remarks on first and second order periodic boundary value problems on etudie les problemes aux valeurs limites periodiques d ordre 1 et 2 suivants u f t u u 0 u 2 et u f t u u 0 u 2 u 0 u 2 on obtient l existence de solutions extremales comme limites d iterees monotones

a geometric treatment of implicit differential algebraic equations abstract a differential geometric approach for proving the existence and uniqueness of implicit differential algebraic equations is presented it provides for a significant improvement of an earlier theory developed by the authors as well as for a completely intrinsic definition of the index of such problems the differential algebraic equation is transformed into an explicit ordinary differential equation by a reduction process that can be abstractly defined for specific submanifolds of tangent bundles here called reducible submanifolds local existence and uniqueness results for differential algebraic equations then follow directly from the final stage of this reduction by means of an application of the standard theory of ordinary differential equations

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

compact families of piecewise constant functions in lp 0 t b abstract a strong compactness result in the spirit of the lions aubin simon lemma is proven for piecewise constant functions in time u with values in a banach space the main feature of our result is that it is sufficient to verify one uniform estimate for the time shifts u u instead of all time shifts u u h for h 0 as required in simon s compactness theorem this simplifies significantly the application of the rothe method in the existence analysis of parabolic problems

periodic solutions of a quartic differential equation and groebner bases we consider first order ordinary differential equations with quartic nonlinearities the aim is to find the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients it is shown that this number is ten when the coefficients are certain cubic polynomials equations with the maximum number of such periodic solutions are also constructed the paper is heavily dependent on computing groebner bases

uniformly attracting solutions of nonautonomous differential equations abstract understanding the structure of attractors is fundamental in nonautonomous stability and bifurcation theory by means of clarifying theorems and carefully designed examples we highlight the potential complexity of attractors for nonautonomous differential equations that are as close to autonomous equations as possible we introduce and study bounded uniform attractors and repellors for nonautonomous scalar differential equations in particular for asymptotically autonomous polynomial and periodic equations our results suggest that uniformly attracting or repelling solutions are the true analogues of attracting or repelling fixed points of autonomous systems we provide sharp conditions for the autonomous structure to break up and give way to a bewildering diversity of nonautonomous bifurcations

abel like differential equations with no periodic solutions we present various criteria for the non existence of positive periodic solutions of generalized abel differential equations with periodic coefficients that can change sign as an application we obtain some families of planar vector fields without limit cycles

nonvanishing of hecke l functions and the bloch kato conjecture in this paper we study the central values of l functions associated to a large class of algebraic hecke characters of imaginary quadratic fields when these central values are nonzero the bloch kato conjecture predicts an exact formula for the algebraic parts of the central values in terms of periods and arithmetic data most notably the selmer groups corresponding to the hecke characters we investigate the nonvanishing of these central values and prove the p part of the bloch kato conjecture in these cases for primes p which split in k

how strong a logistic damping can prevent blow up for the minimal keller segel chemotaxis system in this paper we study the minimal keller segel model with a logistic source and obtain quantitative and qualitative descriptions of the competition between logistic damping and other ingredient especially chemotactic aggregation to guarantee boundedness and convergence more specifically we establish how precisely strong a logistic source can prevent blow up and then we obtain an explicit relationship between logistic damping and other ingredient especially chemotactic aggregation so that convergences are ensured and their respective convergence rates are explicitly calculated out known results in the literature are completed and refined furthermore our findings provide clues on how to produce blowup solutions for ks chemotaxis models with logistic sources

aggregating pattern dynamics in a chemotaxis model including growth abstract a population model including diffusion chemotaxis and growth is studied assuming that the diffusion rate and the chemotactic rate are both very small compared with the growth rate we derive a new equation to describe the time evolution of the aggregating region of biological individuals and show the conditions for the existence and stability of radially symmetric equilibrium solutions of the equation which indicate the aggregation of individuals

