oscillation theorems for differential equations with a nonlinear damping term in this paper we are concerned with a class of nonlinear second order differential equations with a nonlinear damping term passage to more general class of equations allows us to remove a restrictive condition usually imposed on the nonlinearity and as a consequence our results apply to wider classes of nonlinear differential equations two illustrative examples are considered

on oscillation of nonlinear second order differential equation with damping term we present new oscillation criteria for the second order nonlinear differential equation with damping our theorems are stated in general form they complement and extend related results known in the literature the relevance of our results is illustrated with a number of examples to facilitate computations in these examples we use mathematica r a symbolic computer language

lack of strong completeness for stochastic flows it is well known that a stochastic differential equation sde on a euclidean space driven by a brownian motion with lipschitz coefficients generates a stochastic flow of homeomorphisms when the coefficients are only locally lipschitz then a maximal continuous flow still exists but explosion in finite time may occur if in addition the coefficients grow at most linearly then this flow has the property that for each fixed initial condition x the solution exists for all times almost surely if the exceptional set of measure zero can be chosen independently of x then the maximal flow is called strongly complete the question whether an sde with locally lipschitz continuous coefficients satisfying a linear growth condition is strongly complete was open for many years in this paper we construct a two dimensional sde with coefficients which are even bounded and smooth and which is not strongly complete thus answering the question in the negative

the double bubble problem on the flat two torus we characterize the perimeter minimizing double bubbles on all flat two tori and as corollaries on the flat infinite cylinder and the flat infinite strip with free boundary specifically we show that there are five distinct types of minimizers on flat two tori depending on the areas to be enclosed

oscillation results for a second order damped differential equation with nonmonotonous nonlinearity abstract we present new oscillation criteria for a nonlinear second order differential equation with a damping term an essential feature of our results is that we do not require the nonlinearity to be nondecreasing furthermore as opposed to the recent results by s r grace 1989 math nachr 141 117 127 1992 j math anal appl 171 220 241 and s r grace and b s lalli 1990 j math anal appl 149 277 311 we do not impose any additional condition on the damping term p t consequently our theorems apply to new classes of equations which are not covered by the known criteria the relevance of the results obtained is illustrated with a number of carefully selected examples

oscillation analysis of numerical solutions for nonlinear delay differential equations of hematopoiesis with unimodal production rate this paper is concerned with oscillation and non oscillation of numerical solutions for nonlinear delay differential equation of hematopoiesis with unimodal production rate by investigating the roots of characteristic equation some conditions under which the numerical solution is oscillatory and non oscillatory are obtained the properties of non oscillatory numerical solutions are investigated to verify our results we give numerical experiments

sobolev differentiable stochastic flows for sdes with singular coefficients applications to the transport equation in this paper we establish the existence of a stochastic flow of sobolev diffeomorphisms r d x 7 s t x r d s t r for a stochastic differential equation sde of the form dxt b t xt dt dbt s t r xs x r d the above sde is driven by a bounded measurable drift coefficientb r r d r d and a d dimensional brownian motion b more specifically we show that the stochastic flow s t of the sde lives in the space l 2 w 1 p r d w for all s t and all p 1 where w 1 p r d w denotes a weighted sobolev space with weight w possessing a p th moment with respect to lebesgue measure on r d this result is counter intuitive since the dominant culture in stochastic and deterministic dynamical systems is that the flow inherits its spatial regularity from the driving vector fields the spatial regularity of the stochastic flow yields existence and uniqueness of a sobolev differentiable weak solution of the stratonovich stochastic transport equation dtu t x b t x du t x dt p d 1 ei du t x db i t 0 u 0 x u0 x where b is bounded and measurable u0 is c 1 b and ei d 1 a basis for r d it is well known that the deterministic counter part of the above equation does not in general have a solution using stochastic perturbations and our analysis of the above sde we establish a deterministic flow of sobolev diffeomorphisms for classical one dimensional deterministic ode s driven by discontinuous vector fields furthermore and as a corollary of the latter result we construct a sobolev stochastic flow of diffeomorphisms for one dimensional sde s driven by discontinuous diffusion coefficients

