on existence of solutions to differential equations or inclusions remaining in a prescribed closed subset of a finite dimensional space abstract this paper is devoted to existence of trajectories to differential equations and inclusions remaining in a given closed or compact set topological properties of the boundary of this set enables us to conclude the existence of such trajectories in many cases

guiding functions for generalized periodic problems and applications abstract in this paper developing the method of guiding functions for differential inclusions with a generalized periodic condition we obtain the existence theorem for such problems it is shown how the abstract result can be applied to the study of differential games and anti periodic problems

approachability and fixed points for non convex set valued maps this paper studies the approachability of upper semicontinuous nonconvex set valued maps by single valued continuous functions and its applications to fixed point theory the use of approximate selections was initiated by j von neumann 41 in the proof of his well known minimax principle this remarkable property was extended to the normed space setting by cellina s and cellina and lasota 9 in the context of a degree theory for upper semicontinuous convex compact valued maps it also holds for maps with compact contractible values defined on finite polyhedra of iw ma cole11 40 and more generally defined on compact anrs mclennan 39 it was recently taken up by gbrniewicz granas and kryszewski 22 241 in the context of an index theory for non convex maps defined on compact anrs the content of this paper is divided into three parts the first part is devoted to background material concerning the types of spaces and set valued maps studied herein in the second part we define the abstract class d of approachable set valued maps that is the class of maps a x y

equilibria of set valued maps on nonconvex domains we present new theorems on the existence of equilibria or zeros of convex as well as nonconvex set valued maps defined on compact neighborhood retracts of normed spaces the maps are subject to tangency conditions expressed in terms of new concepts of normal and tangent cones to such sets among other things we show that if k is a compact neighborhood retract with nontrivial euler characteristic in a banach space e and 1 k 2e is an upper hemicontinuous set valued map with nonempty closed convex values satisfying the tangency condition x n tk x 0 for all x c k then there exists xo c k such that 0 c 4 xo here tk x denotes a new concept of retraction tangent cone to k at x suited for compact neighborhood retracts when k is locally convex at x tk x coincides with the usual tangent cone of convex analysis special attention is given to neighborhood retracts having lipschitzian behavior called l retracts below this class of sets is very broad it contains compact homeomorphically convex subsets of banach spaces epi lipschitz subsets of banach spaces as well as proximate retracts our results thus generalize classical theorems for convex domains as well as recent results for nonconvex sets

asymptotic boundary value problems in banach spaces abstract a continuation principle is given for solving boundary value problems on arbitrary possibly infinite intervals to caratheodory differential inclusions in banach spaces for this aim the appropriate fixed point index is defined to condensing decomposable multivalued operators in frechet spaces this index extends and unifies the one for compact maps in andres et al trans amer math soc 351 1999 4861 4903 as well as the one for operators in banach spaces in bader ph d thesis university of munich 1995 as an application we prove the existence of an entirely bounded solution of a semilinear evolution inclusion

equilibria of nonconvex valued maps under constraints abstract in the paper the notion of n tangency for set valued maps defined on a subset of a banach space is considered the existence of equilibria of upper semicontinuous map being n tangent to a sleek retract with the nontrivial euler characteristic is established

analysing the efficiency of some modifications of the secant method some modifications of the secant method for solving nonlinear equations are revisited and the local order of convergence is found in a direct symbolic computation to do this a development of the inverse of the first order divided differences of a function of several variables in two points is presented a generalisation of the efficiency index used in the scalar case to several variables is also analysed in order to use the most competitive algorithm

fixed point theorems and morse s lemma for lipschitzian functions abstract we prove a fixed point theorem for set valued mappings defined on a nonempty compact subset x of r n which can be represented by inequality constraints i e x x r n f x 0 f locally lipschitzian and satisfying a nondegeneracy assumption outside of x this class of sets extends significantly the class of convex compact sets with a nonempty interior topological properties of such sets x are proved continuous deformation retract of a ball acyclicity as a consequence of a generalization of morse s lemma for lipschitzian real valued function defined on r n a result also of interest for itself

multiple bounded solutions of differential inclusions the nielsen theory approach abstract the generalized nielsen number is defined for self maps which are composed by operators with r values on compact connected anrs then it is applied to caratheodory differential inclusions with constraints for obtaining the multiplicity criteria more precisely such problems are transformed to those for the lower estimate of fixed points of the related operators with the given properties on bounded compact connected neighbourhood retracts of frechet spaces in this way multiple solutions can be proved e g for the multivalued initial value problem on the half line or in the single valued case for boundary value problems to ordinary differential equations