boundedness and finite time collapse in a chemotaxis system with volume filling effect abstract we consider the elliptic parabolic pde system u t u u u v x t 0 0 v m u x t 0 with nonnegative initial data u 0 having mean value m under homogeneous neumann boundary conditions in a smooth bounded domain r n the nonlinearities and are supposed to generalize the prototypes u u 1 p u u u 1 q 1 with p 0 and q r problems of this type arise as simplified models in the theoretical description of chemotaxis phenomena under the influence of the volume filling effect as introduced by painter and hillen k j painter t hillen volume filling and quorum sensing in models for chemosensitive movement can appl math q 10 2002 501 543 it is proved that if p q 2 n then all solutions are global in time and bounded whereas if p q 2 n q 0 and is a ball then there exist solutions that become unbounded in finite time the former result is consistent with the aggregation inhibiting effect of the volume filling mechanism the latter however is shown to imply that if the space dimension is at least three then chemotactic collapse may occur despite the presence of some nonlinearities that supposedly model a volume filling effect in the sense of painter and hillen

exact multiplicity results for two classes of periodic equations for two classes of first order equations we study the exact multiplicity of periodic solutions if the nonlinearity is a quadric in the unknown function we give conditions for existence of exactly four solutions in another direction we use bifurcation theory to derive exact multiplicity results for quadratic and cubic equations

on discontinuous first order implicit boundary value problems t 0 1 2 under various boundary conditions cf e g 1 5 8 13 29 31 32 thereis little known about the existence of solutions of 1 2 even in the casewhen f is continuous unless it is possible to solve the equation 1 2 forthe highest derivative and then to apply standard existence theory such asleray schauder theory one approach taken for instance in 8 16 is toreduce equation 1 2 to a differential inclusion of the formu

a generalized upper and lower solutions method and multiplicity results for nonlinear first order ordinary differential equations abstract we introduce a generalized upper and lower solutions method for the solvability of first order ordinary differential equations u t t u t u 0 u 1 in order to cover the case when the function satisfies caratheodory conditions this method is then applied to get multiplicity results when the nonlinearity interacts with the real eigenvalue of the linearized problem our proofs are based on differential inequalities and classical leray schauder degree

on uniqueness of limit cycle for the equation ẍ f x φ ẋ ẋ g x 0 abstract in this paper we use the theory of generalized rotated fields to prove a theorem of the uniqueness of limit cycle for the generalized lienard system the damping terms of this system are dependent on x our result extends a famous zhang zhifen s theorem on the uniqueness problem of limit cycle which is contributed the lienard systems whose damping terms are independent of x

lienard equations with bounded trajectories in the lienard plane on considere l equation de lienard x f x x g x 0 on suppose f r r et g r r continues on presente un ensemble de conditions pour l existence d un attracteur global borne pour t pour le systeme x y f x y g x x y r 2

on the determination of the number of periodic or closed solutions of a scalar differential equation with convexity abstract it is well known that a scalar differential equation x f t x where f t x is continuous t periodic in t and weakly convex or concave in x has no one or two t periodic solutions or a connected band of t periodic solutions the last possibility can be excluded if f t x is strictly convex or concave for some t in the period interval in this paper we investigate how the actual number of t periodic solutions for a given equation of this type in principle can be determined if f t x is also assumed to have a continuous derivative f x t x it turns out that there are three cases in each of these cases we indicate the monotonicity properties and the domain of values for the function p s where s is the poincare successor function from these informations the actual number of periodic solutions can be determined since a zero of p represents a periodic solution

planar nonautonomous polynomial equations ii coinciding sectors abstract we give a few sufficient conditions for the existence of periodic solutions of the equation z j 0 n a j t z j where a j s are complex valued we prove the existence of one up to two periodic solutions and heteroclinic ones

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

on a class of keller segel chemotaxis systems with cross diffusion abstract in this paper we study a class of keller segel chemotaxis systems with cross diffusion by using the entropy dissipation method and assuming mainly the chemotactic sensitivity separates the cell density and the chemical signal we first establish the existence of global weak solutions with the effects of cross diffusion included in 3 d then we show there is a critical cross diffusion rate c such that no patterns may be expected for c while patterns are formed for c and their stability is also derived in particular in 1 d patterns are always formed whenever c and the chemotactic coefficient is larger than an expressible bifurcation value and there is another critical cross diffusion rate c c such that cells with cross diffusion rate c c are stable while for cells with c to be stable their degradation rate must be less than a threshold value hence in some sense cross diffusion is harmful to enable pattern formation while it is helpful to stabilize the cells once patterns are formed finally we show that the cross diffusion plays a role in regularizing the cell aggregation phenomenon for large chemotactic coefficient our results provide global dynamics and insights on how the biological parameters especially the cross diffusion affect pattern formations

