on the positive almost periodic type solutions for some nonlinear delay integral equations in this paper we discuss the existence of positive almost periodic type solutions for some nonlinear delay integral equations by constructing a new fixed point theorem in the cone some known results are extended

existence of a positive almost periodic solution for a nonlinear delay integral equation in this paper we discuss the existence of positive almost periodic solution for some nonlinear delay integral equation by constructing a new fixed point theorem in the cone some known results are extended

almost automorphic mild solutions to some partial neutral functional differential equations and applications abstract the paper considers the existence and uniqueness of almost automorphic mild solutions to some classes of first order partial neutral functional differential equations sufficient conditions for the existence and uniqueness of almost automorphic mild solutions to the above mentioned equations are obtained as an application a first order boundary value problem arising in control systems is considered

trees with a large laplacian eigenvalue multiplicity abstract in this paper we study the multiplicity of the laplacian eigenvalues of trees it is known that for trees integer laplacian eigenvalues larger than 1 are simple and also the multiplicity of laplacian eigenvalue 1 has been well studied before here we consider the multiplicities of the other non integral laplacian eigenvalues we give an upper bound and determine the trees of order n that have a multiplicity that is close to the upper bound n 3 2 and emphasize the particular role of the algebraic connectivity

positive almost periodic solutions of some delay integral equations the delay integral equation x t tt f s x s ds which arises in models for the spread of epidemics is studied with the aim of establishing the existence of positive almost periodic solutions for large values of when f t x is uniformly almost periodic in t for x in compact subsets of r under reasonable assumptions on f it is shown that there exist two positive numbers 0 they do exist a priori bounds on the set of positive solutions and uniqueness results are also obtained

pseudo almost periodic solutions of neutral integral and differential equations with applications abstract the existence and uniqueness of pseudo almost periodic solutions to general neutral integral equations with deviations are obtained for this pseudo almost periodic functions in two variables are considered the results extend the corresponding ones to the convolution type integral equations they are used to study pseudo almost periodic solutions of general neutral differential equations and to the so called scalar neutral logistic equation version

on the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces abstract the main purpose of this paper is to deal with almost automorphic and asymptotically almost automorphic solutions of the initial value problem as well as the nonlinear volterra integral equation in banach spaces we obtain a collection of existence results of such solutions to these equations we investigate also a topological structure of such solution sets moreover we prove aronszajn type theorems for solutions of the initial value problem as well as the nonlinear volterra integral equation defined on the whole real line

composition of pseudo almost automorphic and asymptotically almost automorphic functions this paper is concerned with pseudo almost automorphic functions which are more general and complicated than pseudo almost periodic functions and asymptotically almost automorphic functions new results concerning the composition of pseudo almost automorphic functions are established

exponential fitting collocation methods for a class of volterra integral equations abstract in this paper we propose a collocation method for a class of volterra integral equations whose solutions contain periodic functions since the exponential fitting interpolation has an advantage in approximating periodic functions we consider employing it with collocation method to construct our scheme the global convergence analysis of the scheme is also presented based on the interpolation error the theoretical results as well as the superiority of the method are verified in the numerical part

stepanov like almost automorphic functions and monotone evolution equations abstract in this paper we are concerned with a new class of stepanov like almost automorphic s p a a functions with values in a banach space x this class contains the space aa x of all bochner almost automorphic functions we use the results obtained to prove the existence and uniqueness of a weak s p a a solution to the parabolic equation u t a t u f t in a reflexive banach space assuming some appropriate conditions of monotonicity coercitivity of the operators a t and s p almost automorphy of the forced term f t this result extends a known result in the case of almost periodicity an application is also given

asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions abstract of concern is a class of abstract semilinear integrodifferential equations with nonlocal initial conditions under some suitable hypotheses we establish some new theorems about the existence of asymptotically almost automorphic solutions to the integrodifferential equations moreover an example is given to illustrate our results

weighted pseudo almost automorphic solutions for nonautonomous spdes driven by levy noise abstract in this paper we introduce the concept of weighted pseudo almost automorphy in distribution using the theory of evolution family and stochastic analysis method we establish the existence and uniqueness results of almost automorphic solutions in distribution and weighted pseudo almost automorphic solutions in distribution for some semilinear nonautonomous stochastic partial differential equations driven by levy noise when the coefficients of the equations satisfy some suitable conditions moreover we investigate the global exponentially stability of these solutions the results are generalizations of the results in wang and liu 32 chen and lin 11 and liu and sun 26