dynamics of stochastic 2d navier stokes equations abstract in this paper we study the dynamics of a two dimensional stochastic navier stokes equation on a smooth domain driven by linear multiplicative white noise we show that solutions of the 2d navier stokes equation generate a perfect and locally compacting c 1 1 cocycle using multiplicative ergodic theory techniques we establish the existence of a discrete non random lyapunov spectrum for the cocycle the lyapunov spectrum characterizes the asymptotics of the cocycle near an equilibrium stationary solution we give sufficient conditions on the parameters of the navier stokes equation and the geometry of the planar domain for hyperbolicity of the zero equilibrium uniqueness of the stationary solution viz ergodicity local almost sure asymptotic stability of the cocycle and the existence of global invariant foliations of the energy space

perfect cocycles through stochastic differential equations we prove that if is a random dynamical system cocycle for whicht t x is a semimartingale then it is generated by a stochastic differential equation driven by a vector field valued semimartingale with stationary increment helix and conversely this relation is succinctly expressed as semimartingale cocycle exp semimartingale helix to implement it we lift stochastic calculus from the traditional one sided time to two sided timet and make this consistent with ergodic theory we also prove a general theorem on the perfection of a crude cocycle thus solving a problem which was open for more than ten years

oscillation criteria for nonlinear differential equations with damping oscillation criteria are established for the nonlinear second order damped differential equation rx px qf x 0 which are different from most known results

oscillation theorems for certain functional differential equations abstract some new criteria for the oscillation of functional differential equations with a middle term of the form x n t p t x n 1 t h h t x g t are presented

the spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds it is proved that the spaces of index one minimal surfaces and stable constant mean curvature surfaces with genus greater than one in non fixed flat three manifolds are compact in a strong sense given a sequence of any of the above surfaces we can extract a convergent subsequence of both the surfaces and the ambient manifolds in the ck topology these limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds some applications to the isoperimetric problem are given

comparative computational analysis of the cahn hilliard equation with emphasis on c1 continuous methods the numerical treatment of the fourth order cahn hilliard equation is nonstandard using a galerkin method necessitates for instance piecewise smooth and globally c 1 continuous basis functions or a mixed formulation the latter is obtained introducing an auxiliary field which allows to rephrase the cahn hilliard equation as a set of two coupled second order equations in view of this the formulation in terms of the primal unknown appears to be a more intuitive and natural choice but requires a c 1 continuous interpolation therefore isogeometric analysis using a spline basis and natural element analysis are addressed in the present contribution mixed second order finite element methods introducing the chemical potential or alternatively a nonlocal concentration as auxiliary field serve as references to which both higher order methods are compared in terms of accuracy and efficiency

extremal properties of central half spaces for product measures we deal with the isoperimetric and the shift problem for subsets of measure 1 2 in product probability spaces we prove that the canonical central half spaces are extremal in particular cases products of log concave measures on the real line satisfying precise conditions and products of uniform measures on spheres or balls as a corollary we improve the known log sobolev constants for euclidean balls we also give some new results about the related question of estimating the volume of sections of unit balls of lp sums of minkowski spaces

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

on oscillation of nonlinear second order differential equations with damping term the paper is concerned with oscillation of a novel class of nonlinear differential equations with a damping term first it is demonstrated how known oscillation results for another intensively studied class of equations can be translated to the one in question and vice versa advantages and drawbacks of such translation are carefully examined then an oscillation criterion for the new class of equations is established the principal result of the paper is compared to those reported in the literature and an illustrative example to which known oscillation criteria fail to apply is provided

the periodic isoperimetric problem given a discrete group g of isornetries of r 3 we study the g isoperimetric problem which consists of minimizing area modulo g among surfaces in r 3 which enclose a g invariant region with a prescribed volume fraction if g is a line group we prove that solutions are either families of round spheres or right cylinders in the doubly periodic case we prove that for most rank two lattices solutions must be spheres cylinders or planes for the remaining rank two lattices we show among other results an isoperimetric inequality in terms of the topology of the isoperimetric surfaces finally we study the case where g pm3m the group of symmetries of the integer rank three lattice z 3 and other crystallographic groups of cubic type we prove that isoperimetric solutions must be spheres if the prescribed volume fraction is less than 1 6 and we give an isoperimetric inequality for g invariant regions that for instance implies that the area modulo z 3 of a surface dividing the three space in two g invariant regions with equal volume fractions is at least 2 19 the conjectured solution is the classical p schwarz triply periodic minimal surface whose area is 2 34 another consequence of this isoperimetric inequality is that pm3m symmetric surfaces other than families of spheres cannot be isoperimetric for the lattice group z 3

