert space determined by their sum k 1 k 2 need not be subnormal this settles a recent conjecture of gregory t adams nathan s feldman and paul j mcguire in the negative we also discuss some cases for which the answer is affirmative

extension of the leray schauder degree for abstract hammerstein type mappings abstract we introduce a new extension of the classical leray schauder topological degree in a real separable reflexive banach space the new class of mappings for which the degree will be constructed is obtained essentially by replacing the compact perturbation by a composition of mappings of monotone type it turns out that the class contains the leray schauder type maps as a proper subclass the new class is not convex thus preventing the free application of affine homotopies however there exists a large class of admissible homotopies including subclass of affine ones so that the degree can be effectively used we shall construct the degree and prove that it is unique we shall generalize the borsuk theorem of the degree for odd mappings and show that the principle of omitted rays remains valid to illuminate the use of the new degree we shall briefly consider the solvability of abstract hammerstein type equations and variational inequalities

moment problems for multi sequences of operators abstract we give an explicit solution to the scalar moment problem on semi algebraic compact subsets of r n and apply this result to the study of some operator multi sequences

the leray schauder approach to the degree theory for s perturbations of maximal monotone operators in separable reflexive banach spaces abstract the purpose of this paper is to demonstrate the fact that the topological degree theory of leray and schauder may be used for the development of the topological degree theory for bounded demicontinuous s perturbations f of strongly quasibounded maximal monotone operators t in separable reflexive banach spaces certain basic homotopy properties and the extension of this degree theory to possibly unbounded strongly quasibounded perturbations f are shown to hold this work uses the well known embedding of browder and ton and extends the work of berkovits who developed this theory for the case t 0 besides being an interesting mathematical problem the existence of such a degree theory may conceivably become useful in situations where the use of the leray schauder degree via infinite dimensional compactness is necessary

multiple nontrivial solutions for nonlinear periodic problems with the p laplacian abstract we consider a nonlinear periodic problem driven by the scalar p laplacian with a nonsmooth potential hemivariational inequality using the degree theory for multivalued perturbations of s operators and the spectrum of a class of weighted eigenvalue problems for the scalar p laplacian we prove the existence of at least three distinct nontrivial solutions two of which have constant sign

topological degree theories and nonlinear operator equations in banach spaces abstract let x be a real banach space and g 1 g 2 two nonempty open and bounded subsets of x such that 0 g 2 and g 2 g 1 the problem t x c x 0 is considered where t x d t x or x is an accretive or monotone operator with 0 d t and t 0 0 while c x d c x or x can be e g one of the following types a compact b continuous and bounded with the resolvents of t compact c demicontinuous bounded and of type s with t positively homogeneous of degree one d quasi bounded and satisfying a generalized s condition w r t the operator t while t is positively homogeneous of degree one solutions are sought for the problem lying in the set d t c g 1 g 2 these solutions are nontrivial even when c 0 0 the degree theories of leray and schauder browder and skrypnik are used as well as the degree theory by kartsatos and skrypnik for densely defined operators t c the last three degree theories do not assume any compactness conditions the excision and additivity properties of these degree theories are employed and the main results are significant extensions or generalizations of previous results by krasnoselskii guo ding and kartsatos and other authors involving the relaxation of compactness conditions and or conditions on the boundedness of the operator t an application in the field of partial differential equations is also included

positive weak solutions of semilinear second order elliptic inequalities via variational inequalities existence uniqueness and convergence of approximants of positive weak solutions for semilinear second order elliptic inequalities are obtained the nonlinearities involved in these inequalities satisfy suitable upper or lower bound conditions or monotonicity conditions the lower bound conditions are allowed to contain the critical sobolev exponents the methodology is to establish variational inequality principles for demicontinuous pseudo contractive maps in hilbert spaces by considering convergence of approximants and apply them to the corresponding variational inequalities arising from the semilinear second order elliptic inequalities examples on the existence uniqueness and convergence of approximants of positive weak solutions of the semilinear second order elliptic inequalities are given