stability of impulsive functional differential equations abstract in this paper the stability of impulsive functional differential equations in which the state variables on the impulses are related to the time delay is studied by using lyapunov functions and razumikhin techniques some criteria of stability asymptotic stability and practical stability for impulsive functional differential equations in which the state variables on the impulses are related to the time delay are provided some examples are also presented to illustrate the efficiency of the results obtained

on dynamical properties of general dynamical systems and differential inclusions abstract let f be a general dynamical system defined on a complete locally compact metric space x we give a slightly improved version of the lyapunov characterization of asymptotic stability in one of our previous works and provide a short self contained proof for the existence of arbitrarily small positively invariant neighborhoods of compact asymptotically stable sets in the present context based on these results we prove that the uniform attractors of f are connected and are stable with respect to perturbations under appropriate conditions we are also interested in the dynamical properties of differential inclusion x t f x t on r m first we show that if no solutions of the system blow up in finite future time then its reachable mapping f is a general dynamical system then we discuss some asymptotic stability properties of the system in particular we prove that if there exists a nonempty compact connected subset m of r m such that m attracts a neighborhood of itself then the system has a connected uniform attractor a we also prove that a is stable with respect to both internal and external perturbations more precisely we prove that when 0 is sufficiently small the perturbed system x t con f x t b 1 b 1 has a connected uniform attractor a moreover h a a 0 as 0 where h is the hausdorff distance in r m

continuity compactness and degree theory for operators in systems involving p laplacians and inclusions the study of weak solutions for systems of nonlinear partial differential equations of elliptic type with inclusions leads to a multivalued operator of superposition type in sobolev spaces we show that under natural assumptions this operator has the properties which allow to apply degree theory fixed point index for multivalued maps more precisely this operator is upper semicontinuous and compact with nonempty convex compact values for the particular case of systems involving p laplacians we show that there is a homeomorphism transforming the whole system to a situation for which a fixed point index is available

on controllability of duffing equation in this paper by using the topological degree theory for multivalued maps and the method of a priori bounds we study the controllability problem for a system governed by the duffing equation an example is given

representations of epi lipschitzian sets abstract a closed subset m of a banach space e is epi lipschitzian i e can be represented locally as the epigraph of a lipschitz function if and only if it is the level set of some locally lipschitz function f e r for which clarke s generalized gradient does not contain 0 at points in the boundary of m i e such that m x f x 0 0 f x if x bd m this extends the characterization previously known in finite dimension and answers to a standing question

classification of joinings for kleinian groups we classify all locally finite joinings of a horospherical subgroup action on g when g psl2 r or psl2 c and is a geometrically finite zariski dense subgroup of g this generalizes ratner s 1983 joining theorem for the case when is a lattice in g

on the local convergence of a family of two step iterative methods for solving nonlinear equations a local convergence analysis for a generalized family of two step secant like methods with frozen operator for solving nonlinear equations is presented unifying earlier methods such as secant s newton chebyshev like steffensen and other new variants the family of iterative schemes is built up where a profound and clear study of the computational efficiency is also carried out numerical examples and an application using multiple precision and a stopping criterion are implemented without using any known root finally a study comparing the order efficiency and elapsed time of the methods suggested supports the theoretical results claimed

the algorithmic analysis of hybrid systems we present a general framework for the formal specification and algorithmic analysis of hybrid systems a hybrid system consists of a discrete program with an analog environment we model hybrid systems as finite automata equipped with variables that evolve continuously with time according to dynamical laws for verification purposes we restrict ourselves to linear hybrid systems where all variables follow piecewise linear trajectories we provide decidability and undecidability results for classes of linear hybrid systems and we show that standard program analysis techniques can be adapted to linear hybrid systems in particular we consider symbolic model checking and minimization procedures that are based on the reachability analysis of an infinite state space the procedures iteratively compute state sets that are definable as unions of convex polyhedra in multidimensional real space we also present approximation techniques for dealing with systems for which the iterative procedures do not converge

viability theory and fuzzy differential equations new results are presented which guarantee the existence of viable solutions to differential inclusions we show that a viable solution exists if i a tangency condition is satisfied and ii there exists a maximal solution to an appropriate differential equation in addition we discuss the topological structure of the solution set these results will then be used with the stacking theorem to establish new existence criteria for fuzzy differential equations if there exists a maximal solution for an appropriate set of differential equations

