complete solution of the electrostatic equilibrium problem for classical weights abstract in this paper we complete the solution of the electrostatic equilibrium problem for all classical weight functions in particular we solve this problem for classical generalized bessel jacobi on 0 and pseudo jacobi weights in addition we present an elementary unified way of dealing with the electrostatic equilibrium problem which depends on the weights satisfying the pearson differential equation

rotation numbers eigenvalues and the poincare birkhoff theorem using the relation between rotation numbers and eigenvalues we prove the existence of nontrivial t periodic solutions having precise oscillatory properties for a class of asymptotically linear second order differential equations

maslov index poincare birkhoff theorem and periodic solutions of asymptotically linear planar hamiltonian systems abstract we study the relationship between the twist condition in the poincare birkhoff fixed point theorem and the assumptions on the maslov index for asymptotically linear planar hamiltonian systems for this aim we develop a variant of the poincare birkhoff theorem which together with its classical version allows to obtain a lower bound for the number of nontrivial periodic solutions in terms of the gap between the maslov indexes associated to the linearizations of the planar system at zero and at infinity

v subdifferentials of convex operators abstract for any convex operator from a convex set c of a topological vector space e into another one f endowed with a convex cone f a notion of v subdifferential v a of at a c is introduced although it is equivalent to the notion of e sub differential when f r it enjoys many important properties which are not satisfied by the e subdifferential whenever int f o the nonvacuity of v a is proved whenever v is a neighbourhood of zero in f and belongs to a large class of mappings analogous to the class of lower semicontinuous real valued convex functions other properties of v subdifferentials are studied and applications to differentiability of are made

an n dimensional pendulum like equation via topological methods abstract we study the elliptic boundary value problem u g u p x with nonlinear periodic type boundary conditions we prove the existence of a solution under a landesman lazer type condition moreover using the method of upper and lower solutions we study a problem which concerns the range of the semilinear operator su u g u

on the solvability of a nonlinear second order elliptic equation at resonance ii we study the existence of solutions of the neumann problem for semilinear second order elliptic equations at resonance in which the nonlinear terms may grow superlinearly in one of the directions u oo and u oo and sublinearly in the other solvability results are obtained under assulmptions either with or without a landesman lazer condition the proofs are based on degree theoretic arguments

differentiability of relations and differential stability of perturbed optimization problems several new concepts of differentiability of multifunctions are introduced special attention is paid to relations given by perturbed inequalities applications are given to the study of the value or marginal function of a perturbed nonlinear program

multiple periodic solutions for asymptotically linear duffing equations with resonance we investigate multiple periodic solutions of asymptotically linear duffing equation with resonance using index theory and morse theory and obtain a new result

landesman lazer conditions for resonant p laplacian problems with jumping nonlinearities we study the existence of solutions of the p laplacian dirichlet problem 1 p u p u h x u f x u u x 0 1 2 u 0 u 1 0 where r p 1 s s 1sgn for s 1 r the function h 0 1 r r has the form h x a x p a x p x 0 1 r with max 0 and a l1 0 1 and the function f 0 1 r2 r is continuous and satisfies f x k x 1 q 1 x 0 1 r2 for some q 1 p and k l1 0 1 the dominant asymptotic behaviour of equation 1 as u is determined by the coefficients a and we allow a a in which case the problem is said to be jumping if the positively homogeneous problem obtained from 1 2 by setting f 0 has a non trivial solution then the problem is said to be resonant and is said to be a half eigenvalue assuming that the problem 1 2 is both jumping and resonant we will obtain a solution under certain landesman lazer conditions on f

resonance and rotation numbers for planar hamiltonian systems multiplicity results via the poincare birkhoff theorem abstract in the general setting of a planar first order system 0 1 u g t u u r 2 with g 0 t r 2 r 2 we study the relationships between some classical nonresonance conditions including the landesman lazer one at infinity and in the unforced case i e g t 0 0 at zero and the rotation numbers of large and small solutions of 0 1 respectively such estimates are then used to establish via the poincare birkhoff fixed point theorem new multiplicity results for t periodic solutions of unforced planar hamiltonian systems j u u h t u and unforced undamped scalar second order equations x g t x 0 in particular by means of the landesman lazer condition we obtain sharp conclusions when the system is resonant at infinity

nonautonomous nonlinear odes nonresonance conditions and rotation numbers abstract we deal with the t periodic problem associated with a nonlinear scalar differential equation like 1 x f t x 0 where for x 0 and x the nonlinearity f is assumed behave linearly with a time dependent coefficient we prove that the landesman lazer conditions at zero and at infinity possess a rotational effect on small and large in the phase plane solutions of 1 as a consequence we are able to generalize previous multiplicity results in the resonant case i e when the linearizations at zero and or at infinity are resonant through the use of the poincare birkhoff fixed point theorem

