a topological approach to periodic oscillations related to the liebau phenomenon abstract we give some sufficient conditions for existence non existence and localization of positive solutions for a periodic boundary value problem related to the liebau phenomenon our approach is of topological nature and relies on the krasnosel ski guo theorem on cone expansion and compression our results improve and complement earlier ones in the literature

an introduction to mathematical models of coagulation fragmentation processes a discrete deterministic mean field approach abstract we summarise the properties and the fundamental mathematical results associated with basic models which describe coagulation and fragmentation processes in a deterministic manner and in which cluster size is a discrete quantity an integer multiple of some basic unit size in particular we discuss smoluchowski s equation for aggregation the becker doring model of simultaneous aggregation and fragmentation and more general models involving coagulation and fragmentation

precise asymptotic of eigenvalues of resonant quasilinear systems in this work we study the sequence of variational eigenvalues of a system of resonant type involving p and q laplacians on rn with a coupling term depending on two parameters and satisfying p q 1 we show that the order of growth of the kth eigenvalue depends on k o k n

existence theorems for a second order three point boundary value problem abstract letf 0 1 r 2 r be continuous ande c 0 1 let i 0 1 ai r all of theai s having the same sign i 1 2 m 2 0 x t f t x t x t e t t 0 1 e x 0 0 x 1 i 1 m 2 a i x i bc m we discuss existence theorems for the problem e bcm under some sign condition offabout the origin our analysis is based on a nonlinear alternative of leray schauder

the becker doring equations at large times and their connection with the lsw theory of coarsening a systematic procedure is given for obtaining the asymptotic late time behavior of the becker doring equations describing the time evolution of a population of clusters of particles in lowest order of approximation the distribution of the sizes of the largest clusters satisfies the equations of the lifshitz slyozov wagner theory of coarsening

boundedness in nonlinear asymmetric oscillations abstract in this paper the boundedness of all solutions of the nonlinear equation p x p 1 p x p x f x g x e t is discussed where e t c 7 is 2 p periodic f g are bounded c 6 functions p u u p 2 u p 2 are positive constants x max x 0 x max x 0

on a nonlocal boundary value problem for second order linear ordinary differential equations abstract existence and uniqueness criteria are established for the solution of the equation u p 1 t u p 2 t u p 0 t satisfying the boundary conditions u a c 1 u b b a u x d x c 2 where the coefficients pk a b r k 0 1 2 are locally integrable and a b r is a function of bounded variation these criteria include the case when the functions pk a b r k 0 1 2 are not integrable on a b having singularities in a and b

boundedness of solutions for nonlinear reversible system at resonance abstract in this paper we are concerned with the problem of boundedness of solutions for the following nonlinear p laplacian p x p 1 n p p x f x p x g x e t where p u u p 2u p 2 for some n n f g c r are odd functions e t c r is odd and 2 p periodic

from the becker doring to the lifshitz slyozov wagner equations connections between two classical models of phase transitions the becker doring bd equations and the lifshitz slyozov wagner lsw equations are investigated homogeneous coefficients are considered and a scaling of the bd equations is introduced in the spirit of the previous works by penrose and collet goudon poupaud and vasseur convergence of the solutions to these rescaled bd equations towards a solution to the lsw equations is shown for general coefficients an approach in the spirit of numerical analysis allows to approximate the lsw equations by a sequence of bd equations a new uniqueness result for the bd equations is also provided

existence results for some n dimensional nonlocal boundary value problems abstract in this article we study an n th dimensional nonlocal boundary value problem for a second order differential equation of sturm liouville type by using the linear part of the boundary condition an equivalent integral equation is obtained and then the nonlinear alternative fixed point result is applied some special cases are also given

the fredholm alternative at the first eigenvalue for the one dimensionalp laplacian abstract in this work we study the range of the operatoru u p 2 u 1 u p 2 u u 0 u t 0 p 1 we prove that all functionsh c1 0 t satisfying t0 h t sinp pt t dt 0 lie in the range but that ifp 2 andh 0 the solution set is bounded here sin pt t is a first eigenfunction associated to 1 we also show that in this case the associated energy functionalu 1 p t0 u p 1 p t0 u p t0 huis unbounded from below if 1 2 onw1 p0 0 t 1corresponds precisely to the best constant in thelp poincare inequality moreover we show that unlike the linear casep 2 forp 2 the range contains a nonempty open set inl 0 t

