twist periodic solutions for differential equations with a combined attractive repulsive singularity abstract we study the existence of twist periodic solutions of second order differential equations with an attractive repulsive singularity such twist periodic solutions are stable in the sense of lyapunov the proof is based on the third order approximation method in combination with some location information obtained by the averaging method and the method of upper and lower solutions on the reversed order

weak singularities may help periodic solutions to exist abstract in a periodically forced semilinear differential equation with a singular nonlinearity a weak force condition enables the achievement of new existence criteria through a basic application of schauder s fixed point theorem the originality of the arguments relies in that contrary to the customary situation in the available references a weak singularity facilitates the arguments of the proofs

new beams of global bifurcation points for a reaction diffusion system with inequalities or inclusions abstract we consider a reaction diffusion system of activator inhibitor or substrate depletion type which is subject to diffusion driven instability we show that an obstacle e g a unilateral membrane modeled either in terms of inequalities or of inclusions introduces whole beams of new global bifurcation points of spatially non homogeneous stationary solutions which lie in parameter domains which are excluded as bifurcation points for the problem without the obstacle

an influence of unilateral sources and sinks in reaction diffusion systems exhibiting turing s instability on bifurcation and pattern formation abstract we consider a general reaction diffusion system exhibiting turing s diffusion driven instability in the first part of the paper we supplement the activator equation by unilateral integral sources and sinks of the type k u x k d k and k u x k d k these terms measure an average of the concentration over the set k and are active only when this average decreases below or increases above the value of the reference spatially homogeneous steady state which is shifted to the origin we show that the set of diffusion parameters in which spatially heterogeneous stationary solutions can bifurcate from the reference state is smaller than in the classical case without any unilateral integral terms this problem is studied for the case of mixed pure neumann and periodic boundary conditions in the second part of the paper we investigate the effect of both unilateral terms of the type u u and unilateral integral terms on the pattern formation using numerical experiments on the system with well known schnakenberg kinetics

solvability of inclusions of hammerstein type abstract we establish a universal rule for solving operator inclusions of hammerstein type in lebesgue bochner spaces with the aid of some recently proven continuation theorem of leray schauder type for the class of so called admissible multimaps examples illustrating the legitimacy of this approach include the initial value problem for perturbation of m accretive multivalued differential equation the nonlocal cauchy problem for semilinear differential inclusion abstract integral inclusion of fredholm and volterra type and the two point boundary value problem for nonlinear evolution inclusion

unilateral sources and sinks of an activator in reaction diffusion systems exhibiting diffusion driven instability abstract a reaction diffusion system exhibiting turing s diffusion driven instability is considered the equation for an activator is supplemented by unilateral terms of the type s x u s x u describing sources and sinks active only if the concentration decreases below and increases above respectively the value of the basic spatially constant solution which is shifted to zero we show that the domain of diffusion parameters in which spatially non homogeneous stationary solutions can bifurcate from that constant solution is smaller than in the classical case without unilateral terms it is a dual information to previous results stating that analogous terms in the equation for an inhibitor imply the existence of bifurcation points even in diffusion parameters for which bifurcation is excluded without unilateral sources the case of mixed dirichlet neumann boundary conditions as well as that of pure neumann conditions is described

twist periodic solutions of repulsive singular equations motivated by the lazer solimini equation and the brillouin equation which present a repulsive singularity at the origin we will develop in this paper some criterion for the positive periodic solution to be of twist type as an application of the criterion we will give also some quantitative estimates to the region of parameters so that the equations have twist periodic solutions these together with the moser twist theorem imply that the equations have rich dynamics in the neighborhood of the twist periodic solutions 2003 elsevier ltd all rights reserved

the stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator article history in this paper we present a sufficient condition for the stability of the equilibrium of a nonlinear planar system the proof is based on the computation of the corresponding birkhoff normal forms the result does not involve small parameters applications to the relativistic oscillator are also given