fixed point theory and nonlinear problems introduction among the most original and far reaching of the contributions made by henri poincare to mathematics was his introduction of the use of topological or qualitative methods in the study of nonlinear problems in analysis his starting point was the study of the differential equations of celestial mechanics and in particular of their periodic solutions his work on this topic began with his thesis in 1879 and was developed in detail in his great three volume work methodes nouvelles de la mechanique celeste which appeared in the early 1890s and summarized his many memoirs of the intervening period it continued until his memoir shortly before his death in 1912 in which he put forward the unproved fixed point result usually referred to as poincare s last geometric theorem the ideas introduced by poincare include the use of fixed point theorems the continuation method and the general concept of global analysis the writer s acquaintance with poincare s influence came through contact with solomon lefschetz and marston morse both of whom were very explicit as to the role of poincare as an initiator in this direction of mathematical development in 1934 in the foreword to his colloquium volume on the calculus of variations in the large morse had put this forward very forcefully in the first paragraph

moment problems for operator polynomials abstract haviland s theorem states that given a closed subset k in r n each functional l r x r positive on pos k p r x p k 0 admits an integral representation by a positive borel measure schmudgen proved that in the case of compact semialgebraic set k it suffices to check positivity of l on a preordering t having k as the non negativity set further he showed that the compactness of k is equivalent to the archimedianity of t the aim of this paper is to extend these results from functionals on the usual real polynomials to operators mapping from the real matrix or operator polynomials into r m n r or b k

a new topological degree theory for densely defined s l perturbations of multivalued maximal monotone operators in reflexive separable banach spaces abstract let x be a real reflexive separable locally uniformly convex banach space with locally uniformly convex dual space x let t x d t 2 x be maximal monotone with 0 d t and 0 t 0 and c x d c x assume that l d c is a dense linear subspace of x c is of class s l and c x x x x d c where r r is nondecreasing a new topological degree theory is developed for the sum t c the current approach utilizes the approximate degree d t t c g 0 t 0 t t t 1 t j 1 1 g x open and bounded of kartsatos and skrypnik for the single valued mapping t t c the subdifferential for belonging to a large class of proper convex lower semicontinuous functions gives rise to operators t to which this degree theory applies a theoretical application to an existence problem of nonlinear analysis is included

solutions to semilinear elliptic problems with combined nonlinearities abstract we are concerned with the following nonlinear dirichlet problem u h x u q f x u 0 u h 0 1 0 where is a bounded smooth domain in rn n 1 and h x l f x s is asymptotically linear with respect to s at infinity by a variant version of mountain pass theorem we prove that there exist at least two nonnegative solutions under suitable assumptions on f x s our methods also work for the cases where f x s is superlinear or linear in s

moment problems and subnormality abstract we investigate the connection between certain algebraic conditions on a subnormal tuple s and the support of the representing measure s for s

bernstein nagumo conditions and solutions to nonlinear differential inequalities abstract for an open bounded subset of r n with smooth boundary and 1 p we establish w 1 p a priori bounds and prove the compactness of solution sets to differential inequalities of the form div a x u f x u u which are bounded in l the main point in this work is that the nonlinear term f may depend on u and may grow as fast as a power of order p in this variable such growth conditions have been used extensively in the study of boundary value problems for nonlinear ordinary differential equations and are known as bernstein nagumo growth conditions in addition we use these results to establish a sub supersolution theorem

on the existence of periodic solutions for nonlinear evolutions in hilbert spaces a periodic solutions for nonlinear evolution equations of the form du dt au t f t u t t r where f r h 2 is a set valued caratheodory function are considered when both f and a are set valued mappings and not necessarily linear

on the calculation of the transmission line parameters for long tubes using the method of multiple scales the present paper deals with the classical problem of linear sound propagation in tubes with isothermal walls the perturbation technique of the method of multiple scales in combination with matched asymptotic expansions is applied to derive the first order solutions and in addition the second order solutions representing the correction due to boundary layer attenuation the propagation length is assumed to be so large that in order to obtain asymptotic solutions which extend over the whole spatial range the first order corrections to the classical attenuation rates of the different modes come into play as well starting with the case of the characteristic wavelength being large compared to the characteristic dimension of the duct the analysis is then extended to the case where both of these quantities are of the same order of magnitude furthermore the transmission line parameters and the transfer functions relating the sound pressures at the ends of the duct to the axial velocities are calculated