application of implicit function theorems to existence of solutions to ordinary differential equations with nonlocal boundary conditions abstract we consider the problem y a x y y b y 0 0 y 1 g y y 0 fixed in 0 1 as a model of steady state heat conduction in a rod when the heat flux at the end x 1 is determined by observation of the temperature and heat flux at some interior point we establish conditions sufficient for existence uniqueness and positivity of solutions existence and uniqueness for a semi infinite problem with a boundary condition of this form are also examined

convergence rate for some quasilinear eigenvalues homogenization problems abstract in this work we study the homogenization problem for nonlinear eigenvalues of some quasilinear elliptic operators we obtain an explicit order of convergence in k and in e for the variational eigenvalues

existence of positive solutions for three point boundary value problems at resonance with the help of the theorem of a fixed point index for a proper semilinear operators established by cremins we get a existence theorem concerning the existence of positive solution for the second order ordinary differential equation of three point boundary value problems at resonance

positive solutions of multi point boundary value problems at resonance abstract we establish new results on the existence of positive solutions for some multi point boundary value problems at resonance our results are based on a recent leggett williams norm type theorem due to o regan and zima we also derive a new result for a three point problem previously studied by several authors

solvability of a three point nonlinear boundary value problem for a second order ordinary differential equation in case f x u u ps x p x u pjx u with pk 0 l 4 r locally integrable the boundary value problem l lt 1 2 was studied by kiguradze and lomtatidze i the purpose of this paper is to obtain existence and uniqueness theorems for the boundary value problem 1 1 t 1 2 under natural conditions on f using degree theoretic arguments we note that if u is a solution of 1 1 1 2 then there exists at least one 5 e q l such that u c 0 accordingly the boundary value problem

positive solutions for a three point boundary value problem the second order three point boundary value problem x t 2x t h t f t x t t 0 1 x 0 0 x x 1 is considered under some conditions concerning the first eigenvalue of the relevant linear operator where 0 1 is a constant h t is allowed to be singular at t 0 and t 1 the existence of positive solutions is studied by means of fixed point index theory

solvability of the resonant 1 dimensional periodic p laplacian equations abstract in this paper we study the existence of 2 periodic solutions of x p 2 x x p 2 x f t x r where p 1 is an eigenvalue of the corresponding homogeneous equation f is a 2 periodic function some existence results are proved via the morse theory

eigenvalues and the one dimensional p laplacian we consider the boundary value problem p u f t u 0 with p 1 t 0 1 u 0 u 1 0 and with 0 the value of is chosen so that the boundary value problem has a positive solution in addition we derive an explicit interval for such that for any in this interval the existence of a positive solution to the boundary value problem is guaranteed in addition the existence of two positive solutions for in an appropriate interval is also discussed

multiple positive solutions of resonant and non resonant nonlocal boundary value problems abstract we study the existence of positive solutions for equations of the form u t f t u t a e t 0 1 when f t u 2 u 0 for u 0 for some constant 0 and for the perturbed equation u t 2 u t h t u t a e t 0 1 when h t u 0 subject to various non local boundary conditions in particular we establish existence and multiplicity of positive solutions for non perturbed boundary value problems at resonance by considering equivalent non resonant perturbed problems with the same boundary conditions

maximum and anti maximum principles for singular sturm liouville problems the maximum and anti maximum principles are extended to the case of eigenvalue sturm liouville problems r t u p t u lambda m t u h t with bounded interval a b the function r is assumed to be continuous and 0 on a b but the function 1 r is not necessarily integrable on a b the conditions on the functions p m and h depend on the integrability or nonintegrability of 1 r on a c and or c b for some c epsilon a b the weight function m is not necessarily of constant sign

perturbation of the p laplacian by vanishing nonlinearities in one dimension abstract we treat the quasi linear spectral problem p u p 2 u u p 2 u h x u 0 x a u 0 u a 0 at resonance with vanishing nonlinearity h x u g u f x g u 0 as u and f l 0 a f 0 satisfying certain orthogonality related hypotheses 1 p 3 p 2 the parameter r takes an arbitrary resonant value we first establish the boundedness of the set of all weak solutions in the sobolev space w 0 1 p 0 a which then enables us to obtain an existence result by the leray schauder degree theory the boundedness is obtained from a very precise asymptotic estimate valid for large solutions to p which can be applied thanks to a sufficiently fast rate of decay g u 0 as u

