critical point theory for nonsmooth functionals abstract in this paper we develop critical point theory for nonsmooth functional f h 0 1 r defined by f u 1 2 i j 1 a i j x u d i u d j u d x g x u d x the corresponding deformation lemmas are proved with the application to the bifurcation for quasilinear schrodinger equations in mind we extend the obtained results to that for a functional defined on a product space and prove a generalized saddle point theorem

infinitely many solutions for a differential inclusion problem in r n abstract in this paper we consider the differential inclusion problem di p u u p 2 u x f u x x r n u w 1 p r n where 2 n p l 1 r n l r n is radially symmetric and f stands for the generalized gradient of a locally lipschitz function f r r under suitable oscillatory assumptions on the potential f at zero or at infinity we show the existence of infinitely many radially symmetric solutions of di no symmetry requirement on f is needed our approach is based on a non smooth ricceri type variational principle developed by marano and motreanu j differential equations 182 2002 108 120

existence results for obstacle problems for nonlinear hemivariational inequality at resonance in this paper we are interested in studying the existence of solutions to obstacle problems for nonlinear hemivariational inequality at resonance driven by the p laplacian using a variational approach based on the nonsmooth critical point theory for nondifferentiable functionals we prove two existence theorems

the monotone iterative technique for a parabolic boundary value problem with discontinuous nonlinearity let be a bounded lebesgue measurable domain of r n with a lipschitz boundary the following initial boundary value problem is considered u t lu f u h in q bu o on u 0 where q o t o t and 0 0 l is a uniformly elliptic operator b a boundary operator defined by bu u v bu in which v denotes the outer normal derivative on and f represents a discontinuous non linearity

nonlinear hemivariational inequalities at resonance in this paper we examine a semilinear hemivariational inequality at resonance in the first eigenvalue 1 of h01 z we prove two existence theorems for such problems our approach is variational and is based on the nonsmooth critical point theory of chang which uses the subdifferential calculus of clarke for locally lipschitz functions

signless laplacians of finite graphs we survey properties of spectra of signless laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix for regular graphs the whole existing theory of spectra of the adjacency matrix and of the laplacian matrix transfers directly to the signless laplacian and so we consider arbitrary graphs with special emphasis on the non regular case the results which we survey old and new are of two types a results obtained by applying to the signless laplacian the same reasoning as for corresponding results concerning the adjacency matrix b results obtained indirectly via line graphs among other things we present eigenvalue bounds for several graph invariants an interpretation of the coefficients of the characteristic polynomial a theorem on powers of the signless laplacian and some remarks on star complements

elliptic problems with lack of compactness via a new fixed point theorem abstract this paper provides a new fixed point theorem for increasing self mappings g b b of a closed ball b x where x is a banach semilattice which is reflexive or has a weakly fully regular order cone x by means of this fixed point theorem we are able to establish existence results of elliptic problems with lack of compactness

closed geodesics with lipschitz obstacle abstract geodesics on a general subsetmof r nare considered whenmis the closure of an open subset with lipschitz boundary and suitable topological conditions are fulfilled the existence of a nonconstant closed geodesic is proved

the line graphs of lollipop graphs are determined by their spectra abstract haemers et al w h haemers x g liu y p zhang spectral characterizations of lollipop graphs linear algebra appl 428 2008 2415 2423 first investigated the spectral characterizations of the so called lollipop graphs h g l n with order n which is obtained by identifying a vertex of a cycle with order g and a pendent vertex of a path with order l for the graphs with least eigenvalue at least 2 cvetkovic and lepovic d cvetkovic m lepovic cospectral graphs with least eigenvalue at least 2 publ inst math nouv ser 78 92 2005 51 63 introduced the discriminant and the star value of a graph whose relations are investigated in this paper employing this relation and other techniques we prove that all line graphs of lollipop graphs are determined by their adjacency spectra

on the existence of small energy solutions for a sublinear neumann problem abstract in this paper we are concerned with the sublinear problem 0 1 u u p 2 u in u 0 on where r n is a bounded domain and 1 p 2 for p 1 the nonlinearity u p 2 u will be identified by sgn u in contrast to previous work on the dirichlet problem some difficulties arise due to the fact that the associated energy functional is not bounded from below complementing recent work by parini and weth in 15 on least energy solutions we prove that 0 1 has infinitely many solutions with small negative energy

existence of solutions for elliptic hemivariational inequalities abstract this paper is devoted to the existence of solutions concerning the dirichlet problem for quasilinear elliptic hemivariational inequalities at the first eigenvalue using the notion of the generalized gradient of clarke and the property of the first eigenfunction some existence results of solutions have been proved