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

global bifurcation of the p laplacian and related operators abstract we prove some global bifurcation theorems of rabinowitz type for a large class of operators the results apply in particular to equations involving the p laplacian

periodic solutions to impulsive differential inclusions with constraints abstract the existence of a periodic solution to an impulsive differential inclusion being invariant with respect to a non convex set of state constraints is established by the use a lefschetz type fixed point theorem for set valued maps

letter to the editor frozen divided difference scheme for solving systems of nonlinear equations the development of an inverse first order divided difference operator for functions of several variables as well as a direct computation of the local order of convergence of an iterative method is presented a generalized algorithm of the secant method for solving a system of nonlinear equations is studied and the maximum computational efficiency is computed furthermore a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced

existence of generalized equilibria we consider the problem of the existence of equilibria generalized equilibria and 8xed points of a correspondence f de8ned on a compact subset m of rn with values in rn when the set m is neither assumed to be convex nor smooth 1 the general framework of the article is the following let f be a correspondence from m to rn that is a map from m to the set of all the subsets of rn the correspondence f is said to be upper semicontinuous u s c if the set x m f x v is open in m for every open set v rn let n be a normal cone correspondence i e at this stage n is a correspondence from m to rn such that for every x m n x

fixed points and equilibria in nonconvex sets tcentre de recherches mathtmatiques universite de montreal montreal quebec h3c 357 canada psteklov institute of mathematics moscow 117966 russia and ildepartment of mathematics and statistics concordia university montreal quebec h4b lr6 canada received 30 september 1993 received forpublication 18 august 1994 key words and phrases multivalued mapping tangent cones zeros fixed points differential inclusion weak invariance nonsmooth analysis 1 introduction

variants of a classic traub s result we present a generalization of a modification of a classic traub s result which allows constructing from a given iterative method with order of convergence p efficient iterative methods with order of convergence at least pq 1 q p 1 q n

monotone iterative techniques of v lakshmikantham for a boundary value problem for systems of impulsive differential difference equations by means of the monotone iterative techniques of v lakshmikantham a couple of minimal and maximal quasisolutions of a two point boundary value problem for a system of impulsive differential difference equations are constructed

sublevel representations of epi lipschitz sets and other properties epi lipschitz sets in normed spaces are represented as sublevel sets of lipschitz functions satisfying a so called qualification condition canonical representations through the signed distance functions associated with the sets are also obtained new optimality conditions are provided for optimization problems with epi lipschitz set constraints in terms of the signed distance function

the constrained degree and fixed point index theory for set valued maps abstract in the first part of the paper we give a construction of a topological degree theory for set valued tangent vector fields with convex and nonconvex values defined on nonsmooth closed subsets of a banach space the obtained homotopy invariant is an extension of the classical degree for vector fields on manifolds in the second part we propose a fixed point index for inward maps on arbitrary closed convex sets

existence of equilibria of set valued maps on bounded epi lipschitz domains in hilbert spaces without invariance conditions abstract in this paper we provide a new result of the existence of equilibria for set valued maps on bounded closed subsets k of hilbert spaces we do not impose either convexity or compactness assumptions on k but we assume that k has epi lipschitz sections i e its intersection with suitable finite dimensional spaces is locally the epigraph of lipschitz functions in finite dimensional spaces the famous brouwer theorem asserts the existence of a fixed point for a continuous function from a compact convex set k to itself our result could be viewed as a kind of generalization of this classical result in the context of hilbert spaces and when the function or the set valued map does not necessarily map k into itself k is not invariant under the map our approach is based firstly on degree theory for compact and for condensing set valued maps and secondly on flows generated by trajectories of differential inclusions

topological structure of solution sets for impulsive differential inclusions in frechet spaces abstract in this paper we consider the existence of solutions as well as the topological and geometric structure of solution sets for first order impulsive differential inclusions in some frechet spaces both the initial and terminal problems are considered using ingredients from topology and homology the topological structures of solution sets closedness and compactness as well as some geometric properties contractibility acyclicity a r and r are investigated some of our existence results are obtained via the method of taking the inverse system limit on noncompact intervals

existence of periodic solutions of nonlinear systems of differential equations with impulse effect abstract sufficient conditions for the existence of periodic solutions of nonlinear systems of differential equations with impulses in a canonic domain have been found in the paper

