elliptic problems with nonlinearities indefinite in sign abstract let r n n 3 be a bounded open set with smooth boundary and r we study the dirichlet problem formula with 1 q p and h k l nonnegative functions we prove existence nonexistence and multiplicity results depending on and according to the integrability properties of the ratio k p 1 h q 1

a bishop phelps bollobas type theorem for uniform algebras abstract this paper is devoted to showing that asplund operators with range in a uniform banach algebra have the bishop phelps bollobas property i e they are approximated by norm attaining asplund operators at the same time that a point where the approximated operator almost attains its norm is approximated by a point at which the approximating operator attains it to prove this result we use the weak to norm fragmentability of weak compact subsets of the dual of asplund spaces and we need to observe a urysohn type result producing peak complex valued functions in uniform algebras that are small outside a given open set and whose image is inside a stolz region

pairs of positive periodic solutions of nonlinear odes with indefinite weight a topological degree approach for the super sublinear case we study the periodic and neumann boundary value problems associated with the second order nonlinear differential equationwhere is a sublinear function at infinity having superlinear growth at zero we prove the existence of two positive solutions whenand 0 is sufficiently large our approach is based on mawhin s coincidence degree theory and index computations

three positive solutions to an indefinite neumann problem a shooting method abstract we deal with the neumann boundary value problem u a t a t g u 0 0 u t 1 t 0 t u 0 u t 0 where the weight term has two positive humps separated by a negative one and g 0 1 r is a continuous function such that g 0 g 1 0 g s 0 for 0 s 1 and lim s 0 g s s 0 we prove the existence of three solutions when and are positive and sufficiently large

multiple positive solutions of superlinear elliptic problems with sign changing weight we study the existence of multiple positive solutions for a superlinear elliptic pde with a sign changing weight our approach is variational and relies on classical critical point theory on smooth manifolds a special care is paid to the localization of minimax critical points

the bishop phelps bollobas theorem for operators we prove the bishop phelps bollobas theorem for operators from an arbitrary banach space x into a banach space y whenever the range space has property of lindenstrauss we also characterize those banach spaces y for which the bishop phelps bollobas theorem holds for operators from 1 into y s everal examples of classes of such spaces are provided for instance the bishop phelps bollobas theorem holds when the range space is finite dimensional an l1 space for a finite measure a c k space for a compact hausdorff space k or a uniformly convex banach space

denseness for norm attaining operator valued functions abstract in this note we offer a short constructive proof for hilbert spaces of lindenstrauss famous result on the denseness of norm attaining operators specifically we show given any a l h there is a sequence of rank 1 operators k n such that a k n is norm attaining for each n and k n converges in norm to zero we then apply our construction to establish denseness results for norm attaining operator valued functions

monotonicity and complex convexity in banach lattices abstract the goal of this article is to study the relations among monotonicity properties of real banach lattices and the corresponding convexity properties in the complex banach lattices we introduce the moduli of monotonicity of banach lattices we show that a banach lattice e is uniformly monotone if and only if its complexification e c is uniformly complex convex we also prove that a uniformly monotone banach lattice has finite cotype in particular we show that a banach lattice is of cotype q for some 2 q if and only if there is an equivalent lattice norm under which it is uniformly monotone and its complexification is q uniformly pl convex we also show that a real kothe function space e is strictly respectively uniformly monotone and a complex banach space x is strictly respectively uniformly complex convex if and only if kothe bochner function space e x is strictly respectively uniformly complex convex

positive solutions with a complex behavior for superlinear indefinite odes on the real line abstract we show the existence of infinitely many positive solutions defined on the real line for the nonlinear scalar ode u a t a t u 3 0 where a is a periodic sign changing function and the parameter 0 is large such solutions are characterized by the fact of being either small or large in each interval of positivity of a in this way we find periodic solutions having minimal period arbitrarily large and bounded non periodic solutions exhibiting a complex behavior the proof is variational exploiting suitable natural constraints of nehari type

strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices abstract using the variational method it is shown that the set of all strong peak functions in a closed algebra a of c b k is dense if and only if the set of all strong peak points is a norming subset of a as a corollary we can induce the denseness of strong peak functions on other certain spaces in case that a set of uniformly strongly exposed points of a banach space x is a norming subset of p x n then the set of all strongly norm attaining elements in p x n is dense in particular the set of all points at which the norm of p x n is frechet differentiable is a dense g subset in the last part using reisner s graph theoretic approach we construct some strongly norm attaining polynomials on a cl space with an absolute norm then we show that for a finite dimensional complex banach space x with an absolute norm its polynomial numerical indices are one if and only if x is isometric to l n moreover we give a characterization of the set of all complex extreme points of the unit ball of a cl space with an absolute norm

norm attaining compact operators abstract we show examples of compact linear operators between banach spaces which cannot be approximated by norm attaining operators this is the negative answer to an open question posed in the 1970s actually any strictly convex banach space failing the approximation property serves as the range space on the other hand there are examples in which the domain space has a schauder basis