retarded evolution systems driven by fractional brownian motion with hurst parameter h 1 2 abstract this paper is devoted to the study of pathwise solutions of delayed evolution systems driven by a hilbert valued holder continuous function with holder index greater than 1 2 in the first part we shall obtain the existence and uniqueness of a mild solution for such a system which turns out to define a non autonomous dynamical system in the second part we consider retarded stochastic systems driven by a hilbert valued fractional brownian motion with hurst parameter bigger than 1 2 and show that the corresponding mild solutions generate a random dynamical system provided that the canonical version of the fractional brownian motion is chosen as the metric dynamical system

some oscillation criteria for a class of second order nonlinear damped differential equations a class of second order nonlinear damped differential equations is studied and several new oscillation conditions are obtained our results extend improve and unify a number of existing results and handle the cases which are not covered by known criteria

oscillation criteria for second order neutral equations with distributed deviating argument abstract we present new oscillation criteria for the second order neutral delay differential equations with distributed deviating argument r t x t z t a b p t f x g t d 0 t t 0 where z t x t q t x t new oscillation criteria are established which are based on a class of new functions t s l defined in the sequel our results are sharper than some previous results which can be seen by the example at the end of this paper

the stable manifold theorem for non linear stochastic systems with memory i existence of the semiflow abstract we consider non linear stochastic functional differential equations sfde s on euclidean space we give sufficient conditions for the sfde to admit locally compact smooth cocycles on the underlying infinite dimensional state space our construction is based on the theory of finite dimensional stochastic flows and a non linear variational technique in part ii of this article the above result will be used to prove a stable manifold theorem for non linear sfde s

global well posedness of stochastic 2d primitive equations with random initial conditions abstract in this article we consider 2d stochastic primitive equations pes driven by affine linear multiplicative white noise and with random initial conditions we obtain the global well posedness of the stochastic pes when the random initial condition satisfies sufficient malliavin regularity in the proof process of the global existence of solutions to 2d stochastic pes the malliavin calculus plays a key role

the stable manifold theorem for stochastic differential equations type equations are treated starting with the existence of a stochastic flow for a sde we introduce the notion of a hyperbolic stationary trajectory we prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution for stratonovich sdes the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde the proof of the stable manifold theorem is based on ruelle oseledec multiplicative ergodic theory

the substitution theorem for semilinear stochastic partial differential equations abstract in this article we establish a substitution theorem for semilinear stochastic evolution equations see s depending on the initial condition as an infinite dimensional parameter due to the infinite dimensionality of the initial conditions and of the stochastic dynamics existing finite dimensional results do not apply the substitution theorem is proved using malliavin calculus techniques together with new estimates on the underlying stochastic semiflow applications of the theorem include dynamic characterizations of solutions of stochastic partial differential equations spde s with anticipating initial conditions and non ergodic stationary solutions in particular our result gives a new existence theorem for solutions of semilinear stratonovich spde s with anticipating initial conditions

the stable manifold theorem for non linear stochastic systems with memory ii the local stable manifold theorem abstract we state and prove a local stable manifold theorem theorem 4 1 for non linear stochastic differential systems with finite memory viz stochastic functional differential equations sfde s we introduce the notion of hyperbolicity for stationary trajectories of sfde s we then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary trajectory the stable and unstable manifolds are stationary and asymptotically invariant under the stochastic semiflow the proof uses infinite dimensional multiplicative ergodic theory techniques developed by d ruelle together with interpolation arguments

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

a new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces let m be a compact riemannian manifold with a fixed conformal structure then we introduce the concept of conformal volume of m in the following manner for each branched conformal immersion q9 of m into the unit sphere s n we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of s we let vc n qg be the max imum volume of these branched immersions the conformal volume of m is defined to be the infimum of v n q0 where qo ranges over all branched conformal immersions of m into the unit sphere s in this paper we study the case when m is a compact surface and we call the conformal volume of m to be the conformal area of m we demonst ra te that this conformal invariant is non trivial in fact we prove that if there exists a minimal immersion of m into s where coordinate functions are first eigenfunctions then the conformal area of m is given by the area of m with respect to the induced metric this enables us to compute the conformal area for several surfaces for example the conformal area of r p z is 6n and the conformal area of the square torus is 27c 2 we believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces we demonstrate this claim by applying the concept of conformal area to two different branches of surface theory the first application is to study the total curvature of a compact surface in r this problem has a long history fenchel and fary 9 proved that for a closed curve o in r slk 27c where k is its curvature then milnor 12