on the number of closed solutions to an equation ẋ f t x where fxn t x 0 n 1 2 or 3

existence of closed solutions for a polynomial first order differential equation we find new criteria for the existence of closed solutions in a first order polynomial differential equation which contains the abel equation as a particular case such results are applied to the problem of the existence of limit cycles in planar polynomial vector fields

periodic solutions with nonconstant sign in abel equations of the second kind abstract the study of periodic solutions with constant sign in the abel equation of the second kind can be made through the equation of the first kind this is because the situation is equivalent under the transformation x x 1 and there are many results available in the literature for the first kind equation however the equivalence breaks down when one seeks for solutions with nonconstant sign this note is devoted to periodic solutions with nonconstant sign in abel equations of the second kind specifically we obtain sufficient conditions to ensure the existence of a periodic solution that shares the zeros of the leading coefficient of the abel equation uniqueness and stability features of such solutions are also studied

a nonsplit sum of coefficients of modular forms we introduce and study certain truncated sums of hecke eigenvalues of gl2 automorphic forms along quadratic polynomials a power saving estimate is established and new applications to moments of critical l values associated to quadratic fields are derived an application to the asymptotic behavior of the height of heegner points and singular moduli is discussed in detail

central values of hecke l functions of cm number fields 0 introduction it is well known that the zeta function of cm complex multiplication abelian varieties can be given in terms of l functions of associated hecke characters in this paper we prove a formula expressing the central special value of the l function of certain hecke characters in terms of theta functions the formula easily implies that the central value is nonnegative and yields a criterion for its positivity combining this criterion with the work of arthaud and rubin we show that certain cm elliptic curves have mordell weil rank zero over their field of definition let f be a totally real number field of degree t and let be a quadratic hecke character of f of conductor f such that 2f f 1 given a cm extension e of f we consider the twist can of by a canonical hecke character can of

boundedness and global existence in the higher dimensional parabolic parabolic chemotaxis system with without growth source abstract in this paper we are concerned with a general class of quasilinear parabolic parabolic chemotaxis systems with without growth source under homogeneous neumann boundary conditions in a smooth bounded domain r n with n 2 it is recently known that blowup is possible even in the presence of superlinear growth restrictions here we derive new and interesting characterizations on the growth versus the boundedness we show that the hard task of proving the l boundedness of the cell density can be reduced to proving its l r boundedness in other words we show that the l r boundedness of the cell density can successfully guarantee its l boundedness and hence its global boundedness where r n or n 2 depending on whether the growth restriction is essentially linear including no growth or superlinear hence a blowup solution also blows up in l p norm for any suitably large p more detailed information on how the growth source affects the boundedness of the solution is derived these results reveal deep understandings of blowup mechanism for chemotaxis models then we use these criteria to establish uniform boundedness and hence global existence of the underlying models logistic source in 2 d cubic source as initially proposed by mimura and tsujikawa in 3 d n 1 st source in n d with n 4 as a consequence in a chemotaxis growth model blowup is impossible if the growth effect is suitably strong finally we underline that our results remove the commonly assumed convexity on the domain

boundedness in a quasilinear parabolic parabolic keller segel system with subcritical sensitivity abstract we consider the quasilinear parabolic parabolic keller segel system u t d u u s u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a convex smooth bounded domain r n with n 1 it is proved that if s u d u c u with 2 n and some constant c 0 for all u 1 then the classical solutions to the above system are uniformly in time bounded provided that d u satisfies some technical conditions such as algebraic upper and lower growth resp decay estimates as u this boundedness result is optimal according to a recent result by the second author winkler 2010 27 which says that if s u d u c u for u 1 with c 0 and some 2 n n 2 then for each mass m 0 there exist blow up solutions with mass u 0 m in addition this paper also proves a general boundedness result for quasilinear non uniformly parabolic equations by modifying the iterative technique of moser alikakos alikakos 1979 1

center conditions at infinity for abel differential equations an abel differential equation y p x y 2 q x y 3 is said to have a center at a set a a 1 a r of complex numbers if y a 1 y a 2 y a r for any solution y x with the initial value y a 1 small enough the polynomials p q are said to satisfy the polynomial composition condition on a if there exist polynomials p q and w such that p p and q q are representable as p x p w x q x q w x and w a 1 w a 2 w a r we show that for wide ranges of degrees of p and q restricted only by certain assumptions on the common divisors of these degrees the composition condition provides a very accurate approximation of the center one up to a finite number of configurations not accounted for to our best knowledge this is the first general i e not restricted to small degrees of p and q or to a very special form of these polynomials result in the center problem for abel equations as an important intermediate result we show that at infinity according to an appropriate projectivization of the parameter space the center conditions are given by a system of the moment equations of the form as a1 p k q 0 s 2 r k 0 1