existence and uniqueness of c n almost periodic solutions to some ordinary differential equations abstract in this paper we prove the existence and uniqueness of c n almost periodic solutions to the ordinary differential equation x t a t x t f t t r where the matrix a t r m k c is periodic and f r c k is c n almost periodic we also prove the existence and uniqueness of an ultra weak c n almost periodic solution in the case when a t a is independent of t finally we prove also the existence and uniqueness of a mild c n almost periodic solution of the semilinear hyperbolic equation x t a x t f t x considered in a banach space assuming f t x is c n almost periodic in t for each x x satisfies a global lipschitz condition and takes values in an extrapolation space f a 1 associated to a

bounded mild solutions of perturbed volterra equations with infinite delay abstract we study existence and regularity of bounded mild solutions on the real line to perturbed integral equations with infinite delay in the space of almost periodic functions in the bohr sense the space of compact almost automorphic functions the space of almost automorphic functions and the space of asymptotically almost automorphic functions

pseudo almost automorphic mild solutions to semilinear integral equations in a banach space abstract in this paper we consider the existence of pseudo almost automorphic solutions of the semilinear integral equation x t t a t s a x s f s x s d s t r in a banach space x where a l 1 r a is the generator of an integral resolvent family of linear bounded operators defined on the banach space x and f r x x is a pseudo almost automorphic function the main results are proved by using integral resolvent families combined with the theory of fixed points

discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization based on the authors previous work which established theoretical foundations of two conceptual successive convex relaxation methods i e the ssdp successive semidefinite programming relaxation method and the ssilp successive semi infinite linear programming relaxation method this paper proposes their implementable variants for general quadratic optimization problems these problems have a linear objective function c t x to be maximized over a nonconvex compact feasible region f described by a finite number of quadratic inequalities we introduce two new techniques discretization and localization into the ssdp and ssilp relaxation methods the discretization technique makes it possible to approximate an infinite number of semi infinite sdps or semi infinite lps which appeared at each iteration of the original methods by a finite number of standard sdps or standard lps with a finite number of linear inequality constraints we establish given any open convex set u containing f there is an implementable discretization of the ssdp or ssilp relaxation method which generates a compact convex set c such that f c u in a finite number of iterations the localization technique is for the cases where we are only interested in upper bounds on the optimal objective value for a fixed objective function vector c but not in a global approximation of the convex hull of f this technique allows us to generate a convex relaxation of f that is accurate only in certain directions in a neighborhood of the objective direction c this cuts off redundant work to make the convex relaxation accurate in unnecessary directions we establish given any positive number e there is an implementable localization discretization of the ssdp or ssilp relaxation method which generates an upper bound of the objective value within e of its maximum in a finite number of iterations

a massera type theorem for almost automorphic solutions of differential equations abstract in this note we present a massera type theorem for the existence of almost automorphic solutions of periodic linear evolution equations of the form x t a t x t f t where a t is unbounded linear operator depending on t periodically and generates a periodic evolutionary process f is almost automorphic the main results are stated in terms of the almost automorphy of solutions and their carleman spectra

exponential dichotomy and pseudo almost automorphy for partial neutral functional differential equations abstract in this work we study the existence and uniqueness of a pseudo almost automorphic solution for some partial functional differential equations of neutral type we assume that the undelayed part is not necessarily densely defined and satisfies the hille yosida condition the delay part is assumed to be pseudo almost automrophic in time we prove if the homogeneous linear equation has an exponential dichotomy then the nonhomogeneous linear equation has a unique pseudo almost automorphic solution an application is given for some nonlinear equations we present also a new concept of generalized pseudo almost automorphy

almost automorphic and pseudo almost automorphic mild solutions to an abstract differential equation in banach spaces in this paper we prove some existence results for almost automorphic and pseudo almost automorphic mild solutions to a class of abstract differential equations in banach spaces the main technique is based on some composition theorems combined with the contraction mapping theorem finally we present an application to a semilinear partial differential equation with dirichlet conditions

on almost automorphic type solutions of abstract integral equations a bohr neugebauer type property and some applications abstract in the present work we give some sufficient conditions to obtain a unique almost automorphic solution to abstract nonlinear integral equations which are simultaneously of advanced and delayed type and also a unique asymptotically almost automorphic mild solution to abstract integro differential equations with nonlocal initial conditions both situations are posed on banach spaces also we develop a bohr neugebauer type result for the abstract integral equations before that we introduce the notion of bounded functions develop the appropriate abstract theory and discuss the almost periodic situation as applications we study the existence of an asymptotically almost automorphic solution to integro differential equations modeling heat conduction in materials with memory and also the existence of the almost automorphic solution to semilinear parabolic evolution equations with finite delay

