the beginning of the fucik spectrum for the p laplacian has a nontrivial solution u here 1 p 2p u div u p 2 u u max u 0 u uu in 1 1 denoting by 1 2 the first two eigenvalues of 2p on w 1 p 0 0 cf the end of this introduction 7p clearly contains in particular 1 1 2 2 and the two lines 1 r and r 1 this generalized notion of spectrum was introduced for p 2 i e in the linear case in the 1970s by fuc ik 21 and dancer 15 in connection article id jdeq 1999 3645 available online at http www idealibrary com on

constant sign and nodal solutions of coercive p q laplacian problems abstract the existence of a greatest negative a smallest positive and a nodal weak solution to a dirichlet problem with p q laplacian and reaction term that makes coercive the corresponding energy functional is investigated via variational methods besides truncation techniques

existence of a non trivial solution for the p laplacian equation with fucik type resonance at infinity abstract we prove the existence of a non trivial weak solution for the equation p u a u p 1 b u p 1 f x u in u 0 on by variational methods we consider the case where a 1 or b 1 1 denotes the first eigenvalue of p or the case where a is close to b while the nonlinearity f belongs to a class that contains for example u q 2 u 1 q p

positive linear maps and spreads of matrices ii abstract it is shown how positive linear maps can be used to obtain lower bounds for spreads of normal matrices some known results and some new ones are obtained in this way

some inequalities for central moments of matrices abstract in this paper we shall study non commutative central moment inequalities with a focus on whether the commutative bounds are tight in the non commutative case as well we prove that the answer is affirmative for the fourth central moment and several particular results are given in the general case as an application we shall present some lower estimates of the spread of hermitian and normal matrices as well

generalization of fredholm alternative for nonlinear differential operators on considere le probleme aux valeurs limites u t p 2 u t f t u t g t t 0 u 0 u 0 on demontre que si lim u inf f t u u p 2 u lim u sup f t u u p 2 u sont entre des valeurs propres du probleme aux valeurs propres u p 2 u u p 2 u sur 0 u 0 u 0 alors le probleme aux limites est resoluble pour tout g l 1

bounds for the spectrum of normal matrices abstract let a be a normal matrix with eigenvalues 1 2 n and let denote the smallest disc containing these eigenvalues we give some inequalities relating the center and radius of to the entries in a when applied to hermitian matrices our results give lower bounds on the spread maxij i j of a when applied to positive definite hermitian matrices they give lower bounds on the kantorovich ratio maxij i j i j

eigenvalues of the p x laplacian steklov problem abstract consider steklov eigenvalue problem involving the p x laplacian on a bounded domain the open subset of r n with n 2 as follows p x u u p x 2 u in u p x 2 u u p x 2 u on where p x constant we prove that the existence of infinitely many eigenvalue sequences unlike the p laplacian case for a variable exponent p x constant there does not exist a principal eigenvalue and the set of all eigenvalues is not closed under some assumptions finally we present some sufficient conditions for the infimum of all eigenvalues is zero and positive respectively

characterization of birkhoff james orthogonality the birkhoff james orthogonality is a generalization of hilbert space orthogonality to banach spaces we investigate this notion of orthogonality when the banach space has more structures we start by doing so for the banach space of square matrices moving gradually to all bounded operators on any hilbert space then to an arbitrary c algebra and finally a hilbert c module

variance bounds with an application to norm bounds for commutators abstract murthy and sethi m n murthy v k sethi sankhya ser b 27 1965 201 210 gave a sharp upper bound on the variance of a real random variable in terms of the range of values of that variable we generalise this bound to the complex case and more importantly to the matrix case in doing so we make contact with several geometrical and matrix analytical concepts such as the numerical range and introduce the new concept of radius of a matrix we also give a new and simplified proof for a sharp upper bound on the frobenius norm of commutators recently proven by bottcher and wenzel a bottcher d wenzel the frobenius norm and the commutator linear algebra appl 429 2008 1864 1885 and point out that at the heart of this proof lies exactly the matrix version of the variance we have introduced as an immediate application of our variance bounds we obtain stronger versions of bottcher and wenzel s upper bound

on the fucik spectrum of the laplacian on a torus abstract we study the fucik spectrum of the laplacian on a two dimensional torus t 2 exploiting the invariance properties of the domain t 2 with respect to translations we obtain a good description of large parts of the spectrum in particular for each eigenvalue of the laplacian we will find an explicit global curve in the fucik spectrum which passes through this eigenvalue these curves are ordered and we will show that their asymptotic limits are positive on the other hand using a topological index based on the mentioned group invariance we will obtain a variational characterization of global curves in the fucik spectrum also these curves emanate from the eigenvalues of the laplacian and we will show that they tend asymptotically to zero thus we infer that the variational and the explicit curves cannot coincide globally and that in fact many curve crossings must occur we will give a bifurcation result which partially explains these phenomena

some inequalities for positive linear maps it has long been known that an analogue of jensen s inequality holds for positive unital linear maps on matrix algebras provided that instead of ordinary convex functions one restricts to matrix convex functions we show that this restriction is not necessary in the case of 2 2 matrices a noncommutative analogue of the variance is studied and a basic inequality with several applications is established

multiple constant sign and nodal solutions for nonlinear elliptic equations with the p laplacian we consider a nonlinear elliptic equation driven by the p laplacian with dirichlet boundary conditions using variational techniques combined with the method of upper lower solutions and suitable truncation arguments we establish the existence of at least five nontrivial solutions two positive two negative and a nodal sign changing solution our framework of analysis incorporates both coercive and p superlinear problems also the result on multiple constant sign solutions incorporates the case of concave convex nonlinearities