the number of bifurcation points of a periodic one parameter ode with at most two periodic solutions abstract we study the number of bifurcation points of x f t x where f is periodic in t continuous and locally lipschitz continuous with respect to x by assuming that the differential equation has at most two periodic solutions for each r under some additional assumptions we prove that there are at most two bifurcation points and we find sufficient conditions under which this equation has exactly k bifurcation values where k 0 1 2

boundedness and blow up in the higher dimensional attraction repulsion chemotaxis system with nonlinear diffusion abstract this paper deals with the following quasilinear attraction repulsion chemotaxis system u t d u u u v u w x t 0 v t v u v x t 0 0 w u w x t 0 in a bounded domain r n n 2 under homogeneous neumann boundary conditions the parameters and are positive and the diffusion d u is supposed to generalize the prototype d u d 0 u with d 0 0 and r under the assumption 2 n 1 it is proved that the corresponding initial boundary problem possesses a nonnegative globally bounded solution moreover by constructing a lyapunov functional and by assuming that 2 n 1 and n 3 it is proved that there exist radially symmetric solutions which blow up in finite time in b r 0 with r 0 finally it is also shown that blow up may occur if 2 n 1 and 0 when is a ball in r n with n 3

periodic solutions of a quartic nonautonomous equation on considere des equations de la forme z z 4 t z 3 t z 2 t z on donne un ensemble qui a au moins huit solutions periodiques les coefficients sont des polynomes quadratiques en cost et sint

boundedness vs blow up in a chemotaxis system abstract we determine the critical blow up exponent for a keller segel type chemotaxis model where the chemotactic sensitivity equals some nonlinear function of the particle density assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution furthermore we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails

existence of periodic solutions for fully nonlinear first order differential equations in this paper we establish some existence results for periodic solutions of fully nonlinear first order scalar differential equation f t x x 0 the approach here uses the viscosity solutions method developed in recent decades and the classical upper lower solutions method

periodic orbits in complex abel equations abstract this paper is devoted to prove two unexpected properties of the abel equation d z d t z 3 b t z 2 c t z where b and c are smooth 2 periodic complex valuated functions t r and z c the first one is that there is no upper bound for its number of isolated 2 periodic solutions in contrast recall that if the functions b and c are real valuated then the number of complex 2 periodic solutions is at most three the second property is that there are examples of the above equation with b and c being low degree trigonometric polynomials such that the center variety is formed by infinitely many connected components in the space of coefficients of b and c this result is also in contrast with the characterization of the center variety for the examples of abel equations d z d t a t z 3 b t z 2 studied in the literature where the center variety is located in a finite number of connected components

on global bifurcation for quasilinear elliptic systems on bounded domains general second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains are formulated into nonlinear mappings between sobolev spaces it is shown that the linearized mapping is a fredholm operator of index zero this and the abstract global bifurcation theorem of jacobo pejsachowicz patrick j rabier degree theory for c1 fredholm mappings of index 0 j anal math 76 1998 289 319 allow us to carry out bifurcation analysis directly on these elliptic systems at the abstract level we establish a unilateral global bifurcation result that is needed when studying positive solutions finally we supply two examples of cross diffusion population model and chemotaxis model to demonstrate how the theory can be applied

central values of l functions of cubic twists we are interested in finding for which positive integers d we have rational solutions for the equation x 3 y 3 d the aim of this paper is to compute the value of the l function l e d 1 for the elliptic curves e d x 3 y 3 d for the case of p prime p equiv 1 mod 9 two formulas have been computed by rodriguez villegas and zagier we have computed formulas that relate l e d 1 to the square of a trace of a modular function at a cm point this offers a criterion for when the integer d is the sum of two rational cubes furthermore when l e d 1 is nonzero we get a formula for the number of elements in the tate shafarevich group and we show that this number is a square when d is a norm in mathbb q sqrt 3