multibump nodal solutions for an indefinite superlinear elliptic problem abstract we define some nehari type constraints using an orthogonal decomposition of the sobolev space h 0 1 and prove the existence of multibump nodal solutions for an indefinite superlinear elliptic problem

mixedk functionals a measure of smoothness for blending type approximation we consider uniform approximation of bivariate functions f c 2 2 or f c i 2 i e continuous real valued functions which are 2 periodic with respect to both variables or defined on the compact square i 2 we are interested in quantitative results for blending type approximation more precisely we want to approximate f by continuous pseudo polynomials of degree m n we introduce mixed k functionals which will serve similar purposes as the k functionals in the univariate theory

numerical radius attaining compact linear operators abstract we show that there are compact linear operators on banach spaces which cannot be approximated by numerical radius attaining operators

the bishop phelps bollobas theorem for l l1 μ l 0 1 abstract we show that the bishop phelps bollobas theorem holds for all bounded operators from l 1 into l 0 1 where is a finite measure

multiple positive solutions to elliptic boundary blow up problems abstract we prove the existence of multiple positive radial solutions to the sign indefinite elliptic boundary blow up problem u a x a x g u 0 x 1 u x x 1 where g is a function superlinear at zero and at infinity a and a are the positive negative part respectively of a sign changing function a and 0 is a large parameter in particular we show how the number of solutions is affected by the nodal behavior of the weight function a the proof is based on a careful shooting type argument for the equivalent singular ode problem as a further application of this technique the existence of multiple positive radial homoclinic solutions to u a x a x g u 0 x r n is also considered

simultaneously continuous retraction and bishop phelps bollobas type theorem abstract the dual space x of a banach space x is said to admit a uniformly simultaneously continuous retraction if there is a retraction r from x onto its unit ball b x which is uniformly continuous in norm topology and continuous in weak topology we prove that if a banach space resp complex banach space x has a normalized unconditional schauder basis with unconditional basis constant 1 and if x is uniformly monotone resp uniformly complex convex then x admits a uniformly simultaneously continuous retraction it is also shown that x admits such a retraction if x x i c 0 or x x i l 1 where x i is a family of separable banach spaces whose duals are uniformly convex with moduli of convexity i e with inf i i e 0 for all 0 e 1 let k be a locally compact hausdorff space and let c 0 k be the real banach space consisting of all real valued continuous functions vanishing at infinity as an application of simultaneously continuous retractions we show that a pair x c 0 k has the bishop phelps bollobas property for operators if x admits a uniformly simultaneously continuous retraction as a corollary c 0 s c 0 k has the bishop phelps bollobas property for operators for every locally compact metric space s

the bishop phelps bollobas property for operators from c k to uniformly convex spaces abstract we show that the pair c k x has the bishop phelps bollobas property for operators if k is a compact hausdorff space and x is a uniformly convex space

on the existence of positive solutions for semilinear elliptic equations in the annulus abstract we study the existence of positive radial solutions of u g x u 0 in annuli with dirichlet dirichlet neumann boundary conditions we prove that the problems have positive radial solutions on any annulus if is sublinear at 0 and

multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree abstract we study the periodic boundary value problem associated with the second order nonlinear differential equation u c u a t a t g u 0 where g u has superlinear growth at zero and at infinity a t is a periodic sign changing weight c r and 0 is a real parameter our model includes for c 0 the so called nonlinear hill s equation we prove the existence of 2 m 1 positive solutions when a t has m positive humps separated by m negative ones in a periodicity interval and is sufficiently large thus giving a complete solution to a problem raised by g j butler in 1976 the proof is based on mawhin s coincidence degree defined in open possibly unbounded sets and applies also to neumann boundary conditions our method also provides a topological approach to detect subharmonic solutions

generic type results for chaotic dynamics in equations with periodic forcing terms abstract we prove that a class of equations containing the classical periodically forced pendulum problem displays the main features of chaotic dynamics for a set of forcing terms open and dense in suitable spaces the approach is based on global variational methods

subharmonic solutions with prescribed minimal period for nonautonomous hamiltonian systems on etablit un resultat de multiplicite pour une certaine classe de systemes hamiltoniens periodiques en temps

existence multiplicity and dependence on a parameter for a periodic boundary value problem the authors consider the boundary value problem y 2 y g t f y 0 0 t 2 y 0 y 2 y 0 y 2

minimal boolean sum and blending type projections and extensions in this paper useful characterization theorems are presented for minimal boolean sum and blending type projections and extensions 2000 elsevier science ltd all rights reserved

minimal projections in tensor product spaces in this paper we present a condensed survey of recent results on minimal projections in tensor product spaces the idea of a minimal projection is an old one we have a banach space z and a complemented subspace w then the projection constant delta w z is defined by delta w z inf p p is a projection from z onto w

