the monotonicity of the period function for planar hamiltonian vector fields if the hamiltonian system with hamiltonian h x y 12y2 v x has a center at the origin it is shown that the period function for the family of periodic trajectories surrounding the center is monotone when the function v v 2 is convex this theorem is applied to determine existence and uniqueness of solutions for the neumann boundary value problem x a x 2 bx c x 0 x 1 0

periodic solutions of a class of nonlinear difference equations via critical point method in this paper we obtain some new sufficient conditions for the existence of nontrivial m periodic solutions of the following nonlinear difference equation d p n dx n 1 d f n x n n z by using the critical point method where f zxr r is continuous in the second variable m 2 is a given positive integer p n m p n for any n z and f t m z f t z for any t z zxr 1 d 1 and d 0

a generalized goodwin business cycle model in random environment abstract this work develops a generalization of the goodwin lotka volterra models the traditional model describes the evolution of the employment rate and the labor share motivated by recent developments on business cycles and goodwin type models we consider a generalized goodwin business cycle model with noise we aim at investigating the impact of telegraph noise and white noise on the evolution of trajectories and the energy on trajectories of the model it is proved that with the presence of the noises all trajectories exit any compact subset of the open unit square d 0 1 0 1 and the energy along trajectories can be arbitrarily large moreover under certain additional conditions the boundary of d is the stable limit cycle of the perturbed model by white noise noise impact on the period of business cycles is also discussed numerical examples are provided to demonstrate our findings

on fuzzy measure and fuzzy integral on fuzzy set abstract on a fuzzy algebra of fuzzy sets we discuss some structural characteristics of a fuzzy measure the pseudo autocontinuity and some other concepts for a fuzzy measure are introduced and the relations between these cre discussed for the case of fuzzy sets riesz s theorems and lebesgue s theorems for a sequence of measurable functions are proved the concept of the fuzzy integral on a fuzzy set is given and some properties of fuzzy integrals on fuzzy sets are studied finally we prove some convergence theorems of a sequence of fuzzy integrals on fuzzy sets

an existence theorem for a nonlinear difference equation where p 1 and s p is a real sequence we establish conditions under which 1 1 has a positive nondecreasing solution here a solution of 1 1 is a real sequence y y which satisfies 1 1 since 1 1 is a recurrence relation given real initial values y and y it is clear that we can inductively obtain y y since p is a real number greater than 1 the sequence yk may not be real that is not a solution note however that any constant multiple of a solution is again a solution the existence question originated in l where it was shown that hardy s inequality for a series 2 can be viewed as a necessary condition for the existence of a positive nondecreasing solution of 1 1 our existence theorem reduces to a nonoscillation criterion in the case p 2 since then 1 1 changes to a second order linear difference equation see e g 3 a brief outline of the paper is as follows section 2 introduces a riccati type transformation and then develops various necessary conditions for the existence of a positive nondecreasing solution section 3 discusses some of the functional analytic background which is needed in order to apply schauder s theorem section 4 contains the proof of our existence theorem and section 5 describes a comparison theorem which allows us to drop the assumption that the coefficients sk in 1 1 are nonnegative

local stability and lyapunov functionals for n dimensional quasipolynomial conservative systems abstract we present a method for determining the local stability of equilibrium points of conservative generalizations of the lotka volterra equations these generalizations incorporate both an arbitrary number of species including odd dimensional systems and nonlinearities of arbitrarily high order in the interspecific interaction terms the method combines a reformulation of the equations in terms of a poisson structure and the construction of their lyapunov functionals via the energy casimir method these lyapunov functionals are a generalization of those traditionally known for lotka volterra systems examples are given

a new approach for the study of limit cycles abstract we prove that star like limit cycles of any planar polynomial system can also be seen either as solutions defined on a given interval of a new associated planar non autonomous polynomial system or as heteroclinic solutions of a 3 dimensional polynomial system we illustrate these points of view with several examples one of the key ideas in our approach is to decompose the periodic solutions as the sum of two suitable functions as a first application we use these new approaches to prove that all star like reversible limit cycles are algebraic as a second application we introduce a function whose zeroes control the periodic orbits that persist as limit cycles when we perturb a star like reversible center as far as we know this is the first time that this question is solved in full generality somehow this function plays a similar role that an abelian integral for studying perturbations of hamiltonian systems