positive solutions for singular p laplacian equations with sign changing nonlinearities using inequality theory abstract in this paper general existence theorems are presented for the singular equation p u q t f u r t g u 0 t 1 u 0 u 1 0 throughout our nonlinearity is allowed to change sign the singularity may occur at u 0

positive solutions for a three point boundary value problem at resonance abstract this paper deals with the second order three point boundary value problem x t f t x t t 0 1 x 0 0 x x 1 the existence and multiplicity of positive solutions are investigated by means of the fixed point theorem in cones

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

the modified becker doring system with aggregation dominated rate coefficients we consider a system of becker doring equations modified in the manner of dreyer duderstadt 2006 j stat phys 123 55 we analyse the case of size independent rate coefficients in the case of weak fragmentation aggregation dominated we use matched asymptotic expansions to construct approximations to the cluster size distribution as it evolves from purely monomeric initial data to the equilibrium solution in which the typical cluster size is extremely large we compare the results with earlier calculations of a similar limit in the standard becker doring system and find a similar sequence of timescales with similar profile shapes however there are notable differences in the kinetic behaviour within timescales due to the modified system having a fragmentation rate which depends on the total number of clusters present in the system

remarks on nonlocal boundary value problems at resonance we show that it is important to allow the nonlinear term to change sign when discussing existence of a positive solution for multipoint or more general nonlocal boundary value problems in the resonant case when the nonlinear term has a fixed sign we obtain simple necessary and sufficient conditions for the existence of positive solutions

slow passage through the nonhyperbolic homoclinic orbit associated with a subcritical pitchfork bifurcation for hamiltonian systems and the change in action slowly varying conservative systems are analyzed in the case of a reverse subcritical pitchfork bifurcation in which two saddles and a center coalesce before the bifurcation there is a hyperbolic double homoclinic orbit connecting a linear saddle point at the bifurcation a double nonhyperbolic homoclinic orbit connects to a nonlinear saddle point strongly nonlinear oscillations obtained by the method of averaging are not valid near unperturbed homoclinic orbits in the case in which the solution slowly passes through the nonhyperbolic homoclinic orbit associated with the subcritical pitchfork bifurcation the solution consists of a large sequence of nonhyperbolic homoclinic orbits connecting autonomous nonlinear saddle approaches solutions are captured into the left and right well phase jumps and the boundaries of the basins of attraction are computed it is shown that the change in action in the slow passage through the nonhyperbolic homoclinic orbits is much larger than the known change in action f

on the fredholm alternative for the p laplacian in this paper we study the range of the operator u p 1 1 u p 2u email protected u p 2u email protected u p 2u u 0 u t 0 p 1 p p email protected psin p p a 0 b 0 and p 1 1 p a 1 p email protected 1 p 2 the results of this paper generalize the corresponding results obtained in j differential equations 151 1999 386 and the method used in this paper is elementary

on the fredholm alternative for the p laplacian at higher eigenvalues in one dimension abstract we investigate the existence of a weak solution u to the quasilinear two point boundary value problem p u p 2 u k u p 2 u f x 0 x a u 0 u a 0 we assume 1 p 2 0 a and f l 0 a is a given function f 0 the number k stands for the k th eigenvalue of the one dimensional p laplacian let sin p p x a denote the eigenfunction associated with 1 then sin p k p x a is the eigenfunction associated with k it is well known that problem p has a solution when k 1 and f satisfies the orthogonality condition 0 a f x sin p k p x a d x 0 we focus on the case k 2 assuming the orthogonality condition above we obtain new existence results for p which significantly generalize those already known manasevich and p takac 2002 4 we are able to handle more degenerate sufficient conditions on f than any previous work we present a simple example of f to which previous results and methods are not applicable our approach is based on the third and second order taylor expansions of the functions sin p and cos p sin p respectively in the l 1 norm rather than in the standard l norm