oscillation criteria for certain nth order differential equations with deviating arguments abstract oscillation criteria for nth order differential equations with deviating arguments of the form x n 1 t 1 x n 1 t f t x g t 0 n even are established where g c t0 r f c t0 r r and 0 is a constant

oscillation theorems for nonlinear differential equations of second order abstract in this paper we present some new criteria for the oscillation of the differential equation a t x t x t p t x t q t f x t 0 the obtained results extend improve and correlate a number of existing criteria

on the spatial asymptotic behavior of stochastic flows in euclidean space we study asymptotic growth rates of stochastic flows on r d and their derivatives with respect to the spatial parameter under lipschitz conditions on the local characteristics of the generating semimartingales in a first step these conditions are seen to imply moment inequalities for the flow of the form e sup 0t x 0t y p 1 0 t t in a second step we deduce the growth rates from an integrated version of these moment inequalities using the continuity lemma of garsia rodemich and rumsey we provide two examples to show that our results are sharp

oscillation of even order half linear functional differential equations with damping in this paper a class of even order half linear functional differential equations with damping are studied by using the generalized riccati transformation and the integral averaging technique the new oscillation criteria which generalize and improve some known results are obtained for all solutions of the equations

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

interval oscillation criteria for second order nonlinear differential equations with damping new oscillation criteria are established for the nonlinear damped differential equation ry py qf y 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t0 rather than on the whole half line our results are sharper than some previous results and extend and improve the recent interval criteria of kong and huang and can be applied to extreme cases such as t0 q s ds in particular several examples that dwell upon the sharp conditions of our results are also included

oscillation theorems for superlinear second order damped differential equations abstract in this paper we concerned with the oscillation of the second order superlinear differential equation of the form a t y t p t y t q t f y t 0 several new oscillation criteria are established under quite general assumptions our methodology is somewhat different from that of previous authors ch g philos oscillation theorems for linear differential equations of second order arch math basel 53 1989 482 492 ch g philos oscillation criteria for second order superlinear differential equations canad j math anal xli 1989 321 340 ch g philos i k purnaras oscillations in superlinear differential equations of second order j math anal appl 165 1992 1 11 y h yu oscillation criteria for second order nonlinear differential equations with damping acta math appl 16 1993 433 441 in chinese our results generalize and extend some earlier results of yu 1993 example is also given to illustrate the results

oscillation analysis of advertising capital model analytical and numerical studies abstract this paper mainly deals with the oscillation of nonlinear delay differential equation which is used to describe advertising capital model analytically and numerically firstly the condition of oscillation of the analytic solution is presented by the technique of the theory of characteristic secondly the asymptotic behavior of non oscillatory analytic solution is verified thirdly the methods are applied to the mentioned equation some conditions under which the numerical solution oscillates are obtained moreover it is proved that every non oscillatory numerical solution tends to the steady state of the model finally some numerical simulations for verifying the theoretical findings are provided

oscillation criteria for second order nonlinear perturbed differential equations abstract we present new oscillation criteria for the second order nonlinear perturbed differential equations these criteria are of a high degree of generality and they extend and unify a number of existing results

note on wong s paper using the method of interval integral average we establish some new oscillation criteria which improve the oscillation criteria due to james s w wong

oscillation theorems for second order nonlinear neutral differential equations the purpose of this paper is to study the oscillation of the second order neutral differential equations of the form a t z t c q t x b s t 0 where z t x t p t x t t we explore properties of given equations by examining those of associated first order delay equations new comparison theorems essentially simplify the examination of the equations studied as they allow us to deduce the oscillation of the second order delay differential equation by applying the oscillation criteria obtained to the first order delay equations the results obtained are easy to verify

oscillation criteria for second order nonlinear differential equations with damping abstract for a second order nonlinear differential equation with a damping term we obtain new oscillation criteria without an assumption that has been required for related results reported in the literature over the last two decades two approaches refining the standard integral averaging technique suggested in the paper can be used to derive a variety of simpler oscillation theorems for different classes of nonlinear differential equations and both sets of oscillation criteria established in the paper are of independent interest several examples are provided to illustrate the relevance of the new theorems