on the location and period of limit cycles in gause type predator prey systems in this paper an attempt is made to estimate the location and period of the limit cycles of gause type predator prey systems in the case when there is a unique unstable positive equilibrium an annular region which contains all the limit cycles is determined and an upper bound for the period of the limit cycles is given both the annular region and the upper bound of the period are explicitly computable

oscillatory and asymptotic properties of higher order nonlinear neutral difference equations abstract in this paper we study the oscillatory and asymptotic properties of solutions of higher order nonlinear neutral difference equations using krasnoselskii s fixed point theorem schauder s fixed point theorem and lebesgue s dominated convergence theorem we obtain some necessary and sufficient conditions for all bounded solutions to be oscillatory or tend to zero our results extend and improve some known results

extension and representation of comonotonically additive functionals the domain of comonotonically additive and monotone functionals on the class of continuous functions with compact support is extended in a natural way a necessary and sufficient condition for which the extended functional is comonotonically additive and monotone is presented it is shown that the extended functional can be represented by a choquet integral when the extended functional is comonotonically additive and monotone

oscillatory and asymptotic behavior of second order nonlinear difference equations in this paper we are dealing with the oscillatory and asymptotic behaviour of solutions of second order nonlinear difference equations of the form rn xn f n xn 0 n n n0 1 we obtain the following results a if k n0 l rk any nonoscillatory solution of 1 must belong to one of the following four types k k k 0 k 0 b if k n0 l rk any nonoscillatory solution of 1 must belong to one of the following three types k0 k k0 c necessary and sufficient conditions for 1 to have a nonoscillatory solution which belongs to k k k 0 k0 or k are given depending on whether f is a superlinear or sublinear function all these results include and improve b szmanda s results in bull polish acad sci math 34 nos 3 4 1986 133 141

geometric methods for two point nonlinear boundary value problems on considere une classe d equations differentielles de la forme x f x g x avec des conditions aux limites soit de dirichlet soit de neumann on etudie l existence de la solution triviale

the monotonicity of period function for codimension four quadratic system q4 abstract in this paper we study the codimension four quadratic system q 4 z iz 4 z 2 2 z 2 b ic z 2 where b and c are real constants i 2 1 z x iy b ic 2 it is proved that the period function of periodic trajectories for q 4 is monotone the content of this paper can be viewed as a contribution to the proof of chicone s conjecture cf mr 94h 58072

an extension of sugeno integral abstract in this paper we extend the concept of sugeno fuzzy integral from nonnegative fuzzy measurable functions to extended real valued fuzzy measurable functions and discuss the lost genuine properties for this extension several necessary and sufficient conditions of absolute s integrability for extended real valued fuzzy measurable functions are given moreover the space s of all fuzzy measurable functions will be proved to be a pseudo metric space under a necessary and sufficient condition finally as an application of this extension the pettis integral will be established for this kind of fuzzy integral

a second order nonlinear difference equation oscillation and asymptotic behavior discrete analogues are investigated for well known results on oscillation growth and asymptotic behavior of solutions of y q t y 0 for q t 0 and for q t 0 the analogue of atkinson s oscillation criterion is shown to be true for 2yn 1 qn yn 0 but the analogue for atkinson s nonoscillation criterion is shown to be false

an analytic numerical method for computation of the liapunov and period constants derived from their algebraic structure we consider the problem of computing the liapunov and the period constants for a smooth differential equation with a nondegenerate critical point first we investigate the structure of both constants when they are regarded as polynomials on the coefficients of the differential equation second we take advantage of this structure to derive a method to obtain the explicit expression of the above mentioned constants although this method is based on the use of the runge kutta fehlberg methods of orders 7 and 8 and the use of richardson s extrapolation it provides the real expression for these constants