existence of one signed periodic solutions of some second order differential equations via a krasnoselskii fixed point theorem this paper is devoted to study the existence of periodic solutions of the second order equation x 00 f t x where f is a caratheodory function by combining some new properties of green s function together with krasnoselskii fixed point theorem on compression and expansion of cones as applications we get new existence results for equations with jumping nonlinearities as well as equations with a repulsive or attractive singularity in this latter case our results cover equations with weak singularities and are compared with some recent results by i rachunkova m tvrdyand i vrkoc r 2002 elsevier science usa all rights reserved

bifurcation for a reaction diffusion system with unilateral and neumann boundary conditions abstract we consider a reaction diffusion system of activator inhibitor or substrate depletion type which is subject to diffusion driven instability if supplemented by pure neumann boundary conditions we show by a degree theoretic approach that an obstacle e g a unilateral membrane modeled in terms of inequalities introduces new bifurcation of spatial patterns in a parameter domain where the trivial solution of the problem without the obstacle is stable moreover this parameter domain is rather different from the known case when also dirichlet conditions are assumed in particular bifurcation arises for fast diffusion of activator and slow diffusion of inhibitor which is the difference from all situations which we know

bifurcation points for a reaction diffusion system with two inequalities abstract we consider a reaction diffusion system of activator inhibitor or substrate depletion type which is subject to diffusion driven instability we show that obstacles e g a unilateral membrane for both quantities modeled in terms of inequalities introduce a new bifurcation of spatially non homogeneous steady states in the domain of stability of the trivial solution of the corresponding classical problem without obstacles

the linearization of periodic hamiltonian systems with one degree of freedom under the diophantine condition abstract in this paper we are concerned with the periodic hamiltonian system with one degree of freedom where the origin is a trivial solution we assume that the corresponding linearized system at the origin is elliptic and the characteristic exponents of the linearized system are i with be a diophantine number moreover if the system is formally linearizable then it is analytically linearizable as a result the origin is always stable in the sense of liapunov in this case

lyapunov stability of elliptic periodic solutions of nonlinear damped equations abstract based on the third order approximation developed by ortega we present the formula of the first twist coefficient for a nonlinear damped equation as an example we consider the lyapunov stability of a superlinear damped differential equation

smooth continuation of solutions and eigenvalues for variational inequalities based on the implicit function theorem the implicit function theorem is applied in a nonstandard way to abstract variational inequalities depending on a possibly infinite dimensional parameter in this way results on smooth continuation of solutions as well as of eigenvalues and eigenvectors are established under certain particular assumptions the abstract results are applied to a linear second order elliptic eigenvalue problem with nonlocal unilateral boundary conditions schrodinger operator with the potential as the parameter

a relationship between the periodic and the dirichlet bvps of singular differential equations in this paper a relationship between the periodic and the dirichlet boundary value problems for second order ordinary differential equations with singularities is established this relationship may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations using this relationship we give in this paper an existence result of positive periodic solutions to singular differential equations when the singular forces satisfy some strong force condition at the singularity 0 and some linear growth condition at infinity

global bifurcation for quasivariational inequalities of reaction diffusion type abstract we consider a reaction diffusion system with implicit unilateral boundary conditions introduced by u mosco we show that global continua of stationary spatially nonhomogeneous solutions bifurcate in the domain of parameters where bifurcation in the case of classical boundary conditions is excluded the problem is formulated as a quasivariational inequality and the proof is based on the leray schauder degree

periodic solutions for second order differential equations with a singular nonlinearity abstract this paper deals with the existence of positive t periodic solutions for the following differential equation x a t x f t x c t where a c l 1 0 t and f car 0 t r r the existence results are obtained by using a fixed point theorem in cones

existence and stability of periodic solutions for second order semilinear differential equations with a singular nonlinearity it is proved that a periodically forced second order equation with a singular nonlinearity in the origin with linear growth in infinity possesses a t periodic stable solution for high values of the mean value of the forcing term the method of proof combines a rescaling argument with the analysis of the first twist coefficient of the

the necessity of the carathe odory conditions for the lower semicontinuity in measure of the mulitvalued nemytskii operator