isoperimetric inequalities in crystallography the study of the isoperimetric problem in the presence of crystallographic sym metries is an interesting unsolved question in classical differential geometry given a space group q we want to describe among surfaces dividing euclidean 3 space into two invariant regions with prescribed volume fractions those which have the least area per unit cell of the group we know that this periodic isoperimetric problem always has solutions and any of these solutions is a smooth triply periodic surface with constant mean curvature 3 8 12 20 21 the explicit description of these surfaces which we call q isoperimetric surfaces remains open in most of the cases let us consider for instance the following two simple situations assume the prefixed volume fractions are both equal to 1 2 if q is the cubic integer lattice z3 then it was proved by hadwiger 10 that the isoperimetric surface is up to symmetries the family of parallel planes x n 2 n e z if g is the group of symmetries of 1 pm3m in crystallographic notation then it is conjectured that the isoperimetric surface is the classical p schwarz triply periodic minimal sur face from the point of view of applications it is also important to obtain explicit isoperimetric inequalities for periodic regions with prescribed symmetries triply periodic minimal surfaces and related surfaces are widely accepted in crys tallography and materials science see for instance the papers 6 13 17 29 30 they appear as interface models in mesoscale self assembled phenomena some of these are reminiscent of the familiar soap bubbles such as lipid water systems but others are of a different nature such as diblock copolymers roughly speaking these interfaces are periodic and separate two immiscible liquid materials the size of the periods varies between tens and hundreds of nanometers and the more inter esting cases usually present cubic symmetry mesoscale interfaces can be treated as periodic surfaces minimizing under volume fraction constraints a suitable energy whose dominant term is in several cases the area of the interface itself so the pe riodic isoperimetric problem is the simplest geometric model to explain the shapes of these interfaces among the surfaces appearing most frequently in this context are periodic arrays of planes spheres and cylinders and small perturbations of constant mean curvature p q pm3m and d q fd3m schwarz surfaces and the g schoen gyroid i4i32

oscillation behavior of even order nonlinear neutral differential equations with variable coefficients in this paper some sufficient conditions are obtained for the oscillation of all solutions of even order nonlinear neutral differential equations with variable coefficients our results improve and generalize known results in particular the results are new even when n 2

non oscillatory solutions for a nonlinear neutral delay differential equation abstract we study the existence of non oscillatory bounded solutions for a second order nonlinear neutral delay differential equation some sufficient conditions for the existence of non oscillatory bounded solutions with positive lower bound or negative upper bound are established and the non oscillatory solutions with decay as t are discussed

oscillation theorems for second order neutral differential equations the aim of this paper is to study the oscillation of the second order neutral differential equations e r t x t p t x t t q t x s t 0 the obtained results are based on the new comparison theorems that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation the obtained comparison principles essentially simplify the examination of the studied equations

oscillation of damped pde with p laplacian in unbounded domains abstract in this paper new oscillation theorems for the damped partial differential equation pde with p laplacian div a x u p 2 u b x u p 2 u c x u p 2 u 0 are established in unbounded domains our theorems as two special cases when a i or a i b x 0 improve and complement some existing results in the literature to illustrate our main results we give some corollaries and examples

stochastic flows for nonlinear second order parabolic spde the existence of stochastic flows in l 2 spaces is proved for a stochastic reaction diffusion equation of second order in a bounded domain

regularity of isoperimetric hypersurfaces in riemannian manifolds we add to the literature the well known fact that an isoperimetric hypersurface s of dimension at most six in a smooth riemannian manifold m is a smooth submanifold if the metric is merely lipschitz then s is still holder differentiable

interval oscillation of a general class of second order nonlinear differential equations with nonlinear damping abstract the paper is concerned with the oscillation of a class of general type second order differential equations with nonlinear damping terms several new interval oscillation criteria are established for such a class of differential equations under quite general assumptions examples are also given to illustrate the results in particular it is shown that under some very mild conditions on k 1 k 2 and f the equation k 1 x x cos tk 2 x x x sin tf x 0 is oscillatory which extends the result obtained by kwong and wong j math anal appl 288 2003 15

oscillation theorems for nth order delay differential equations abstract oscillation criteria are obtained for a class of functional differential equations including the generalized emden fowler equation x n q t x g t sgn x g t 0 0 as a special case the purpose of this paper is to extend and improve known fundamental oscillation criteria for superlinear sublinear and linear differential equations with retarded arguments

spatial estimates for stochastic flows in euclidean space we study the behavior for large x of kunita type stochastic flows t x on r d driven by continuous spatial semimartingales for this class of flows we prove new spatial estimates for large x under very mild regularity conditions on the driving semimartingale random field it is expected that the results would be of interest for the theory of stochastic flows on noncompact manifolds as well as in the study of nonlinear filtering stochastic functional and partial differential equations some examples and counterexamples are given