measurable rigidity of actions on infinite measure homogeneous spaces ii we consider the problems of measurable isomorphisms and join ings measurable centralizers and quotients for certain classes of ergodic group actions on infinite measure spaces our main focus is on systems of algebraic origin actions of lattices and other discrete subgroups g on homoge neous spaces g h where h is a sufficiently rich unimodular subgroup in a semi simple group g we also consider actions of discrete groups of isometries isom x of a pinched negative curvature space x acting on the space of horospheres hor x for such systems we prove that the only measurable isomorphisms joinings quotients etc are the obvious algebraic or geometric ones this work was inspired by the previous work of shalom and steger but uses completely different techniques which lead to more general results 1 introduction and statement of the main results

existence and stability of periodic solution of a lotka volterra predator prey model with state dependent impulsive effects according to biological and chemical control strategy for pest we investigate the dynamic behavior of a lotka volterra predator prey state dependent impulsive system by releasing natural enemies and spraying pesticide at different thresholds by using poincare map and the properties of the lambert w function we prove that the sufficient conditions for the existence and stability of semi trivial solution and positive periodic solution numerical simulations are carried out to illustrate the feasibility of our main results

periodicity on a class of neutral impulsive delay system we present some sufficient conditions for the existence of positive periodic solutions of a class of neutral impulsive delay differential system when our results applied to the differential system without impulse a known result in the literature is corrected moreover one example is given to illustrate the results obtained

eigenvalues of the laplacian the first betti number and the congruence subgroup problem theorem 1 1 let f be an arithmetic lattice in the real lie group so n 1 n 2 if n 7 then assume r does not come from the twisted forms 3 6d4 if n 3 and f comes from the units of a quaternion algebra over a number field l with a unique complex embedding then assume l has a subfield of index 2 then for some finite index congruence subgroup fo of r the commutator quotient is infinite equivalently 1 fo the first betti number of the manifold fo hn is nonzero when 1 so n 1 so n is the n dimensional hyperbolic space

the ergodic theory of discrete isometry groups on manifolds of variable negative curvature this paper studies the ergodic theory at infinity of an arbitrary discrete isometry group acting on any hadamard manifold h of pinched variable negative curvature most of the results obtained by sullivan in the constant curvature case are generalized to the case of variable curvature we describe connections between measures supported on the limit set of dynamics of the geodesic flow and the geometry of m h we explore the relationship between the growth exponent of the group the hausdorff dimension of the limit set and the topological entropy of the geodesic flow the equivalence of various descriptions of an analogue of the hopf dichotomy is proved as applications we settle a question of j feldman and m ratner about the horocycle flow on a finite volume surface of negative curvature and obtain an asymptotic formula for the counting function of lattice points at the end of this paper we apply our results to the study of some rigidity problems more applications to mostow rigidity for discrete subgroups of rank 1 noncompact semisimple lie groups with infinite covolume will be published in subsequent papers by the author notations h is a hadamard manifold with pinched sectional curvature k 2 k k 1 k2 k1 0 is a non elementary discrete isometry group acting on h freely and properly discontinuously which is called fuchsian h is the ideal boundary of h m h is a complete riemannian manifold sh or sm is the unit tangent bundle g is the geodesic flow on sh or sm v t is the unique geodesic in sh resp sm with the initial velocity v 0 v sh resp sm v resp v is the asymptotic class of the geodesic v t resp v t l is the limit set of on h l is the radical limit set of in l is the nonwandering set of the geodesic flow on sm received by the editors may 7 1995 1991 mathematics subject classification primary 58f17 secondary 58f11 58f15 20h10 research at msri supported by nsf grant dms 8505550 also partially supported by nsf grant dms 9403870 and sfb 170 c 1996 american mathematical society

application of a coincidence index to some classes of impulsive control systems we suggest the oriented coincidence index theory for pairs consisting of nonlinear zero index fredholm operators and multivalued maps which may be represented as compositions of multimaps with aspheric values we consider sequentially finite dimensional compact and condensing cases the theory developed is applied to the study of a feedback impulsive control system

qualitative analysis and applications of a kind of state dependent impulsive differential equations this paper analyzes a certain type of impulsive differential equations ides several useful theorems for its periodic solutions and their stabilities are given the key idea is that a periodically time dependent ide can be transformed into the state dependent ide as applications of our theory the optimization problems in population dynamics are studied that is the maximum sustainable yields of single population models with periodically impulsive constant harvesting are discussed furthermore we apply these results to the studies of the order 1 periodic solutions and their stability of a single population model with stage structure in which the mature is impulsively proportionally harvested while the immature is impulsively added with the constant