the bishop phelps bollobas theorem for operators on l1 μ in this paper we show that the bishop phelps bollobas theorem holds for l l1 l1 for all measures and and also holds for l l1 l for every arbitrary measure and every localizable measure finally we show that the bishop phelps bollobas theorem holds for two classes of bounded linear operators from a real l1 into a real c k if is a finite measure and k is a compact hausdorff space in particular one of the classes includes all bochner representable operators and all weakly compact operators

superlinear indefinite equations on the real line and chaotic dynamics abstract in this paper we are concerned with a differential equation of the form x c x q t g x 0 t a b where a b q has infinitely many zeros in a b and g is superlinear we prove the existence of solutions with prescribed nodal properties in the intervals of negativity and positivity of q when c 0 and q is periodic we show that the equation under consideration exhibits chaotic like dynamics

the effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction diffusion equations abstract in this paper we analyze how the dynamics of a class of superlinear indefinite reaction diffusion equations varies as the nodal behavior of a coefficient changes to perform this analysis we use both theoretical and numerical tools the analysis aids the numerical study and the numerical study confirms and completes the analysis the numerics in addition provides us with some further results for which at first glance analytical tools are not available yet our main analytical result shows that the problem possesses a unique positive solution which is linearly asymptotically stable if the trivial state is linearly unstable and the model admits some positive solution this result is a relevant feature for superlinear indefinite problems since our numerical computations show how these models can have an arbitrarily large number of positive solutions if the trivial state is unstable

a note on a superlinear indefinite neumann problem with multiple positive solutions abstract we prove the existence of three positive solutions for the neumann problem associated to u a t u 1 0 assuming that a t has two positive humps and 0 t a t d t is large enough actually the result holds true for a more general class of superlinear nonlinearities

multiple positive solutions for a superlinear problem a topological approach we study the multiplicity of positive solutions for a two point boundary value problem associated to the nonlinear second order equation u f x u 0 we allow x f x s to change its sign in order to cover the case of scalar equations with indefinite weight roughly speaking our main assumptions require that f x s s is below 1 as s 0 and above 1 as s in particular we can deal with the situation in which f x s has a superlinear growth at zero and at infinity we propose a new approach based on topological degree which provides the multiplicity of solutions applications are given for u a x g u 0 where we prove the existence of 2n 1 positive solutions when a x has n positive humps and a x is sufficiently large

high multiplicity of positive solutions for superlinear indefinite problems with homogeneous neumann boundary conditions abstract we prove that a class of superlinear indefinite problems with homogeneous neumann boundary conditions admits an arbitrarily high number of positive solutions provided that the parameters of the problem are adequately chosen the sign changing weight in front of the nonlinearity is taken to be piecewise constant which allows us to perform a sharp phase plane analysis firstly to study the sets of points reached at the end of the regions where the weight is negative and then to connect such sets through the flow in the positive part moreover we study how the number of solutions depends on the amplitude of the region in which the weight is positive using the latter as the main bifurcation parameter and constructing the corresponding global bifurcation diagrams

varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems in this work we analyze the existence multiplicity and stability of positive solutions for a class of indefinite superlinear elliptic boundary value problems the main contribution of this paper consists in the change of mind inherent to the fact of adding the superlinear amplitude e as an unfolding parameter this change of mind allows us to unify many previous results obtained separately in the literature it helps us to realize the global structure of the set of positive steady states and provides us with a great variety of new general results our techniques can be applied to much more general equations and systems

the bishop phelps bollobas property for operators between spaces of continuous functions abstract we show that the space of bounded linear operators between spaces of continuous functions on compact hausdorff topological spaces has the bishop phelps bollobas property a similar result is also proved for the class of compact operators from the space of continuous functions vanishing at infinity on a locally compact and hausdorff topological space into a uniformly convex space and for the class of compact operators from a banach space into a predual of an l 1 space

norm attaining bilinear forms onl1 0 1 abstract we show that the set of norm attaining bilinear forms onl1 0 1 is not dense in the space of all continuous bilinear forms

the determination of minimal projections and extensions in equations are derived which are shown to be necessary and sufficient for finite rank projections in l1 to be minimal more generally these equations are also necessary and sufficient to determine operators of minimal norm which extend a fixed linear action on a given finite dimensional subspace of l1 and thus may be viewed as an extension of the hahn banach theorem to higher dimensions in the l1 setting these equations are solved in terms of an l1 best approximation problem and the required orthogonality conditions moreover this solution has a simple geometric interpretation questions of uniqueness are considered and a number of examples are given to illustrate the usefulness of these equations in determining minimal projections and extensions including the minimal l1 projection onto the quadratics

a multilinear lindenstrauss theorem abstract we show that the set of n linear mappings on a product of n banach spaces such that all their arens extensions attain their norms at the same element is norm dense in the space of all bounded n linear mappings