sun wong type theorems for second order damped elliptic equations abstract the oscillation criteria of sun and wong y g sun j s w wong oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities j math anal appl 334 2007 549 560 are extended to the forced second order damped elliptic differential equation with mixed nonlinearities i j 1 n d i a i j x d j y i 1 n b i x d i y c x y e x when n 1 our results include and improve those of nasr a h nasr sufficient conditions for the forced super linear second order differential equations with oscillatory potential proc amer math soc 126 1998 123 125 for a forced second order superlinear differential equation wong j s w wong oscillation criteria for a forced second order linear differential equations j math anal appl 231 1999 235 240 for a forced second order linear differential equation sun and wong y g sun j s w wong oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities j math anal appl 334 2007 549 560 for a forced second order differential equation with mixed nonlinearities when n 2 our results are more general and improve the oscillation criteria of zhuang r k zhuang annual oscillation criteria for second order nonlinear elliptic differential equations j comput anal math 217 2008 268 276 for a forced second order undamped elliptic differential equation

existence of positive entire solutions for a class of quasilinear elliptic equations d lt d r d d liix ax 1 6 i j n and i is a real constant by an entire solution of 1 1 we mean a function ii of class c rn which satisfies 1 1 in rn in recent years the existence of positive entire solutions of 1 1 has been studied under various structure hypotheses for l and f by many authors see e g 1 8 l 15 17 22 241 however as far as we are aware little is known about the existence of nonconstant entire solutions of eq 1 1 subject to the condition

the expansion of a semigroup and a riesz basis criterion problems on the expansion of a semigroup and a criterion for being a riesz basis are discussed in the present paper suppose that a is the generator of a c0 semigroup on a hilbert space and a 1 a 2 a with 2 a is consisted of isolated eigenvalues distributed in a vertical strip it is proved that if 2 a is separated and for each 2 a the dimension of its root subspace is uniformly bounded then the generalized eigenvectors associated with 2 a form an l basis under different conditions on the riesz projection the expansion of a semigroup is studied in particular a simple criterion for the generalized eigenvectors forming a riesz basis is given as an application a heat exchanger problem with boundary feedback is investigated it is proved that the heat exchanger system is a riesz system in a suitable state hilbert space

oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities abstract we present new oscillation criteria for the second order forced ordinary differential equation with mixed nonlinearities p t x q t x i 1 n q i t x i sgn x e t where p t q t q i t e t c 0 p t is positive and differentiable 1 m 1 m 1 n no restriction is imposed on the forcing term e t to be the second derivative of an oscillatory function when n 1 our results reduce to those of el sayed m a el sayed an oscillation criterion for a forced second order linear differential equation proc amer math soc 118 1993 813 817 wong j s w wong oscillation criteria for a forced second linear differential equations j math anal appl 231 1999 235 240 sun ou and wong y g sun c h ou j s w wong interval oscillation theorems for a linear second order differential equation comput math appl 48 2004 1693 1699 for the linear equation nazr a h nazr sufficient conditions for the oscillation of forced super linear second order differential equations with oscillatory potential proc amer math soc 126 1998 123 125 for the superlinear equation and sun and wong y g sun j s w wong note on forced oscillation of nth order sublinear differential equations j math anal appl 298 2004 114 119 for the sublinear equation

a nonsmooth robinson s inverse function theorem in banach spaces in a recent paper izmailov math program ser a 147 581 590 2014 derived an extension of robinson s implicit function theorem for nonsmooth generalized equations in finite dimensions which reduces to clarke s inverse function theorem when the generalized equation is just an equation pales j math anal appl 209 202 220 1997 gave earlier a generalization of clarke s inverse function theorem to banach spaces by employing ioffe s strict pre derivative in this paper we generalize both theorems of izmailov and pales to nonsmooth generalized equations in banach spaces

annual oscillation criteria for second order nonlinear elliptic differential equations in this paper we present some new oscillation criteria for the second order nonlinear elliptic differential email protected a x y q x f y e x email protected email protected where w is an exterior domain in r n these criteria are different from most known ones in the sense that they are based on the information only on a sequence of annuals of w in r n rather than on the whole exterior domain w also information about the distribution of the zero of solutions is obtained

interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential new oscillation criteria established in the paper for a forced second order nonlinear equation with oscillatory potential p t y t q t f y t g t are different from the most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results are of a high degree of generality and sharper than some previous results and handle the cases which are not covered by known results moreover these results can also be applied to the extreme such as t 0 q s ds

new kamenev type oscillation criteria for second order nonlinear differential equations with damping abstract some new oscillation criteria are established for the nonlinear damped differential equation r t y p t y q t f y 0 that are different from most known ones in the sense that they are based on a class of new functions t s r defined in the sequel our results are sharper than some previous results which can be seen by the examples at the end of this paper

inexact newton methods and dennis more theorems for nonsmooth generalized equations in this paper we study local convergence of inexact newton methods of the form big f x k a k x k 1 x k f x k 1 big cap r k x k neq emptyset with a k in cal h x k for solving the generalized equation f x f x ni 0 in banach spaces where the function f is continuous but not necessarily smooth and f is a set valued mapping with closed graph the mapping cal h plays the role of a generalized set valued derivative of f which in finite dimensions may be represented by clarke s generalized jacobian while in banach spaces it may be identified with ioffe s strict prederivative the set valued mappings r k represent inexactness we utilize conditions divided into three groups the first concerns the kind of nonsmoothness of the function f the second involves metric regularity properties of an approximation of the mapping f f and the third is about the sequence of mappings r k under various combinations of these conditions we show linear superlinear or

interval criteria for oscillation of second order linear ordinary differential equations abstract new oscillation criteria are established for the equation py qy 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results are more natural according to the sturm separation theorem and sharper than some previous results and can be applied to extreme cases such as t 0 q t dt

oscillation theorems for elliptic equations with damping the second order elliptic equation with damping i j 1 dd i a i j x d jy i 1 db i x d iy p x f y 0is considered in an exterior domain w r d where the coefficients b i x i 1 d and p x are not be nonnegative in w by the generalized partial riccati transformation and the integral averaging technique some new oscillation theorems are established

strongly regular nonsmooth generalized equations this note presents an implicit function theorem for generalized equations simultaneously generalizing robinson s implicit function theorem for strongly regular generalized equations and clarke s implicit function theorem for equations with lipschitz continuous mappings

some new oscillation theorems for second order nonlinear elliptic equations with damping abstract in this paper some new oscillation criteria are obtained for second order elliptic differential equations with damping of the form a x y b t x y q x f y 0 x where is an exterior domain in r n these criteria are different from most known ones in the sense that they are based on a class of new functions of the form r s l defined in the sequel

perturbations of attractors of differential equations abstract in this paper we study small c 1 perturbations of a differential equation that has a hyperbolic attractor k we show that if k has a suitable lipschitz property and if the perturbation is small enough then there is a homeomorphism h k k y where k y is a hyperbolic attractor for the perturbed equation examples are included

interval criteria for oscillation of second order differential equations with mixed nonlinearities abstract in this paper we establish interval criteria for oscillation of the second order forced ordinary differential equations with mixed nonlinearities p t x q t x i 1 n q i t x i sgn x e t where p t q t qi t e t are continuous functions defined on 0 p t is positive and differentiable 1 m 1 m 1 n 0 n m 1 no restriction is imposed on the potentials q t qi t and e t to be nonnegative

oscillation of second order elliptic equations with damping terms abstract some oscillation criteria are given for the second order elliptic differential equation with damping terms i j 1 n d i a i j x d j y i 1 n b i x d i y c x y 0 x r 0 the results are extensions of modified riccati technique and include earlier results of noussair and swanson xu and zhang et al

oscillation and asymptotics for second order half linear differential equations this paper discusses a class of second order half linear differential equations by using the generalized riccati technique and the averaging technique new oscillation criteria are obtained for all solutions of the equations to be oscillatory asymptotic behavior for forced equations is also discussed

on semiregularity of mappings abstract there are two basic ways of weakening the definition of the well known metric regularity property by fixing one of the points involved in the definition the first resulting property is called metric subregularity and has attracted a lot of attention during the last decades on the other hand the latter property which we call semiregularity can be found under several names and the corresponding results are scattered in the literature we provide a self contained material gathering and extending the existing theory on the topic we demonstrate a clear relationship with other regularity properties for example the equivalence with the so called openness with a linear rate at the reference point is shown in particular cases we derive necessary and or sufficient conditions of both primal and dual type we illustrate the importance of semiregularity in the convergence analysis of an inexact newton type scheme for generalized equations with not necessarily differentiable single valued part