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

orthogonality of matrices and some distance problems abstract if a and b are matrices such that a zb a for all complex numbers z then a is said to be orthogonal to b we find necessary and sufficient conditions for this to be the case some applications and generalisations are also discussed

lower bounds for the spread of a matrix a characterization of the spread of a normal matrix is used to derive several simple lower bounds for the spread comparisons are then made with several known bounds

construction of pseudo gradient vector field and sign changing multiple solutions involving p laplacian existence of multiple and sign changing solutions for a problem involving p laplacian and jumping nonlinearities are considered via the construction of descent flow in c01 sign changing and multiple solutions are obtained under additional assumption on the nonlinearity the uniqueness of positive negative solution theorem is included too

positive linear maps and perturbation bounds of matrices abstract we show how positive unital linear maps can be used to obtain lower bounds for the maximum distance between the eigenvalues of two normal matrices some related bounds for the spread and condition number of hermitian matrices are also discussed here

characterizations and lower bounds for the spread of a normal matrix abstract the spread of an n n matrix a with eigenvalues 1 n is defined by spr a max j k j k we prove that if a is normal then spr a max x ax y ay x y c n x y 1 max x ay y ax x y c n x y 1 re x y 0 max x ay y ax x y c n x y 1 x y 0 max spr z a z a 2 z c z 1 2 max x a 2 x x ax 2 x a ax x ax 2 1 2 x c n x 1 we also present several lower bounds for spr a given by these characterizations

existence and asymptotic behavior of solutions for a class of nonlinear elliptic equations with neumann condition abstract in this paper we study the existence of nonconstant solutions u e and their asymptotic behavior as e 0 for the following class of nonlinear elliptic equations in radial form e 2 r u u r f u in 0 r u 0 u r 0 where are given real numbers and 0 r we use a version of the mountain pass theorem and the main difficulty is to prove that the solutions so obtained are not constants for that matter we have to carryout a careful analysis of the solutions of some dirichlet problems associated with the neumann problem

the fucik spectrum of the p laplacian and jumping nonlinear problems abstract we prove the existence of a nontrivial curve in the fucik spectrum of the p laplacian using the sequence of minimax eigenvalues constructed by cohomological index furthermore we study the solvability of jumping nonlinear problems p u u p 1 u p 1 f x u 0 x

analytical and numerical results for the fucik spectrum of the laplacian nontrivial solutions of u u vu 0 in rn with zero dirichlet condition on are studied the collection of the pairs v is the so called fucik spectrum f a new variational formulation for parts of f is presented and analyzed based on this formulation a minimization algorithm for the computation of a part of f is developed alternatively an approach using an implicit function argument is discussed analytically and via newton s method also numerically by combining the variational minimization method with newton s method a new bifurcation result is first observed numerically and then proved rigorously by replacing the variational minimization method by the mountain pass algorithm higher curves in f are found numerically several numerical results are discussed including a further example of the previously recorded phenomenon of crossing of fucik curves originating from different eigenvalues

multiple solutions with precise sign for nonlinear parametric robin problems we consider a parametric nonlinear robin problem driven by the p laplacian we show that if the parameter 2 the second eigenvalue of the robin p laplacian then the problem has at least three nontrivial solutions two of constant sign and the third nodal in the semilinear case p 2 we show that we can generate a second nodal solution our approach uses variational methods truncation and perturbation techniques and morse theory in the process we produce two useful remarks about the first two eigenvalues of the robin p laplacian

some remarks on the fucik spectrum of the p laplacian and critical groups let be a bounded domain in n n 1 for 1 p the fucik spectrum of the p laplacian on w 1 p 0 is defined as the set p of those points a b 2 such that pu a u p 1 b u p 1 u w 1 p 0 1 1 has a nontrivial solution here pu div u p 2 u and u max u 0 it is known that the first eigenvalue 1 of p on w 1 p 0 is positive simple and admits a positive eigenfunction 1 w 1 p 0 c1 see lindqvist 8 so p contains the two lines 1 and 1 a first nontrivial curve c2 in p through 2 2 where 2 is the second eigenvalue of p was recently constructed and variationally characterized by a

nonlinear elliptic equations with an asymptotically linear reaction term abstract we examine nonlinear elliptic problems driven by the p laplacian and with a reaction term which asymptotically at exhibits a p 1 linear growth 1 p using tools from critical point theory and from morse theory we prove existence theorems and we provide information about the sign of the solutions