topological degree for s mappings with maximal monotone perturbations and its applications to variational inequalities abstract this paper is concerned with the topological degree for mappings of class s with maximal monotone perturbations several results and remarks concerning the evaluation of this degree are given in particular it is shown that the local degree for the generalized gradient of nonsmooth functional at the local minimizer is equal to one as applications two examples of elliptic variational inequalities are given where the multiple existence of solutions is discussed

limit cycles for discontinuous quadratic differential systems with two zones abstract in this paper we study the maximum number of limit cycles given by the averaging theory of first order for discontinuous differential systems which can bifurcate from the periodic orbits of the quadratic isochronous centers x y x 2 y x x y and x y x 2 y 2 y x 2 x y when they are perturbed inside the class of all discontinuous quadratic polynomial differential systems with the straight line of discontinuity y 0 comparing the obtained results for the discontinuous with the results for the continuous quadratic polynomial differential systems this work shows that the discontinuous systems have at least 3 more limit cycles surrounding the origin than the continuous ones

multiplicity of solutions for a quasilinear elliptic problem via the cohomological index abstract in this paper we study a quasilinear elliptic problem with the dirichlet boundary condition in a bounded domain involving the operator a u p u q u assuming that the nonlinearity has a concave convex behavior we obtain some multiplicity results more precisely we obtain five nontrivial solutions two by a minimization argument two by the mountain pass theorem and the other by a cohomological linking theorem

limit cycles of discontinuous piecewise polynomial vector fields abstract when the first average function is non zero we provide an upper bound for the maximum number of limit cycles bifurcating from the periodic solutions of the center x y x 2 y 2 2 m and y x x 2 y 2 2 m with m 1 when we perturb it inside a class of discontinuous piecewise polynomial vector fields of degree n with k pieces the positive integers m n and k are arbitrary the main tool used for proving our results is the averaging theory for discontinuous piecewise vector fields

a new topological degree theory for perturbations of the sum of two maximal monotone operators abstract let x be an infinite dimensional real reflexive banach space with dual space x and g x open and bounded assume that x and x are locally uniformly convex let t x d t 2 x be maximal monotone and strongly quasibounded s x d s x maximal monotone and c x d c x strongly quasibounded w r t s and such that it satisfies a generalized s condition w r t s assume that d s l d t d c where l is a dense subspace of x and 0 t 0 s 0 0 a new topological degree theory is introduced for the sum t s c with degree mapping d t s c g 0 the reason for this development is the creation of a useful tool for the study of a class of time dependent problems involving three operators this degree theory is based on a degree theory that was recently developed by kartsatos and skrypnik just for the single valued sum s c as above

existence of periodic solutions for first order evolution equations without coercivity abstract in this paper we study the existence problem of periodic solutions for the following first order nonlinear evolution equation u t a t u t f t u t 0 t r u t t u t t r in a hilbert space h where a is a monotone type operator and f is a nonlinear operator existence results are obtained without assuming the coercivity condition

the propagation constant in lossy circular tubes near the cutoff frequencies of higher order modes the propagation constants of higher order modes in lossy circular ducts are investigated with particular interest to the frequency range around the corresponding cutoff frequencies the analytic expressions for the propagation constants published up to now contain poles at each cutoff frequency in this article it is shown that no poles exist the transition between the frequency range of evanescent and propagating modes is rapid but continuous an explicit formula for the propagation constant is derived the theory is experimentally checked with an indirect measuring method that is able to give results in close vicinity to the cutoff frequencies the measured attenuation due to the transmission line losses is partially somewhat higher than theoretically expected the deviations are less than about 30