existence and multiplicity of solutions for four point boundary value problems at resonance abstract this paper deals with the second order four point boundary value problem x t f t x t 0 t 1 x 0 ax x 1 bx where f 0 1 r r is continuous 0 1 0 a 1 1 0 b 1 a 1 b 1 a s 1 b 0 by using the coincidence degree theory the method of lower and upper solutions is established as applications some existence and multiplicity results are obtained

on the solvability of nonlinear sturm liouville problems abstract the focus of this paper is the study of nonlinear differential equations subject to general non local boundary conditions to establish sufficient conditions for the existence of solutions we use properties of the nonlinearities and their relationship with the eigenvalues of an associated linear sturm liouville problem

iterative methods for a singular boundary value problem we consider a second order nonlinear ordinary differential equation of the form y 1 q xy q 0 x 1 where q 0 with the boundary conditions y 0 y 1 0 this problem arises in boundary layer equations for the flow of a power law fluid over an impermeable semi infinite flat plane we show that classical iterative schemes such as the picard and newton methods converge to the solution of this problem in spite of the singularity of the solution if we choose an adequate initial approximation moreover we observe that these methods are more efficient than the methods used before and may be applied to a larger range of values of q numerical results for different values of q are given and compared with the results obtained by other authors

metastable states for the becker doring cluster equations the becker doring equations in whichc l t can represent the concentration ofl particle clusters or droplets in say a condensing vapour at timet are begin array 20 c dc l t mathord left vphantom dc l t dt right kern nulldelimiterspace dt j l 1 t j l t l 2 3 end array with j l t a l c 1 t c l t b l 1 c l 1 t and eitherc1 const case a or rho sum limits 1 infty lc l const case b the equilibrium solutions arec l q l z l where q l prod limits 2 l a r 1 mathord left vphantom a r 1 b r right kern nulldelimiterspace b r the density of the saturated vapour defined as rho s sum limits 1 infty lq l z s l wherez s is the radius of convergence of the series is assumed finite it is proved here that subject to some further plausible conditions on the kinetic coefficientsa l andb l there is a class of metastable solutions of the equations withc1 z s small and positive which take an exponentially long time to decay to their asymptotic steady states an exponentially long time means one that increases more rapidly than any negative power of the given value ofc1 z s or in caseb s as the latter tends to zero the main ingredients in the proof are i a time independent upper bound on the solution of the kinetic equations this upper bound is a steady state solution of case a of the equations of the type used in the becker doring theory of nucleation and ii an upper bound on the total concentration of particles in clusters greater than a certain critical size which with suitable initial conditions remains exponentially small until the time becomes exponentially large

solvability conditions for non local boundary value problems for two dimensional half linear differential systems abstract in this paper we consider two non local boundary value problems for two dimensional half linear differential systems we prove general fredholm type theorems which allow one to derive new efficient solvability criteria for the problems studied

fredholm alternative for the p laplacian in higher dimensions abstract in this paper we characterize the set of all right hand sides h c formula for which the boundary value problem p u 1 u p 2 u h in u 0 on has at least one weak solution u w01 p here 1 0 is the first eigenvalue of the p laplacian in particular we prove that for h 1 0 this problem is solvable and the energy functional e h u 1 p u p 1 p u p hu is unbounded from below

asymptotic behaviour of solutions to the generalized becker doring equations for general initial data we prove the following asymptotic behaviour for solutions to the generalized becker doring system for general initial data under a detailed balance assumption and in situations where density is conserved in time there is a critical density s such that solutions with an initial density s converge strongly to the equilibrium with density and solutions with initial density s converge in a weak sense to the equilibrium with density s this extends the previous knowledge that this behaviour happens under more restrictive conditions on the initial data the main tool is a new estimate on the tail of solutions with density below the critical density

existence result for a singular nonlinear boundary value problem at resonance we study the existence of solutions of the second order boundary value problems u t 2u t a t g u t h t a e t 0 1 u acloc 0 1 u 0 u 1 0 where g r r is continuous a h z lloc1 0 1 01t z t dt the proof of the main result is based upon the lyapunov schmidt procedure and the connectivity properties of the solution set of parametrized families of compact vector fields