multiple solutions for an asymptotically linear duffing equation with newmann boundary value conditions abstract we investigate multiple solutions of an asymptotically linear duffing equation satisfying newmann boundary value conditions with resonance using index theory and morse theory and obtain some new results

local monotonicity and full stability for parametric variational systems this paper introduces and characterizes new notions of lipschitzian and holderian full stability of solutions to general parametric variational systems defined via partial subdifferential of prox regular functions acting in finite dimensional and hilbert spaces these notions which postulate certain quantitative properties of single valued localizations of solution maps are closely related to local strong maximal monotonicity of associated set valued mappings based on advanced tools of variational analysis and generalized differentiation we derive verifiable characterizations of the local strong maximal monotonicity and full stability notions under consideration via some positive definiteness conditions involving second order constructions of variational analysis

characterizations of the range of neumann problem for semilinear elliptic equations on etudie le probleme de neumann au gu i j 1 n a ij x u x i x j g u f ce bu x j 1 n a ij x u x i cos x j 0 xe ou est un domaine de r n a frontiere lisse est la normale exterieure unite de t a ij x e 1 a est un operateur uniformement fortement elliptique dans et g t r r est une fonction continue et bornee on donne un theoreme d existences des solutions faibles

multivariate inequalities of chernoff type for classical orthogonal polynomials abstract in this paper we present two sided chernoff type inequalities for the error of the best approximation of a smooth d variate function by polynomials of total degree less than k in the l w 2 norm it is supposed that the d variate weight function w has one dimensional classical component weights which satisfy pearson differential equations similarly as in the univariate case the leading coefficients of the multivariate classical polynomials orthonormal with respect to the weight function w play an important role in the presented estimates

a system of coupled pendulii abstract we study the existence of solutions for a coupled system of n dimensional pendulum equations under generalized periodic type conditions we obtain existence results under appropriate conditions using topological degree methods and a shooting type argument

index theory nontrivial solutions and asymptotically linear second order hamiltonian systems abstract in this paper we consider the existence and multiplicity of solutions of second order hamiltonian systems we propose a generalized asymptotically linear condition on the gradient of hamiltonian function classify the linear hamiltonian systems prove the monotonicity of the index function and obtain some new conditions on the existence and multiplicity for generalized asymptotically linear hamiltonian systems by global analysis methods such as the leray schauder degree theory the morse theory the ljusternik schnirelman theory etc

computations of critical groups and applications to some differential equations at resonance abstract in this paper we first consider the computations of the critical groups of a functional with generalized ahmad lazer paul type conditions both at zero and infinity then the abstract results are applied to investigate some new cases of the existence of nontrivial solutions of the second order hamiltonian systems and elliptic boundary value problems which may be resonant both at zero and infinity

some weighted polynomial inequalities abstract the author considers a weight function of sub exponential type with zeros and singularities in 1 1 some inequalities of remez bernstein markov and nikolsky type are proved the results hold for polynomials and generalized polynomials moreover an upper and a lower bound for the christoffel functions are derived

double resonance with landesman lazer conditions for planar systems of ordinary differential equations abstract we prove the existence of periodic solutions for first order planar systems at resonance the nonlinearity is indeed allowed to interact with two positively homogeneous hamiltonians both at resonance and some kind of landesman lazer conditions are assumed at both sides we are thus able to obtain as particular cases the existence results proposed in the pioneering papers by lazer and leach 1969 27 and by frederickson and lazer 1969 18 our theorem also applies in the case of asymptotically piecewise linear systems and in particular generalizes fabry s results in fabry 1995 10 for scalar equations with double resonance with respect to the dancer fucik spectrum

a note on a modified version of the poincare birkhoff theorem abstract we show by constructing a suitable area preserving homeomorphism without fixed points that the twist condition considered at zero and at infinity in a recent modified version of the poincare birkhoff fixed point theorem cannot be interchanged

asymptotically sharp inequalities for polynomials involving mixed laguerre norms abstract the paper concerns best constants in markov type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself the norm of the polynomial and its derivative is taken in l 2 on the half line with the weight t e t and t e t respectively under the assumption that is larger than the order of the derivative we determine the leading term of the asymptotics of the constants as the degree of the polynomial goes to infinity thus proving a conjecture raised by bottcher and dorfler in 2010 4