semilocal convergence of secant like methods for differentiable and nondifferentiable operator equations abstract from well known secant like methods we observe that we can construct a new family of secant like methods that includes the secant method and kurchatov s method we analyse the local orders of convergence and the efficiencies of the methods of the family and study the semilocal convergence for differentiable and nondifferentiable operators finally we apply our results to conservative problems

the bolzano mean value theorem and partial differential equations abstract we study the existence of solutions to abstract equations of the form 0 a u f u u k e where a is an abstract differential operator acting in a banach space e k is a closed convex set of constraints being invariant with respect to resolvents of a and a perturbation f satisfies a certain tangency condition such problems are closely related to the so called poincare miranda theorem being the multi dimensional counterpart of the celebrated bolzano intermediate value theorem in fact our main results should be regarded as infinite dimensional variants of bolzano and miranda poincare theorems along with single valued problems we deal with set valued ones yielding the existence of the so called constrained equilibria of set valued maps the abstract results are applied to show existence of strong steady state solutions to some weakly coupled systems of drift reaction diffusion equations or differential inclusions of this type in particular we get the existence of strong solutions to the dirichlet neumann and periodic boundary problems for elliptic partial differential inclusions under the presence of state constraints of different type certain aspects of the bernstein theory for bvp for second order ode are studied too no assumptions concerning structural coupling monotonicity cooperativity are undertaken

on the solution sets of differential inclusions and the periodic problem in banach spaces abstract in this paper the topological structure of the solution set of a constrained semilinear differential inclusion in a banach space e is studied it is shown that the set of all mild solutions with values in a closed and in general thin subset d e is an r set provided natural boundary conditions and appropriate geometrical assumptions on d which hold e g when d is convex are satisfied applications to the periodic problem and to the existence of equilibria are given

spaces and maps approximation and fixed points abstract a general approximation property for topological spaces is studied in relation with fixed point theory for set valued maps a particular instance of this property is the admissibility in the sense of klee examples of convex sets of topological spaces equipped with a local topological convexity structure as well as general classes of approximative neighborhood retracts are shown to have this approximation property a general topological principle on the preservation of the fixed point property under this space approximation is proved it allows the passage from basic classes of spaces to more elaborate ones for general classes of nonconvex set valued maps

impulse differential inclusions a viability approach to hybrid systems impulse differential inclusions are introduced as a framework for modeling hybrid phenomena connections to standard problems in the area of hybrid systems are discussed conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion for sets that violate these conditions methods are developed for approximating their viability and invariance kernels that is the largest subset that is viable or invariant under the action of the impulse differential inclusion the results are demonstrated on examples

homotopy invariants for tangent vector fields on closed sets abstract in the paper we construct a topological degree theory for single and convex valued tangent vector fields defined on locally compact closed subsets of a banach space the obtained homotopy invariant is an extension of the classical degree for vector fields on manifolds the degree allows to study vector fields on sets which are neither smooth nor convex and may be applied to study continuation and bifurcation of equilibria in parameterized families of closed sets

rigidity of group actions on homogeneous spaces iii consider homogeneous g h and g f for an s algebraic group g a lattice acts on the left strictly conservatively the following rigidity results are obtained morphisms factors and joinings defined apriori only in the measurable category are in fact algebraically constrained arguing in an elementary fashion we manage to classify all the measurable commuting with the action assuming ergodicity we find they are algebraically defined

on a generalization of the miranda theorem and its application to boundary value problems abstract in this paper we have proved a multi valued version of the miranda theorem involving an admissible map in the sense of gorniewicz examples of its application to boundary value problems have been given

differential inclusions and target problems where and how solutions associated to a differential inclusion can or cannot enter a given target is studied for this purpose partitions of the target boundary are associated with the dynamic of the system the behaviour of these solutions is qualitatively described in terms of viability and invariance kernels of sets these kernels determine points such that there exist respectively all solutions starting at these points remain in a given set of constraints the sets that are reached in finite time by viable solutions to the system are also studied finally some applications to control systems with one target are provided and the concept of semipermeable barriers will be generalized

boundary value problems for nonlinear impulsive hamiltonian systems we study two point boundary value problems for nonlinear impulsive hamiltonian systems spectral analysis of the corresponding linear impulsive hamiltonian system and a fixed point theorem are employed to obtain an existence and uniqueness result for solutions of the nonlinear problem two examples are given in which the main condition is made explicit