metric properties of the group of area preserving diffeomorphisms area preserving cliffeoinorpliisms of the 2 disk which are identity near the boundary form a group d2 wllich can be equipped usin g tlhe l2nlorm on its lie algebra with a right invariant metric witll tllis metric the diameter of d2 is infinite in this paper we sl iow that d2 contains quasiisometric embeddings of any finitely generated free group and any finitely generated abelian free group

a note on multi point boundary value problems abstract multi point boundary value problems for a second order ordinary differential equation are considered in this note an existence result is obtained with the help of coincidence degree theory

on solvability and unique solvability of two point singular boundary value problems abstract for the differential equation u f t u u where the function f a b r 2 r has nonintegrable singularities with respect to the time variable at t a and t b new unimprovable sufficient conditions of solvability and unique solvability of the boundary value problems u a 0 u b 0 and u a 0 u b 0 are established

optimal conditions of solvability of nonlocal problems for second order ordinary differential equations abstract for the differential equation u f t u in regular as well as in singular cases there are established optimal sufficient conditions of existence for solutions satisfying nonlocal boundary conditions of the type a b u i 1 s d i s c i i 1 2

quasi isometrically embedded subgroups of braid and diffeomorphism groups we show that a large class of right angled artin groups in particular those with planar complementary defining graph can be embedded quasi isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary with respect to the norm metric this extends results of benaim and gambaudo who gave quasi isometric embeddings of and for all as a consequence we are also able to embed a variety of gromov hyperbolic groups quasi isometrically in pure braid groups and in the group examples include hyperbolic surface groups some hnn extensions of these along cyclic subgroups and the fundamental group of a certain closed hyperbolic 3 manifold

the fredholm alternative for the one dimensional p laplacian abstract in this paper we consider the solvability of the boundary value problem p u 1 p u f u h t u 0 u t 0 where p 1 p u u p 2 u by using generalized polar coordinates transformation method we improve and generalize some results obtained recently in j differential equations 151 1999 386

on certain boundary value problems for second order linear ordinary differential equations with singularities pour l equation differentielle u p 1 t u p 2 t a p 0 t a coefficients localement integrables p k a b r k 0 1 2 et pour chacune des conditions aux limites suivantes u a lim t 5 u h p 2 t u a u b u a u b u t 0 ou a t 0 b et p 2 t exp t a b 2 p 2 7b t d 7b t on etablit des conditions d existence et d unicite d une solution

a nodal inverse problem for a quasi linear ordinary differential equation in the half line abstract in this paper we study an inverse problem for a quasi linear ordinary differential equation with a monotonic weight in the half line first we find the asymptotic behavior of the singular eigenvalues and we obtain a weyl type asymptotics imposing an appropriate integrability condition on the weight then we investigate the inverse problem of recovering the coefficients from nodal data we show that any dense subset of nodes of the eigenfunctions is enough to recover the weight

a singular non local problem at resonance abstract we study a second order non local problem using the coincidence degree theory we show the existence of a solution whose derivative is singular at the right end point of the interval which is a new result for any resonant non local problem under the non local condition we find a general way to ensure that the differential operator is a fredholm operator of index zero

the generalized quasilinearization method for second order three point boundary value problems abstract in this paper we obtain the existence and uniqueness results for general second order three point boundary value problems by applying the method of upper and lower solutions together with leray schauder degree as well as obtaining monotone iteration schemes which converge quadratically to the unique solution of some specific second order three point boundary value problem by using the method of quasilinearization

characteristic classes of surface bundles let m be a c manifold if one wants to classify all differentiable fibre bundles over a given manifold which have m as fibres then one must study the topology of the corresponding structure group diffm the group of all diffeomorphisms of m equipped with the c topology it is easy to see that the connected component diffo m of the identity of diff m is an open subgroup therefore the natural topology on the quotient group m diff m diffo m the diffeotopy group of m is the discrete topology hence we have a fibration

path lifting properties and embedding between raags abstract for a finite simplicial graph let g denote the right angled artin group on the complement graph of in this article we introduce the notions of induced path lifting property and semi induced path lifting property for immersions between graphs and obtain graph theoretical criteria for the embeddability between right angled artin groups we recover the result of s h kim and t koberda that an arbitrary g admits a quasi isometric group embedding into g t for some finite tree t the upper bound on the number of vertices of t is improved from 2 2 m 1 2 to m 2 m 1 where m is the number of vertices of we also show that the upper bound on the number of vertices of t is at least 2 m 4 lastly we show that g c m embeds in g p n for n 2 m 2 where c m and p n denote the cycle and path graphs on m and n vertices respectively