existence of positive almost automorphic solutions to neutral nonlinear integral equations abstract this paper is concerned with neutral nonlinear delay integral equations we establish an existence theorem for positive almost automorphic solutions to the equations which extend some existing results even in the case of almost periodicity some examples are given to illustrate our results

on favard s theorems abstract in this paper we improve and extend the classical favard s theorems i e favard s theorem of the module containment favard s theorem of linear differential equations we study favard s theory of linear differential equations with piecewise constant argument an example shows that the new module containment is necessary in the study of differential equations with piecewise constant argument the equivalence between almost automorphic functions and n almost periodic ones is studied

massera type theorem for almost automorphic solutions of functional differential equations of neutral type abstract in this work we study the existence of almost automorphic solutions for functional differential equations of neutral type we prove that the existence of a bounded solution on r implies the existence of an almost automorphic solution

a variation of constants formula for abstract functional differential equations in the phase space abstract for linear functional differential equations with infinite delay in a banach space a variation of constants formula is established in the phase space as an application one applies it to study the admissibility of some spaces of functions whose spectra are contained in a closed subset of the real line

almost automorphic solutions of nonautonomous evolution equations abstract in this paper we establish some new theorems about the existence of almost automorphic solutions to nonautonomous evolution equations u t a t u t f t and u t a t u t f t u t in banach spaces as we will see our results allow for a more general a t to some extent an example is also given to illustrate our results in addition by means of an example we show that one cannot ensure the existence of almost automorphic solutions to u t a t u t f t even if the evolution family u t s generated by a t is exponentially stable and f a a x

almost periodic and almost automorphic solutions to semilinear parabolic boundary differential equations abstract in this paper we study the existence and uniqueness of almost periodic and almost automorphic solutions to the semilinear parabolic boundary differential equations sbde x t a m x t h t x t t r l x t t x t t r where a a m ker l generates a hyperbolic analytic semigroup on a banach space x the functions h and are defined on some intermediate subspaces x 0 1 and take values in x and in a boundary space x respectively

the fredholm alternative for parabolic evolution equations with inhomogeneous boundary conditions abstract we study the fredholm properties of parabolic evolution equations on r with inhomogeneous boundary values these problems are transformed into evolution equations with inhomogeneities taking values in certain extrapolation spaces assuming that the underlying homogeneous problem is asymptotically hyperbolic we show the fredholm alternative for these equations the results are applied to parabolic partial differential equations

on state dependent delay partial neutral functional integro differential equations in this paper the existence of mild solutions for a class of abstract neutral integro differential equations with state dependent delay is studied

the existence of positive almost periodic type solutions for some neutral nonlinear integral equation abstract this paper presents the conditions of existence of positive almost periodic type solutions for some neutral nonlinear integral equation by using a fixed point theorem in the cone some known results are extended

almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional applications to heat and wave equations with nonlinearities in this work we study the existence of bounded and almost automorphic solutions for evolution equations in banach spaces we suppose that the linear part is the infinitesimal generator of a compact c0 semigroup of bounded linear operators and the nonlinear part is an almost automorphic function with respect to the second argument we give sufficient conditions ensuring the existence of an almost automorphic solution when there is at least one bounded solution on r we use the subvariant functional method to show that every k minimizing mild solution is compact almost automorphic applications are provided for both heat and wave equations with nonlinearities in several functional spaces

on the almost automorphy of bounded solutions of differential equations with piecewise constant argument abstract in this paper we give sufficient spectral conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x t a x t f t t r where a is a bounded linear operator in x and f is an x valued almost automorphic function

semi fredholm operators and periodic solutions for linear functional differential equations abstract we deal with the inhomogeneous linear periodic equation with infinite delay of the formdx dt ax t b t xt f t whereais the generator of ac0 semigroup on a banach space assuming that it has a bounded solution we obtain several criteria on the existence and the uniqueness of periodic solutions for the equation in the general phase space b and in the concrete phase space b ucg the key of our approach is the employment of the perturbation theory of semi fredholm operators to show that the period map satisfies the condition of the fixed point theorem by chow and hale funkcial ekvac 17 1974 31 38