periodic solutions of a logistic type population model with harvesting abstract we consider a bifurcation problem arising from population biology d u t d t f u t e h t where f u is a logistic type growth rate function e 0 h t is a continuous function of period t such that 0 t h t d t 0 we prove that there exists an e 0 0 such that the equation has exactly two t periodic solutions when 0 e e 0 exactly one t periodic solution when e e 0 and no t periodic solution when e e 0

well posedness for chemotaxis dynamics with nonlinear cell diffusion abstract this paper investigates the well posedness of a reaction diffusion system of chemotaxis type with a nonlinear diffusion coefficient and a dynamics growth death of the cell population b and a stationary equation for the chemoattractant c with respect to other works in which a nonlinear diffusion for cells has been considered we treat here two distinct cases for this diffusion coefficient the first in which it is a positive bounded function on r and the other in which it may display a singularity at a finite value of the cell density essentially the latter model is new and describes the saturation of the cell population in the neighborhood of a critical value for its diffusion coefficient the chemotactic sensitivity function is supposed to depend both on the cell and chemoattractant densities for homogeneous neumann boundary conditions for the cell population and chemoattractant we prove the existence of a local in time solution when the l 2 norm of the initial datum b 0 is sufficiently small and compute the maximum time interval for which the solution is bounded and smooth under a stronger assumption related to the chemotactic sensitivity we show that there exists a global in time solution for arbitrarily large initial data in a case when the diffusion coefficient is singular we focus on a model expressed by a variational inequality describing the saturation of the cell population at the blowing up diffusion value here the proof requires the study of an intermediate problem with robin boundary conditions which may be interesting by itself in all situations uniqueness follows on a time interval included not necessarily strictly in that of the solution existence under sufficient conditions

existence and uniqueness of initial value problems for a class of second order differential equations etude du probleme de la valeur initiale pour les equations differentielles du second ordre de la forme x t g x t x t x t qui ne peuvent pas etre resolues pour des derivees d ordre plus elevees

local and global bifurcation in periodic scalar odes abstract following the lines of yu ilyashenko and w li we propose a definition of local and global bifurcations for the family of differential equations x f t x where f c r 3 is locally lipschitz continuous with respect to x and t periodic in t then we prove that a value of the parameter is a global bifurcation value if and only if there exists a local bifurcation point including bifurcation points at infinity for this value of the parameter

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

the set of periodic scalar differential equations with cubic nonlinearities we study the structure induced by the number of periodic solutions on the set of differential equations x f t x where f c3 r2 is t periodic in t fx3 t x 0 for every t x r2 and f t x as x uniformly on t we find that the set of differential equations with a singular periodic solution is a codimension one submanifold which divides the space into two components equations with one periodic solution and equations with three periodic solutions moreover the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension two submanifold

exact multiplicity for periodic solutions of a first order differential equation abstract consider the differential equation 1 x f t x h t where h t is a 1 periodic continuous function and f t x c3 is concave convex in x and 1 periodic in t we obtain the complete structure of 1 periodic solutions by means of singularity theory more precisely we show that the image of singularities f consists of a codimension 1 manifold that divides the c r z into two open sets a1 a3 i equation 1 has a unique 1 periodic solution for h t a1 ii the equation has exactly three 1 periodic solutions for h t a3 furthermore if the image of cusp singularities f c is a codimension 1 manifold of f the differential equation has exactly two solutions for h t f f c and has a unique 1 periodic solution for h t f c

implicit differential equations near a singular point abstract this paper presents general and precise results on existence and number of solutions to implicitly defined ordinary differential equations in the vicinity of a singular point where the equation is not equivalent to an explicit one unlike in most if not in all other studies devoted to this problem in the literature it is not assumed that either rank drop occurs on an entire neighborhood of the singular point or that the equation is a scalar one attention is confined to the simplest but most frequently encountered kind of singular points it is shown that such standard singular points split into two complementary classes those from which exactly two distinct solutions emanate and those at which exactly two distinct solutions terminate whether a point of the latter class is eventually encountered is unaffected by slight modifications of the nonsingular initial condition and further evolution of the system governed by the singular ode then requires using a suitable jump condition according to appropriate physical criteria points of the former class are also shown to play a more subtle but equally important role in the dynamics the phenomena described here are relevant in various problems from the sciences such as phase transitions or plasticity they should also be relevant in some aspects of the classical domain of application of differential algebraic equations singular perturbations of ode s