multiple positive solutions for boundary value problems of second order delay differential equations with one dimensional p laplacian abstract we consider the boundary value problems p x t q t f t x t x t 1 x t 0 p s s p 2 s p 1 t 0 1 subject to some boundary conditions by using a generalization of the leggett williams fixed point theorem due to avery and peterson we provide sufficient conditions for the existence of at least three positive solutions to the above problems

on a thermodynamically consistent modification of the becker doring equations abstract recently dreyer and duderstadt have proposed a modification of the becker doring cluster equations which now have a different lyapunov function we start with existence and uniqueness results for the modified equations next we derive an explicit criterion for the existence of equilibrium states and solve the minimization problem for the lyapunov function finally we discuss the long time behavior in the case that equilibrium solutions do exist

positive solutions of m point boundary value problems the second order m point boundary value problem x h x f x 0 0 x 1 0 0 1 i 1m 2ai i is considered under some conditions concerning the first eigenvalue of the relevant linear operator where i 0 1 with 0 1 2 m 2 1 ai 0 with i 1m 2ai 1 h x is allowed to be singular at x 0 and x 1 the existence of positive solutions and multiple positive solutions is obtained by means of fixed point index theory similarly conclusions hold for some other m point boundary value conditions

landesman lazer type results for second order hamilton jacobi bellman equations we study the boundary value problem f d2u du u x u f x u in u 0on where the second order differential operator f is of hamilton jacobi bellman type f is sub linear in u at infinity and rn is a regular bounded domain we extend the well known landesman lazer conditions to study various bifurcation phenomena taking place near the two principal eigenvalues associated to the differential operator we provide conditions under which the solution branches extend globally along the eigenvalue gap we also present examples illustrating the results and hypotheses

lyapunov type inequalities for partial differential equations in this work we present a lyapunov inequality for linear and quasilinear elliptic differential operators in n dimensional domains we also consider singular and degenerate elliptic problems with ap coefficients involving the p laplace operator with zero dirichlet boundary condition as an application of the inequalities obtained we derive lower bounds for the first eigenvalue of the p laplacian and compare them with the usual ones in the literature

nonnegative solutions to nonlocal boundary value problems for systems of second order differential equations dependent on the first order derivatives abstract in this paper we consider nonlocal boundary value problems for systems of second order differential equations with dependence on the first order derivatives and deviating arguments by using a fixed point theorem due to avery and peterson sufficient conditions are obtained that guarantee the existence of at least three nonnegative solutions to such problems we investigate our problem both for delayed and advanced arguments i i and also for the case when i t i t t t 0 1 in all cases arguments i i can change the character on 0 1 so in some subinterval i of 0 1 they can be delayed in i and advanced in 0 1 i some remarks concern also the case when differential equations do not depend on the first order derivatives examples illustrate some results

on the existence of positive solution for a kind of multi point boundary value problem at resonance abstract this paper concerns the existence of positive solution for a class of second order m point boundary value problems under different resonant conditions by using the leggett williams norm type theorem due to o regan and zima we obtain the existence of positive solution an example is given to demonstrate the main results

the becker doring cluster equations basic properties and asymptotic behaviour of solutions existence and uniqueness results are established for solutions to the becker doring cluster equations the density is shown to be a conserved quantity under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined denoting the set of equilibrium solutions byc 0 s the principal result is that if the initial density 0 s then the solution converges strongly toc o while if 0 s the solution converges weak toc s in the latter case the excess density 0 s corresponds to the formation of larger and larger clusters i e condensation the main tools for studying the asymptotic behaviour are the use of a lyapunov function with desirable continuity properties obtained from a known lyapunov function by the addition of a special multiple of the density and a maximum principle for solutions

asymptotic behaviour of solutions to the becker doring equations for arbitrary initial data on determine le comportement asymptotique des solutions des equations d amas de becker doring sous certaines hypotheses de decroissances sur les donnees initiales faites dans un article anterieur