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

local c1 ω minimizers versus local w1 p ω minimizers of nonsmooth functionals abstract we study not necessarily differentiable functionals of the form j u 1 p u p d x 1 p u p d x j 1 x u d x j 2 x u d with 1 p involving locally lipschitz functions j 1 r r as well as j 2 r r we prove that local c 1 minimizers of j must be local w 1 p minimizers of j

on the signless laplacian spectral characterization of the line graphs of lollipop graphs abstract let h g k n denote the lollipop graph on n vertices obtained by identifying a vertex of the cycle c g of order g and an end vertex of the path p k 1 of order k 1 in this paper we prove that all line graphs l h g k n of lollipop graphs are determined by their laplacian spectrum furthermore we show that all l h g k n but the case that g k 3 are determined by their signless laplacian spectrum and also give the q cospectral class of l h r r n r 3

two algorithms for the minimum enclosing ball problem given cal a a 1 dots a m subset mathbb r n and epsilon 0 we propose and analyze two algorithms for the problem of computing a 1 epsilon approximation to the radius of the minimum enclosing ball of cal a the first algorithm is closely related to the frank wolfe algorithm with a proper initialization applied to the dual formulation of the minimum enclosing ball problem we establish that this algorithm converges in o 1 epsilon iterations with an overall complexity bound of o mn epsilon arithmetic operations in addition the algorithm returns a core set of size o 1 epsilon which is independent of both m and n the latter algorithm is obtained by incorporating away steps into the former one at each iteration and achieves the same asymptotic complexity bound as the first one while the asymptotic bound on the size of the core set returned by the second algorithm also remains the same as the first one the latter algorithm has the potential to compute even smaller core sets in practice since in contrast to the former one it allows dropping points from the working core set at each iteration our analysis reveals that the leading terms in the asymptotic complexity analysis are reasonably small in contrast to the first algorithm we also establish that the second algorithm asymptotically exhibits linear convergence which provides further insight into our computational results indicating that the latter algorithm indeed terminates faster with smaller core sets in comparison with the first one we also discuss how our algorithms can be extended to compute an approximation to the minimum enclosing ball of more general input sets without sacrificing the iteration complexity and the bound on the core set size in particular we establish the existence of a core set of size o 1 epsilon for a much wider class of input sets we adopt the real number model of computation in our analysis

periodic problems with asymmetric nonlinearities and nonsmooth potentials abstract we consider a nonlinear periodic boundary value problem driven by the ordinary scalar p laplacian and with a nonsmooth locally lipschitz potential which exhibits an asymmetric behaviour at and at using the fucik spectrum and a variational approach based on the nonsmooth critical point theory we obtain a solution for the problem using a general minimax principle concerning linking sets

on some elliptic hemivariational and variational hemivaritional inequalities abstract the existence of solutions to an elliptic hemivariational inequality involving the p laplacian is first established the variational hemivariational case which includes the above one is then examined when p 2 both of them are chiefly treated through mountain pass like critical point results for non differentiable functions

on the second largest laplacian eigenvalues of graphs the second largest laplacian eigenvalue of a graph is the second largest eigenvalue of the associated laplacian matrix in this paper we study extremal graphs for the extremal values of the second largest laplacian eigenvalue and the laplacian separator of a connected graph respectively all simple connected graphs with second largest laplacian eigenvalue at most 3 are characterized it is also shown that graphs with second largest laplacian eigenvalue at most 3 are determined by their laplacian spectrum moreover the graphs with maximum and the second maximum laplacian separators among all connected graphs are determined