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

generalizations of browder s degree theory the starting point of this paper is the recent important work of f e browder who extended degree theory to operators of monotone type the degree function of browder is generalized to maps of the form t f g where t is maximal monotone f is of class s bounded and g is an u s c compact multifunction it is also generalized to maps of the form f ng with f of class s and ng the nemitsky operator of a multifunction g x r satisfying various types of sign conditions some examples are also included to illustrate the abstract results

degree theory for generalized variational inequalities and applications in this paper a degree theory for finite dimensional generalized variational inequalities is built and employed to prove some results on solution existence and solution stability

existence of solutions for quasilinear elliptic equations abstract using variational methods we study the existence and multiplicity of solutions of the dirichlet problem for the equation div a u p u p 2 u f x u

on the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive banach spaces let x be a real reflexive banach space with dual x and g c x open and bounded and such that 0 g let t x d t 2 x be maximal monotone with 0 d t and 0 t 0 and c x d c x with 0 e d c and c 0 0 a general and more unified eigenvalue theory is developed for the pair of operators t c further conditions are given for the existence of a pair a x 0 x d t c n g such that tx cx 0 the implicit eigenvalue problem with c x in place of cx is also considered the existence of continuous branches of eigenvectors of infinite length is investigated and a fredholm alternative in the spirit of necas is given for a pair of homogeneous operators t c no compactness assumptions have been made in most of the results the degree theories of browder and skrypnik are used as well as the degree theories of the authors involving densely defined perturbations of maximal monotone operators applications to nonlinear partial differential equations are included

toeplitz algebras subnormal tuples and rigidity on reproducing c z1 zd modules this paper mainly considers toeplitz algebras subnormal tuples and rigidity concerning reproducing c z1 zd modules by making use of arveson s boundary representation theory we find there is more rigidity in several variables than there is in single variable we specialize our attention to reproducing c z1 zd modules with u invariant kernels by examining the spectrum and the essential spectrum of the d tuple mz1 mzd and deducing an exact sequence of c algebras associated with toeplitz algebra finally we deal with toeplitz algebras defined on arveson submodules and rigidity of arveson submodules

existence of positive solutions for a class of p q elliptic problems with critical growth on rn this paper is concerned with the existence of positive solutions to the class of p q elliptic problems with critical growth type div a u p u p 2 u b u p u p 2u f u u 2u u z 0 x rn where is a positive parameter a r r is a function of c1 class and b f r r are continuous functions

the bolzano mean value theorem and partial differential equations abstract we study the existence of solutions to abstract equations of the form 0 a u f u u k e where a is an abstract differential operator acting in a banach space e k is a closed convex set of constraints being invariant with respect to resolvents of a and a perturbation f satisfies a certain tangency condition such problems are closely related to the so called poincare miranda theorem being the multi dimensional counterpart of the celebrated bolzano intermediate value theorem in fact our main results should be regarded as infinite dimensional variants of bolzano and miranda poincare theorems along with single valued problems we deal with set valued ones yielding the existence of the so called constrained equilibria of set valued maps the abstract results are applied to show existence of strong steady state solutions to some weakly coupled systems of drift reaction diffusion equations or differential inclusions of this type in particular we get the existence of strong solutions to the dirichlet neumann and periodic boundary problems for elliptic partial differential inclusions under the presence of state constraints of different type certain aspects of the bernstein theory for bvp for second order ode are studied too no assumptions concerning structural coupling monotonicity cooperativity are undertaken