existence of positive almost automorphic solutions to nonlinear delay integral equations abstract this paper is concerned with the existence of positive almost automorphic solutions to some nonlinear delay integral equations we first establish a new fixed point theorem for mixed monotone operator in a cone and then with its help we obtain existence theorems of positive almost automorphic solutions some examples are given to illustrate our results as one will see even in the case of almost periodicity our theorems extend some earlier results and moreover the approach dealing with the integral equation arising in an epidemic problem in this paper is also new

lowest rank solutions of continuous and discrete lyapunov equations over symmetric cone abstract the low rank solutions of continuous and discrete lyapunov equations are of great importance but generally difficult to compute in control system analysis and design fortunately mesbahi and papavassilopoulos 1997 30 showed that with the semidefinite cone constraint the lowest rank solutions of the discrete lyapunov inequality can be efficiently solved by a linear semidefinite programming in this paper we further show that the lowest rank solutions of both the continuous and discrete lyapunov equations over symmetric cone are unique and can be exactly solved by their convex relaxations the symmetric cone linear programming problems therefore they are polynomial time solvable since the underlying symmetric cone is a more general algebraic setting which contains the semidefinite cone as a special case our results also answer an open question proposed by recht fazel and parrilo 2010 36

stepanov like pseudo almost automorphic mild solutions to nonautonomous evolution equations abstract we obtain new existence and uniqueness theorems of pseudo almost automorphic mild solutions to several nonautonomous evolution equations with stepanov like pseudo almost automorphic terms two explanation examples are given

generalized linear complementarity problems it has been shown by lemke that if a matrix is copositive plus on n then feasibility of the corresponding linear complementarity problem implies solvability in this article we show under suitable conditions that feasibility of ageneralized linear complementarity problem i e defined over a more general closed convex cone in a real hilbert space implies solvability whenever the operator is copositive plus on that cone we show that among all closed convex cones in a finite dimensional real hilbert space polyhedral cones are theonly ones with the property that every copositive plus feasible glcp is solvable we also prove a perturbation result for generalized linear complementarity problems

weighted stepanov like pseudo almost automorphy and applications abstract in this paper we propose a new class of functions called weighted stepanov like pseudo almost automorphic functions which generalize in a natural fashion the concept of almost automorphy and its various extensions we systematically explore the properties of the weighted stepanov like pseudo almost automorphic functions in general banach space including a composition result as an application we establish some sufficient criteria for the existence uniqueness of the weighted pseudo almost automorphic solution to a class of partial neutral functional differential equations and also to a class of nonlinear volterra integral equations of convolution type with infinite delay in banach space some interesting examples are presented to illustrate the main findings

complete second order linear differential equations with almost periodic solutions abstract this paper characterizes equations of the type u t bu t au t 0 in a banach space where a b are densely defined closed linear operators and whose solutions are all almost periodic

almost automorphic solutions for some partial functional differential equations abstract in this work we study the existence of almost automorphic solutions for some partial functional differential equations we prove that the existence of a bounded solution on r implies the existence of an almost automorphic solution our results extend the classical known theorem by bohr and neugebauer on the existence of almost periodic solutions for inhomegeneous linear almost periodic differential equations we give some applications to hyperbolic equations and lotka volterra type equations used to describe the evolution of a single diffusive animal species

positive almost periodic solutions of some neutral integral equation abstract in this paper we investigate the existence of positive almost periodic solutions for some neutral integral equation by constructing a new fixed point theorem in the cone some new results are obtained

converse problems of fourier expansion and their applications abstract let f c r h have a countable frequency set freq f and satisfy parseval s equality we show that if f satisfies one of the following conditions a uniformly continuous and freq f has a unique limit point at infinity b indefinite integral is lipschitz freq f converges fast in some sense c in the case of euclidean space h all the coefficients are positive then f is pseudo almost periodic example is given to show that the conclusion cannot be improved the results are applied to the theory of riesz fischer and the optimal control theory

complementarity problems over locally compact cones this paper proves an existence result for a complementarily problem on a locally convex space where the mapping is copositive positive homogeneous and of monotone type on a locally compact cone a perturbation theorem is proved that extends a result of mangasarian and doverspike proved for n times n matrices on the nonnegative orthant

pseudo almost periodic solutions for some parabolic evolution equations with stepanov like pseudo almost periodic forcing terms abstract the aim of this work is to study the existence and uniqueness of pseudo almost periodic solutions to a class of semilinear parabolic evolution equations with inhomogeneous boundary conditions under the assumption that the forcing terms of the equation are pseudo almost periodic in the sense of stepanov we also provide a new composition result of pseudo almost periodic functions in the sense of stepanov an example is given for illustration