properties of the period function for some hamiltonian systems and homogeneous solutions of a semilinear elliptic equation abstract in this work we study the period function t of solutions to the conservative equation x t f x t 0 we present conditions on f that imply the monotonicity and convexity of t as a consequence we obtain the criterium established by c chicone and find conditions easier to apply we also get a condition obtained by cima gasull and manosas about monotonicity and following some of their calculations present results on the period function of hamiltonian systems where h x y f x n 1 y n using the monotonicity of t we count the homogeneous solutions to the semilinear elliptic equation u u 1 in two dimensions

the general pettis sugeno integral of vector multifunctions relative to a vector fuzzy multimeasure abstract in this paper we define and study the general pettis sugeno integral of a vector multifunction relative to a vector fuzzy multimeasure both taking values in a locally convex space x ordered by a closed convex pointed cone x with nonempty interior several classic properties of this integral and some estimation and comparative results with the pettis lebesgue integral are established

the bounded convergence in measure theorem for nonlinear integral functionals in this paper we introduce a new notion of the perturbation of nonlinear functionals to formulate a functional form of the convergence theorems for nonlinear integrals in nonadditive measure theory as its direct consequences we obtain the bounded convergence in measure theorems for typical nonlinear integrals which show that the autocontinuity of a nonadditive measure is equivalent to the validity of the bounded convergence in measure theorems for the choquet the sugeno and the shilkret integrals as well as their symmetric and asymmetric extensions

on the number of critical points of the period in the study of subharmonic bifurcations from a planar hamiltonian system it is often assumed that the period of a one parameter family of periodic orbits of the unperturbed system is a monotone function of the parameter see e g hale and taboas s chow and hale es however this hypothesis is in many instances difficult to verify for example it is well known that every orbit in the first quadrant of the vollterra lotka equation is periodic it is only in recent years that the monotonicity of the periods of these periodic orbits has been shown hsu 9 and waldvogel 151 such problems also occur in the study of bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation u f u o uer 1 1 on some interval in 14 smoller and wasserman showed that if f u u u u u cc

some properties of fuzzy integrable function space l 1 μ abstract we introduce definitions of fuzzy orlicz class l and fuzzy mean convergence we show that fuzzy mean convergence is equivalent to convergence in fuzzy measure and l l1 for any orlicz function this conclusion refines a result of d ralescu toward a general theory of fuzzy variable j math anal appl 86 1982 176 193 moreover a necessary and sufficient condition for l1 s is given it also improves a result of d ralescu and g adams the fuzzy integral j math anal appl 75 1980 562 570

the period function for quadratic integrable systems with cubic orbits this paper is concerned with the monotonicity of the period function for families of quadratic systems with centers whose orbits lie on cubic planar curves it is proved that each such system has a period function with at most one critical point

the cyclicity and period function of a class of quadratic reversible lotka volterra system of genus one abstract in this paper we investigate a class of quadratic reversible lotka volterra system of genus one z i z z 2 b z 2 with b 3 5 it is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two moreover we prove that the period function of its period trajectories is monotone increasing

a remark on the period of the periodic solution in the lotka volterra system abstract an important consideration in the nonlinear predator prey problem of lotka volterra type is the determination of the period this paper gives a general expression for the period in terms of the given parameters in the lotka volterra system we also discuss the qualitative behavior of the period related to the energy level of the lotka volterra system

monotone solutions of a class of nonlinear difference equations abstract comparison theorems and existence criteria are summarized and new ones are derived for the monotone solutions of a class of nonlinear difference equations which arises from hardy s inequality

study of perturbed lotka volterra systems via abelian integrals abstract we consider the lotka volterra system as a hamiltonian one and study a special perturbation of it for this perturbed system we get results on the number of limit cycles the main tools used are abelian integrals and degenerate hopf bifurcation

new criteria for the monotonicity of the ratio of two abelian integrals abstract new criteria to determine the monotonicity of the ratio of two abelian integrals are given when two abelian integrals have the forms h f 1 x y d x and h f 2 x y d x or the forms h f 1 x y d x and h f 2 x y d x and h are ovals belonging to the level set x y h x y h where h x y has the form y 2 2 x or x y 2 2 x we give new criteria which are defined directly by the functions which appear in the above abelian integrals and prove that the monotonicity of the criteria implies the monotonicity of the ratios of the abelian integrals the new criteria are applicable in a large class of problems some of which simplify the existing proofs and some of which generalize known results