nonlinear algebraic systems with discontinuous terms abstract using a multiple critical points theorem for locally lipschitz continuous functionals we establish the existence of at least three distinct solutions for a parametric discrete differential inclusion problem involving a real symmetric and positive definite matrix applications to tridiagonal fourth order and partial difference inclusions are presented

a variational approach to discontinuous problems with critical sobolev exponents abstract we employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form u h x h u a uq u2 1 in rn where a 0 are parameters h x is both nonnegative and integrable on rn h is the heaviside function 2 is the critical sobolev exponent and 0 q

spectral characterization of graphs with index at most 2 5 abstract a graph is said to be determined by the adjacency spectrum ds for short if there is no other nonisomorphic graph with the same spectrum all connected graphs with index at most 2 5 are known in this paper we show that with few exceptions all of these graphs are ds

spectral characterizations of signed lollipop graphs abstract let g be a signed graph where g is the underlying simple graph and e g is the sign function on the edges of g in this paper we consider the spectral characterization problem extended to the adjacency matrix and laplacian matrix of signed graphs after giving some basic results we study the spectral determination of signed lollipop graphs and we show that any signed lollipop graph is determined by the spectrum of its laplacian matrix

boundary value problems for discontinuous perturbations of singular ϕ laplacian operator abstract systems of differential inclusions of the form u f t u t 0 t where with strictly convex is a homeomorphism of the ball b a r n onto r n are considered under dirichlet periodic and neumann boundary conditions here f t x stands for the generalized clarke gradient of f t at x r n using nonsmooth critical point theory we obtain existence results under some appropriate conditions on the potential f examples of applications concerning filippov type solutions are also provided

graphs with maximal signless laplacian spectral radius abstract by the signless laplacian of a simple graph g we mean the matrix q g d g a g where a g d g denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of g it is known that connected graphs g that maximize the signless laplacian spectral radius q g over all connected graphs with given numbers of vertices and edges are degree maximal for a maximal graph g with n vertices and r distinct vertex degrees r r 1 1 it is proved that q g q h for some maximal graph h with n 1 respectively n vertices and the same number of edges as g if either g has precisely two dominating vertices or there exists an integer i 2 i r 2 respectively if there exist positive integers i l with l 2 i r 2 such that i r 1 i n 1 respectively i r 1 i l r l 1 graphs that maximize q g over the class of graphs with m edges and m k vertices for k 0 1 2 3 are completely determined

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

the characteristic polynomial of the laplacian of graphs in a b linear classes in this work we deal with the characteristic polynomial of the laplacian of a graph we present some general results about the coefficients of this polynomial we present families of graphs for which the number of edges m is given by a linear function of the number of vertices n in some of these graphs we can find certain coefficients of the above named polynomial as functions just of n

spectral characterizations of lollipop graphs abstract the lollipop graph denoted by h n p is obtained by appending a cycle c p to a pendant vertex of a path p n p we will show that no two non isomorphic lollipop graphs are cospectral with respect to the adjacency matrix it is proved that for p odd the lollipop graphs h n p and some related graphs h n p are determined by the adjacency spectrum and that all lollipop graphs are determined by its laplacian spectrum

nonsmooth version of fountain theorem and its application to a dirichlet type differential inclusion problem abstract in this paper we establish a nonsmooth version of fountain theorem by adopting the framework of nonsmooth analysis theory which is a generalization of theorem 3 6 of willem 1996 2 then we present an application of this theorem to a dirichlet type differential inclusion problem

on a signless laplacian spectral characterization of t shape trees let m be an associated matrix of a graph g the adjacency laplacian and signless laplacian matrix two graphs are said to be cospectral with respect to m if they have the same m spectrum a graph is said to be determined by m spectrum if there is no other non isomorphic graph with the same spectrum with respect to m it is shown that t shape trees are determined by their laplacian spectra moreover among them those are determined by their adjacency spectra are characterized in this paper we identify graphs which are cospectral to a given t shape tree with respect to the signless laplacian matrix subsequently t shape trees which are determined by their signless laplacian spectra are identified