pseudo almost automorphic mild solutions to nonautonomous differential equations and applications abstract in this paper we introduce a new concept of bi almost automorphic functions and obtain new existence and uniqueness theorems for pseudo almost automorphic mild solutions to several nonautonomous differential equations moreover two examples are given to illustrate the general theorems

a cone programming approach to the bilinear matrix inequality problem and its geometry we discuss an approach for solving the bilinear matrix inequality bmi based on its connections with certain problems defined over matrix cones these problems are among others the cone generalization of the linear programming lp and the linear complementarity problem lcp referred to as the cone lp and the cone lcp respectively specifically we show that solving a given bmi is equivalent to examining the solution set of a suitably constructed cone lp or cone lcp this approach facilitates our understanding of the geometry of the bmi and opens up new avenues for the development of the computational procedures for its solution

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

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

positive almost periodic solutions of nonlinear discrete systems with finite delay the existence of positive almost periodic solutions of nonlinear discrete systems with finite delay is obtained by using a variant of hilbert s projective metric on a subset of the space of almost periodic sequences

existence of positive pseudo almost periodic solutions to a class of neutral integral equations abstract in this work a new fixed point theorem in partially ordered banach spaces is established and then used to prove the existence of positive pseudo almost periodic solutions to a class of neutral integral equations our existence theorem extends some recent results due to e ait dads p cieutat l lhachimi positive pseudo almost periodic solutions for some nonlinear infinite delay integral equations mathematical and computer modelling 49 2009 721 739 to a more general class of integral equations

stepanov like pseudo almost periodicity and its applications to some nonautonomous differential equations abstract the paper revisits the concept of s p pseudo almost periodicity recently introduced by the author in particular we study the existence of pseudo almost periodic solutions to some nonautonomous differential equations in the case when the semilinear forcing term is both continuous and s p pseudo almost periodic for p 1 applications include the existence of pseudo almost periodic solutions to the heat equation with a forcing term which is s p pseudo almost periodic and jointly continuous

four positive periodic solutions to two species parasitical system with harvesting terms by using mawhin s continuation theorem of coincidence degree theory we establish the existence of four positive periodic solutions for two species parasitical system with harvesting terms an example is given to illustrate the effectiveness of our results

existence of asymptotically almost automorphic solutions to some abstract partial neutral integro differential equations abstract the paper establishes the existence and uniqueness of asymptotically almost automorphic mild solution to an abstract partial neutral integro differential equation with unbounded delay an example is given to illustrate our results

almost automorphic solutions to nonautonomous semilinear evolution equations in banach spaces abstract this paper is concerned with almost automorphy of the solutions to a nonautonomous semilinear evolution equation u t a t u t f t u t in a banach space with a stepanov like almost automorphic nonlinear term we establish a composition theorem for stepanov like almost automorphic functions furthermore we obtain some existence and uniqueness theorems for almost automorphic solutions to the nonautonomous evolution equation by means of the evolution family and the exponential dichotomy some results in this paper are new even if a t is time independent

on the two conjectures of graffiti we study the laplacian eigenvalues of trees on n vertices with independence number and describe all extremal graphs that attain the maximal laplacian spectral radius and algebraic connectivity then the results are used to confirm two conjectures of graffiti wow conjectures 584 and 636 on the relationship between the laplacian eigenvalues and the independence number of a graph

positive solutions of nonlinear integral equations arising in infectious diseases on considere l equation integrale non lineaire x t t t f s x s ds on utilise la theorie d indice du point fixe et la technique monotone pour etablir trois theoremes sur les solutions le premier affirme l existence de deux solutions non negatives non triviales de periode les deux autres concernent l existence et l unicite des solutions positives non triviales sous certaines conditions

on some interconnections between strict monotonicity globally uniquely solvable and p properties in semidefinite linear complementarity problems in the setting of semidefinite linear complementarity problems on sn the implications strict monotonicity p2 gus p are known here p and p2 properties for a linear transformation l sn sn are respectively defined by x sn xl x l x x 0 x 0 and x 0 y 0 x y l x l y x y 0 x y gus refers to the global unique solvability in semidefinite linear complementarity problems correspond ing to l in this article we show that the reverse implications hold for any self adjoint linear transformation and for normal lyapunov and stein transformations by introducing the concept of a principal subtransformation of a linear transformation we show that l sn sn has the p2 property if and only if for every n n real invertible matrix q every principal subtransformation of l has the p property where l x qtl qxqt q based on this we show that p2 gus and p properties coincide for the two sided multiplication transfor mation