towards a functional calculus for subnormal tuples the minimal normal extension in this paper the study of a functional calculus for subnormal ntuples is initiated and the minimal normal extension problem for this functional calculus is explored this problem is shown to be equivalent to a mean approximation problem in several complex variables which is solved an analogous uniform approximation problem is also explored in addition these general results are applied together with the area and the the coarea formula from geometric measure theory to operators on bergman spaces and to the tensor product of two subnormal operators the minimal normal extension of the tensor product of the bergman shift with itself is completely determined an n tuple of commuting operators s s1 sn on a hilbert space z is subnormal if there is an n tuple n nl nn of commuting normal operators on a hilbert space z that contains z such that for 1 j n nj c and sj njlt for any commuting n tuple of operators s there is a notion of spectrum the taylor spectrum of s 30 also see 12 denoted by v s this spectrum is a nonempty compact subset of un in this general theory it is possible to define s for any function analytic in a neighborhood of v s this functional calculus generalizes the usual riesz functional calculus for a single operator when s is assumed to be a subnormal n tuple however a much richer functional calculus is possible in this case 0 s can be defined for functions 0 that are weak limits of analytic functions and 0 s becomes a subnormal operator the central question in this development is what are the properties of this operator 0 s and what are the relations between the operator and the function x more generally if s is an n tuple and xl xq are analytic functions defined in a neighborhood of the taylor spectrum of s then 0 s 0l s xq s is a commuting q tuple of operators if s is subnormal then 0 s can be defined for x 0l xq consisting of functions that are weak limits received by the editors january 31 1989 and in revised form july 6 1989 1980 mathematics subject classification 1985 revision primary 47b20 47a60 secondary 32a35 47b38 the results here were delivered in a talk at the american mathematical society summer institute in operator theory held at the university of new hampshire in july 1988 during the preparation of this paper partial support was furnished by national science foundation grant dms 87 00835 32 1991 american mathematical society 0002 9947 91 1 00 25 per page

an operator valued poisson kernel abstract versions of the cauchy and poisson formulas in the unit ball of c n are defined by means of appropriate operator valued kernels connections with some problems of dilation theory are then made

existence and multiplicity results for some quasilinear elliptic equation with weights using variational methods we show the existence and multiplicity of solutions of singular boundary value problems of the type div x a u u x f x u in u 0on where the numbers and as well as the functions a and f satisfy certain conditions

on the normalized eigenvalue problems for nonlinear elliptic operators abstract this paper solves the following form of normalized eigenvalue problem a u c u 0 0 and u d where d is a bounded open set in a real infinite dimensional banach space x and both x and its dual x are locally uniformly convex a is an unbounded maximal monotone operator on x the operators c is defined and continuous only on r d such that zero is not in the weak closure of a subset of c u c u this research reveals the fact that such eigenvalue problems do not depend on any properties of c located in r d this remarkable discovery extends theorem 4 in a g kartsatos i v skrypnik normalized eigenvalues for nonlinear abstract and elliptic operators j differential equations 155 1999 443 475 and is applied to the nonlinear elliptic operators under degenerate and singular conditions

partial differential equations involving subcritical critical and supercritical nonlinearities abstract we establish the existence of at least one nonnegative solution for the problem p div a u u f u in u 0 on where a and f satisfy conditions near zero here the novelty is that we do not need restrictions on the nonlinearities at infinity therefore we can consider subcritical critical and supercritical cases

berezin transforms on noncommutative varieties in polydomains abstract let q be a set of polynomials in noncommutative indeterminates z i j i 1 k j 1 n i in this paper we study noncommutative varieties v q h x x i j d h g x 0 for all g q where d h is a regular polydomain in b h n 1 n k and b h is the algebra of bounded linear operators on a hilbert space h under natural conditions on q we show that there is a universal model s s i j such that g s 0 g q acting on a subspace of a tensor product of full fock spaces we characterize the variety v q h and its pure part in terms of the universal model and a class of completely positive linear maps we obtain a characterization of those elements in v q h which admit characteristic functions and prove that the characteristic function is a complete unitary invariant for the class of completely non coisometric elements we study the universal model s its joint invariant subspaces and the representations of the universal operator algebras it generates the variety algebra a v q the hardy algebra f v q and the c algebra c v q using noncommutative berezin transforms associated with each variety we develop an operator model theory and dilation theory for large classes of varieties in noncommutative polydomains this includes various commutative cases which are closely connected to the theory of holomorphic functions in several complex variables and algebraic geometry

three nontrivial solutions for the p laplacian with a nonsmooth potential abstract we consider a nonlinear elliptic problem driven by the partial p laplacian and with a nonsmooth potential hemivariational inequality using variational techniques based on nonsmooth analysis and degree theoretic arguments for operators of the monotone type we establish the existence of at least three distinct nontrivial smooth solutions