on uniqueness of kuhn tucker multipliers in nonlinear programming recently fujiwara han and mangasarian introduced a new constraint qualification which is a slight tightening of the well known mangasarian fromovitz constraint qualification we show that this new qualification is a necessary and sufficient condition for the uniqueness of kuhn tucker multipliers we also show that it implies the satisfaction of second order necessary optimality conditions at a local minimum

metric regularity and stability of optimal control problems for linear systems this paper studies stability properties of the solutions of optimal control problems for linear systems the analysis is based on an adapted concept of metric regularity the strong bi metric regularity which is introduced and investigated in the paper it allows one to give a more precise description of the effect of perturbations on the optimal solutions in terms of a holder type estimate and to investigate the robustness of this estimate the holder exponent depends on a natural number k which is known as the controllability index of the reference solution an inverse function theorem for strongly bi metrically regular mappings is obtained which is used in the case k 1 for proving stability of the solution of the considered optimal control problem under small nonlinear perturbations moreover a new stability result with respect to perturbations in the matrices of the system is proved in the general case k geq 1

lyusternik graves theorems for the sum of a lipschitz function and a set valued mapping in a paper of 1950 graves proved that for a function f acting between banach spaces and an interior point bar x in its domain if there exists a continuous linear mapping a which is surjective and the lipschitz modulus of the difference f a at bar x is sufficiently small then f is linearly open at bar x this is an extension of the banach open mapping principle from continuous linear mappings to lipschitz functions a closely related result was obtained earlier by lyusternik for smooth functions in this paper we obtain lyusternik graves theorems for mappings of the form f f where f is a lipschitz continuous function around bar x and f is a set valued mapping roughly we give conditions under which the mapping f f is linearly open at bar x for bar y provided that for each element a of a certain set of continuous linear operators the mapping f bar x a cdot bar x f is linearly open at bar x for bar y in the case when f is the zero mapping as

ordinary differential equations in the oscillation theory of partial half linear differential equation abstract in the paper we study the damped half linear partial differential equation div a x u p 2 u b x u p 2 u c x u p 2 u 0 using radialization method we derive general oscillation results which allow to deduce new oscillation criteria for this equation from oscillation criteria for ordinary differential equations using careful radialization we improve several known oscillation criteria

oscillation of higher order forced nonlinear differential equations abstract some new criteria are established for the oscillation of higher order forced nonlinear differential equations by introducing a class of new auxiliary functions no restriction is imposed on the forcing term as is generally assumed with the help of the new auxiliary functions the main results in this paper are different from those in the paper y g sun s h saker forced oscillation of higher order nonlinear differential equations appl math comput 173 2006 1219 1226 and are more effective than many existing results

strong lipschitz stability of stationary solutions for nonlinear programs and variational inequalities the stationary solution map x of a canonically perturbed nonlinear program or variational condition is studied we primarily aim at characterizing x to be locally single valued and lipschitz near some stationary point x0 of an initial problem where our focus is on characterizations which are explicitly given in terms of the original functions and assigned quadratic problems since such criteria are closely related to a nonsingularity property of the strict graphical derivative of x explicit formulas for this derivative are presented too it turns out that even for convex polynomial problems our stability and the aubin property does not depend only on the derivatives up to some fixed order of the problem functions at x0 this is in contrast to various other stability concepts further we clarify completely the relations to kojima s strong stability and present simplifications for linearly and certain linearly quadratically constrained problems convex programs and for the map of global minimizers as well