lyapunov exponents of linear stochastic functional differential equations ii examples and case studies we give several examples and examine case studies of linear stochastic functional differential equations the examples fall into two broad classes regular and singular according to whether an underlying stochastic semi flow exists or not in the singular case we obtain upper and lower bounds on the maximal exponential growth rate overline lambda 1 sigma of the trajectories expressed in terms of the noise variance roughly speaking we show that for small sigma overline lambda 1 sigma behaves like sigma 2 2 while for large sigma it grows like log sigma in the regular case it is shown that a discrete oseledec spectrum exists and upper estimates on the top exponent lambda 1 are provided these estimates are sharp in the sense that they reduce to known estimates in the deterministic or nondelay cases

the isoperimetric problem for lens spaces we solve the isoperimetric problem in the lens spaces with large fundamental group namely we prove that the isoperimetric surfaces are either geodesic spheres or quotients of clifford tori we also show that the isoperimetric problem in the lens spaces l 3 1 and l 3 2 follows from the proof of the willmore conjecture by marques and neves

integral averaging techniques for the oscillation of second order nonlinear differential equations abstract in this paper we enlist some known and establish some new oscillation criteria for the second order nonlinear differential equations of the form if a t x t t su p t t q t f x t 0 where the coefficients p t and q t are not assumed to be nonnegative for all large values of t these criteria are obtained by using integral averaging techniques and can be applied to some special cases where other classical oscillation results are not applicable a systematic study is attempted to extend improve and correlate a number of existing results

oscillation and asymptotic behavior of higher order delay differential equations with p laplacian like operators abstract the p laplace equations have some applications in continuum mechanics on the basis of this background detail we study oscillation and asymptotic behavior of solutions to two classes of higher order delay damped differential equations with p laplacian like operators some new criteria are presented that improve the related contributions to the subject several examples are provided to illustrate the relevance of new theorems

the topology and geometry of embedded surfaces of constant mean curvature si m est une surface proprement plongee dans r 3 avec une courbure moyenne constante non nulle alors m n est pas homeomorphe a une surface fermee avec un seul point elimine soit m une surface proprement plongee dans r 3 a courbure moyenne constante non nulle si m est homeomorphe a une surface fermee avec 2 points elimines alors m est une distance bornee a partir d une ligne droite soit m une surface proprement plongee de courbure moyenne non nulle qui est homeomorphe a une surface fermee avec 3 points elimines alors m reste une distance bornee a partir d un plan

averaging technique and nonlinear second order elliptic oscillation abstract by using the refinement of the standard integral averaging technique we obtain new oscillation criteria for certain nonlinear second order damped elliptic differential equations we do not impose any additional conditions on the damping terms except for the continuity in particular when n 1 our results extend those of rogovchenko and tuncay yu v rogovchenko f tuncay oscillation criteria for second order nonlinear differential equations with damping nonlinear analysis 69 2008 208 221 for second order nonlinear differential equations with damping to damped elliptic differential equations when n 2 our results improve the oscillation criteria of xu jia ma z xu b jia d ma oscillation theorems for elliptic equations with damping appl math comput 156 2004 93 106 for second order elliptic equations with damping four examples are provided to illustrate the relevance of the main results

oscillation criteria for second order nonlinear neutral differential equations by refining the standard integral averaging technique we obtain new oscillation criteria for a class of second order nonlinear neutral differential equations of the form r t x t p t x email protected q t f x t x s t 0 assumptions in our theorems are less restrictive whereas the proofs are significantly simpler compared to those in the recent paper by shi and wang 18 and related contributions to the subject examples are provided to illustrate the relevance of new theorems

on kamenev type oscillation theorems for second order differential equations with damping abstract oscillation criteria for the nonlinear second order ordinary differential equation with damping x p t x f x 0 are given the results are extensions of integral averaging technique of kamenev and include earlier results of yeh yan and chen

a generalized formula of ito and some other properties of stochastic flows a stochastic differential equation with smooth coefficients is considered which defines a continuous flow t of c 8 mappings of rd in rd if zt is a continuous semi martingale t zt s is proved to be a semi martingale for which an ito type formula is established it is shown that a s for any t t is an onto diffeomorphism if zt is a continuous semi martingale t 1 zt is proved to be a semi martingale whose ito decomposition is explicitly found