the hahn and meixner polynomials of an imaginary argument and some of their applications the hahn and meixner polynomials belonging to the classical orthogonal polynomials of a discrete variable are analytically continued in the complex plane both in variable and parameter this leads to the origination of two systems of real polynomials orthogonal with respect to a continuous measure the meixner polynomials of an imaginary argument obtained in this manner turned out to be known in the literature as the pollaczek polynomials the orthogonality relation for the hahn polynomials with respect to a continuous measure is apparently new a close connection between the hahn polynomials of an imaginary argument and representations of the lorentz group so 3 1 is considered

multiple periodic solutions for asymptotically linear ordinary differential equations with double resonance abstract in this paper using index theory and morse theory we investigate multiple periodic solutions of asymptotically linear ordinary differential equation with double resonance at infinity and with resonance at origin and obtain new results

chain rules for nonsmooth functions abstract a systematic method is presented for the derivation of chain rules for compositions of functions f f where f is nondecreasing this method is valid for directional derivatives and subgradients associated with any tangent cone having a short list of properties some major special cases are examined in detail in particular calculus rules are derived for rockafellar s epi derivatives and clarke generalized gradients

extremal problems inequalities and classical orthogonal polynomials in this survey paper we give a short account on characterizations for very classical orthogonal polynomials via extremal problems and the corresponding inequalities besides the basic properties of the classical orthogonal polynomials we consider polynomial inequalities of landau and kolmogoroff type some weighted polynomial inequalities in l 2 norm of markov bernstein type as well as the corresponding connections with the classical orthogonal polynomials

subdifferentials of nonconvex vector valued functions abstract the formula of clarke s subdifferential for the sum of two real valued locally lipschitz functions has been extended by rockafellar to the case where one of the two functions is directionally lipschitzian we introduce in this paper a notion of cone related to the epigraph of a function and with the aid of this cone we extend the directionally lipschitzian behaviour to vector valued functions and we study the sum of two vector valued functions

nonlinear second order elliptic partial differential equations at resonance on etudie la resolubilite de problemes aux valeurs limites pour des equations aux derivees partielles elliptiques d ordre 2 semi lineaires de type resonance dans lesquelles la perturbation non lineaire ne satisfait pas necessairement la condition de landesman lazer ou l hypothese de monotonicite

on the use of morse index and rotation numbers for multiplicity results of resonant bvps abstract motivated by a recent series of papers by k li and co workers 14 15 16 17 18 we consider the problem of the existence and multiplicity of solutions for the neumann or periodic bvps associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behavior of its solutions near zero and near infinity may be compared with the behavior of the solutions of two suitable linear systems one considered near zero and the other near infinity we show how a rotation number approach together with the poincare birkhoff theorem and a recent variant of it allows to obtain multiplicity results in terms of the gap between the morse indexes of the referred linear systems at zero and at infinity these systems may also be resonant when the gaps are sufficiently large our multiplicity results improve the ones obtained by variational methods in the quoted papers also our approach allows a description of the solutions obtained in terms of their nodal properties

weighted l2 analogs of bernstein s inequality and classical orthogonal polynomials abstract let p n be the class of algebraic polynomials of degree at most n some weighted l 2 analogues of the bernstein s inequality for polynomials p p n are investigated and a connection with the classical orthogonal polynomials is given

which subgradients have sum formulas on donne une formule de somme generale valable pour un cone tangent avec une courte liste de proprietes

gauss hermite interval quadrature rule the existence and uniqueness of the gaussian interval quadrature formula with respect to the hermite weight function on r is proved similar results have been recently obtained for the jacobi weight on 1 1 and for the generalized laguerre weight on 0 numerical construction of the gauss hermite interval quadrature rule is also investigated and a suitable algorithm is proposed a few numerical examples are included

an elliptic singular system with nonlocal boundary conditions abstract we study the existence of solutions for the nonlinear second order elliptic system u g u f x where g c r n s r n with s r n bounded using topological degree methods we prove an existence result under a geometric condition on g moreover we analyze the particular case of an isolated repulsive singularity under a nirenberg type condition we prove the existence of a sequence of solutions of appropriate approximated problems that converges to a generalized solution

a chain rule for first and second order epiderivatives and hypoderivatives a chain rule is established for contingent and adjacent epiderivatives and hypoderivatives of compositions g h where h is assumed to be hadamard directionally differentiable corollaries include a formula for the contingent and adjacent cones of an equality constraint set defined by a hadamard directionally differentiable function an analogous chain rule for second order contingent and adjacent epiderivatives and hypoderivatives is also developed

multiple solutions for asymptotically linear duffing equations with neumann boundary value conditions ii abstract we study multiple solutions of asymptotically linear duffing equations satisfying neumann boundary value conditions with resonance by using index theory and lusternik schnirelmann theory and obtain some new results