uniqueness of periodic orbits of some vector fields with codimension two singularities abstract we consider the question of uniqueness of limit cycles of some normal form equations of vector fields with singularities of codimension two by using suitable choices of scaling on the normal form equations we give simple proofs on uniqueness of periodic orbits

the cyclicity of the period annulus of the quadratic hamiltonian systems with non morsean point abstract in this paper we investigate the quadratic hamiltonian systems with non morsean point it is proved that in this situation the cyclicity of the period annulus under quadratic perturbations is equal to two

orbits in a stochastic goodwin lotka volterra model abstract this paper examines the cycling behavior of a deterministic and a stochastic version of the economic interpretation of the lotka volterra model the goodwin model we provide a characterization of orbits in the deterministic highly non linear model we then study a stochastic version with brownian noise introduced via a heterogeneous productivity factor existence conditions for a solution to the system are provided we prove that the system produces cycles around a unique equilibrium point in finite time for general volatility levels using stochastic lyapunov techniques for recurrent domains numerical insights are provided

first derivative of the period function with applications abstract given a centre of a planar differential system we extend the use of the lie bracket to the determination of the monotonicity character of the period function as far as we know there are no general methods to study this function and the use of commutators and lie bracket was restricted to prove isochronicity we give several examples and a special method which simplifies the computations when a first integral is known

a criticality result for polycycles in a family of quadratic reversible centers abstract we consider the family of dehomogenized loud s centers x y x 1 x x d x 2 f y 2 y where d f r 2 and we study the number of critical periodic orbits that emerge or disappear from the polycycle at the boundary of the period annulus this number is defined exactly the same way as the well known notion of cyclicity of a limit periodic set and we call it criticality the previous results on the issue for the family x r 2 distinguish between parameters with criticality equal to zero regular parameters and those with criticality greater than zero bifurcation parameters a challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters which form a set b of codimension 1 in r 2 in the present paper we succeed in proving that a subset of b has criticality equal to one

a criterion for determining the monotonicity of the ratio of two abelian integrals abstract a new method to determine the monotonicity of the ratio of two abelian integrals is given we find two criterion functions defined directly by the functions which appear in the abelian integrals the monotonicities of the criterion functions imply the monotonicity of the ratio of the abelian integrals by using this criterion several known results can be deduced easier and some new results are obtained

the period in the lotka volterra system is monotonic abstract periodic solutions in a class of hamiltonian systems with one degree of freedom containing the lotka volterra system are considered their period is represented as an integral appropriate for numerical evaluation as well as for theoretical investigations the monotonicity of the period in the lotka volterra system follows as a simple consequence

bounded and monotone properties of solutions of second order quasilinear forced difference equations abstract the authors consider the quasilinear difference equation p n 1 y n 1 q n y n r n and obtain results on the asymptotic behavior of solutions of including sufficient conditions for all solutions to be bounded or unbounded some results on the existence and behavior of nonincreasing solutions of are also obtained examples are inserted to illustrate the results

an explicit expression of the first liapunov and period constants with applications in this paper we study systems in the plane having a critical point with pure imaginary eigenvalues and we search for effective conditions to discern whether this critical point is a focus or a center in the case of it being a center we look for additional conditions in order to be isochronous we stress that the essential differences between the techniques used in this work and the more usual ones are basically two the elimination of the integration constants when we consider primitives of functions see also remark 3 2 and the fact that we maintain the complex notation in the whole study thanks to these aspects we reach with relative ease an expression of the first three liapunov constants v3 v5 andv7 and of the first two period ones p2andp4 for a general system as far as we know this is the first time that a general and compact expression ofv7has been given moreover the use of a computer algebra system is only needed in the computation ofv7andp4 these results are applied to give a classification of centers and isochronous centers for certain families of differential equations

periodic solutions of a forced lotka volterra equation abstract for a 2 species predator prey model the effect of a periodic forcing dependent on two parameters is studied a description is given for the local bifurcation curves of periodic solutions near a fixed family of solutions of the free equation