strong metric subregularity of mappings in variational analysis and optimization abstract although the property of strong metric subregularity of set valued mappings has been present in the literature under various names and with various equivalent definitions for more than two decades it has attracted much less attention than its older siblings the metric regularity and the strong metric regularity the purpose of this paper is to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications we show that the strong metric subregularity of a mapping f acting between metric spaces is stable under perturbations of the form f f where f is a function with a small calmness constant this result is parallel to the lyusternik graves theorem for metric regularity and to the robinson theorem for strong regularity where the perturbations are represented by a function f with a small lipschitz constant then we study perturbation stability of the same kind for mappings acting between banach spaces where f is not necessarily differentiable but admits a set valued derivative like approximation strong metric q subregularity is also considered where q is a positive real constant appearing as exponent in the definition rockafellar s criterion for strong metric subregularity involving injectivity of the graphical derivative is extended to mappings acting in infinite dimensional spaces a sufficient condition for strong metric subregularity is established in terms of surjectivity of the frechet coderivative and it is shown by a counterexample that surjectivity of the limiting coderivative is not a sufficient condition for this property in general then various versions of newton s method for solving generalized equations are considered including inexact and semismooth methods for which superlinear convergence is shown under strong metric subregularity as applications to optimization a characterization of the strong metric subregularity of the kkt mapping is obtained as well as a radius theorem for the optimality mapping of a nonlinear programming problem finally an error estimate is derived for a discrete approximation in optimal control under strong metric subregularity of the mapping involved in the pontryagin principle

oscillation criteria for second order forced elliptic differential equations with mixed nonlinearities in the case of oscillatory potentials we establish some oscillation criteria for the second order forced elliptic differential equation with mixed nonlinearities div a x y c x y e x when n 1 our results extend those of kong q kong interval criteria for oscillation of second order linear ordinary differential equations j math anal appl 229 1999 258 270 for certain second order linear ordinary differential equations sun and meng y g sun f w meng interval criteria for oscillation of second order differential equations with mixed nonlinearities appl math comput 2007 doi 10 10 1016 j amc 2007 08 042 for certain second order differential equations with mixed nonlinearities yang q yang interval oscillation criteria for a second order nonlinear ordinary differential equations with oscillatory potential appl math comput 135 2003 49 64 for a forced second order superlinear ordinary differential equation when n 2 our theorems are more general and improve the oscillation criteria of zhuang r k zhuang annual oscillation criteria for second order nonlinear elliptic differential equations j comput anal math 2007 doi 10 1016 j cam 2007 06 031 for a forced second order undamped elliptic differential equation

on the stability of weakly hyperbolic invariant sets abstract the dynamical object which we study is a compact invariant set with a suitable hyperbolic structure stability of weakly hyperbolic sets was studied by v a pliss and g r sell see 1 2 they assumed that the neutral unstable and stable linear spaces of the corresponding linearized systems satisfy lipschitz condition they showed that if a perturbation is small then the perturbed system has a weakly hyperbolic set k y which is homeomorphic to the weakly hyperbolic set k of the initial system close to k and the dynamics on k y is close to the dynamics on k at the same time it is known that the lipschitz property is too strong in the sense that the set of systems without this property is generic hence there was a need to introduce new methods of studying stability of weakly hyperbolic sets without lipschitz condition these new methods appeared in 16 17 18 19 20 they were based on the local coordinates introduced in 18 and the continuous on the whole weakly hyperbolic set coordinates introduced in 19 in this paper we will show that even without lipschitz condition there exists a continuous mapping h such that h k k y

approximation dynamics and the stability of invariant sets abstract we introduce the concept of a weakly normally hyperbolic set for a system of ordinary differential equations this concept includes the notion of a hyperbolic flow as well as that of a normally hyperbolic invariant manifold moreover it has the property that it is closed under finite set products consequently the theory presented here can be used for the study of perturbations of the dynamics of coupled systems of weakly normally hyperbolic sets our main objective is to show that under a small c 1 perturbation a weakly normally hyperbolic set k is preserved by a homeomorphism where the image k y is a compact invariant set with a related hyperbolic structure for the perturbed equation in addition the homeomorphism is close to the identity in c 0 1 and the perturbed dynamics on k y are close to the original dynamics on k

generalizations of the dennis more theorem a theorem of dennis and more is generalized to characterize the q superlinear convergence of quasi newton methods applied to nonsmooth equations and generalized equations under strong metric subregularity

quadratic growth and critical point stability of semi algebraic functions we show that quadratic growth of a semi algebraic function is equivalent to strong metric subregularity of the subdifferential a kind of stability of generalized critical points in contrast this equivalence can easily fail outside of the semi algebraic setting as a consequence we derive necessary conditions and sufficient conditions for optimality in subdifferential terms