compact almost automorphic solutions to semilinear cauchy problems with non dense domain in this work we study the existence and uniqueness of compact almost automorphic solutions to a first order differential equation with a linear part dominated by a hille yosida type operator with non dense domain

almost automorphic solutions for abstract functional differential equations for abstract linear functional differential equations with an almost automorphic forcing term we establish a result on the existence of almost automorphic solutions which extends the classical theorem due to massera on the existence of periodic solutions for linear periodic ordinary differential equations

compact almost automorphic solutions to integral equations with infinite delay abstract given a l 1 r and the generator a of an l 1 integrable resolvent family of linear bounded operators defined on a banach space x we prove the existence of compact almost automorphic solutions of the semilinear integral equation u t t a t s a u s f s u s d s for each f r x x compact almost automorphic in t for each x x and satisfying lipschitz and holder type conditions in the scalar linear case we prove that a l 1 r positive nonincreasing and log convex is sufficient to obtain the existence of compact almost automorphic solutions

positive almost periodic solutions of a nonlinear integral equation from the theory of epidemics abstract a generalization of a nonlinear integral equation governing the spread of certain infectious diseases with almost periodic contact rate is shown to have almost periodic positive solutions

almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations abstract in this paper applying the theory of semigroups of operators to evolution family and banach fixed point theorem we prove the existence and uniqueness of an a almost automorphic weighted pseudo almost automorphic mild solution of the semilinear evolution equation x t a t x t f t x t in banach space under conditions

permanental roots and the star degree of a graph abstract for undirected graphs without loops or multiple edges we define the star degree of a graph and prove that it is equal to the multiplicity of the root 1 of per xi b where b d a considering bipartite graphs we prove that per xi b per xi l where l d a and consequently that the star degree of a bipartite graph can also be characterized by the multiplicity of the root 1 of per xi l

cones of matrices and successive convex relaxations of nonconvex sets let f be a compact subset of the n dimensional euclidean space rn represented by finitely or infinitely many quadratic inequalities we propose two methods one based on successive semidefinite programming sdp relaxations and the other on successive linear programming lp relaxations each of our methods generates a sequence of compact convex subsets ck k 1 2 of rn such that a the convex hull of f subseteq c k 1 subseteq c k monotonicity b cap k 1 infty c k text the convex hull of f asymptotic convergence our methods are extensions of the corresponding lovasz schrijver lift and project procedures with the use of sdp or lp relaxation applied to general quadratic optimization problems qops with infinitely many quadratic inequality constraints utilizing descriptions of sets based on cones of matrices and their duals we establish the exact equivalence of the sdp relaxation and the semi infinite convex qop relaxation proposed originally by fujie and kojima using this equivalence we investigate some fundamental features of the two methods including a and b above

pseudo almost periodicity of some nonautonomous evolution equations with delay abstract this paper is concerned with pseudo almost periodicity of the solutions to the nonautonomous evolution equation with delay u t a t u t f t u t h some sufficient conditions which ensure the existence and uniqueness of pseudo almost periodic mild solutions to the evolution equation with delay are given an example is shown to illustrate our results

pseudo almost automorphic solutions to some neutral partial functional differential equations in banach spaces abstract this work aims to investigate the existence and uniqueness of pseudo almost automorphic solutions for some neutral partial functional differential equations in banach spaces recall that the new concept of pseudo almost automorphy generalizes the one of the pseudo almost periodicity and it has been recently introduced in the literature here we assume that the undelayed part is not necessarily densely defined and satisfies the well known hille yosida condition the delayed parts are assumed to be pseudo almost automorphic with respect to the first argument and lipschitz continuous with respect to the second argument

existence results for fractional neutral integro differential equations with state dependent delay in this paper we study the existence of mild solutions for a class of abstract fractional neutral integro differential equations with state dependent delay the results are obtained by using the leray schauder alternative fixed point theorem an example is provided to illustrate the main results

on the rank minimization problem over a positive semidefinite linear matrix inequality we consider the problem of minimizing the rank of a positive semidefinite matrix subject to the constraint that an affine transformation of it is also positive semidefinite our method for solving this problem employs ideas from the ordered linear complementarity theory and the notion of the least element in a vector lattice this problem is of importance in many contexts for example in feedback synthesis problems and such an example is also provided

a lower bound for the laplacian eigenvalues of a graph proof of a conjecture by guo we show that if j is the jth largest laplacian eigenvalue and dj is the jth largest degree 1 j n of a connected graph on n vertices then j dj j 2 1 j n 1 this settles a conjecture due to guo