eigenvalues and bifurcation for problems with positively homogeneous operators and reaction diffusion systems with unilateral terms abstract reaction diffusion systems satisfying assumptions guaranteeing turing s instability and supplemented by unilateral terms of type v and v are studied existence of critical points and sometimes also bifurcation of stationary spatially non homogeneous solutions are proved for rates of diffusions for which it is excluded without any unilateral term the main tool is a general result giving a variational characterization of the largest eigenvalue for positively homogeneous operators in a hilbert space satisfying a condition related to potentiality and existence of bifurcation for equations with such operators the originally non variational non symmetric system is reduced to a single equation with a positively homogeneous potential operator and the abstract results mentioned are used

twist periodic solutions of second order singular differential equations abstract we study the twist periodic solutions of second order singular differential equations such twist periodic solutions are stable in the sense of lyapunov and present much interesting dynamical features around them the proof is based on the third order approximation method the estimates of periodic solutions of ermakov pinney equations and the estimates on rotation numbers of hill equations play an important role in the analysis

bifurcation points and eigenvalues of inequalities of reaction diffusion type on demontre certaines assertions concernant les points de bifurcation et les valeurs propres de systemes de reaction diffusion avec des conditions unilaterales

some crandall rabinowitz type results and applications to reaction diffusion systems abstract for a small lipschitz perturbation of a smooth equation the existence of exactly two bifurcation points near a simple eigenvalue is shown the result is applied to reaction diffusion systems subject to turing s diffusion driven instability under small unilateral obstacles

degree instability and bifurcation of reaction diffusion systems with obstacles near certain hyperbolas abstract for a reaction diffusion system which is subject to turing s diffusion driven instability and which is equipped with unilateral obstacles of various types the nonexistence of bifurcation of stationary solutions near certain critical parameter values is proved the result implies assertions about a related mapping degree which in turn implies for small obstacles the existence of a new branch of bifurcation points spatial patterns induced by the obstacle

on periodic solutions of second order differential equations with attractive repulsive singularities abstract sufficient conditions for the existence of a solution to the problem u t g t u t h t u t f t for a e t 0 u 0 u u 0 u are established

existence of nonnegative and nonpositive solutions for second order periodic boundary value problems we are interested in nonnegative and nonpositive solutions of the boundary value problem u f t u u 0 u 1 u 0 u 1 where f fulfils the caratheodory conditions on 0 1 r we generalize the results reached by m n nkashama j santanilla and l sanchez and present estimates for solutions in addition we apply our existence theorems to periodic boundary value problems for nonlinear duffing equations whose right hand sides have a repulsive or attractive singularity at the origin we extend or generalize existence results by a c lazer and s solimini and other authors moreover we get some multiplicity results and in the case of a repulsive singularity we also admit a weak singularity in constrast to the previous papers on this subject our proofs are based on the method of lower and upper functions and topological degree arguments and the results are tested on examples

periodic solutions for second order singular damped differential equations abstract we study the existence of positive periodic solutions for second order singular damped differential equations by combining the analysis of the sign of green s functions for the linear damped equation together with a nonlinear alternative principle of leray schauder recent results in the literature are generalized and significantly improved

unilateral regulation breaks regularity of turing patterns we consider a reaction diffusion system undergoing turing instability and augment it by an additional unilateral source term we investigate its influence on the turing instability and on the character of resulting patterns the nonsmooth positively homogeneous unilateral term v has favorable properties but the standard linear stability analysis cannot be performed we illustrate the importance of the nonsmoothness by a numerical case study which shows that the turing instability can considerably change if we replace this term by its arbitrarily precise smooth approximation however the nonsmooth unilateral term and all its approximations yield qualitatively similar patterns although not necessarily developing from small disturbances of the spatially homogeneous steady state further we show that the unilateral source breaks the approximate symmetry and regularity of the classical patterns and yields asymmetric and irregular patterns moreover a given system with a unilateral source produces spatial patterns even for diffusion parameters with ratios closer to 1 than the same system without any unilateral term

degree and global bifurcation for elliptic equations with multivalued unilateral conditions abstract a bifurcation problem for an elliptic multivalued boundary value problem with a real parameter is considered the existence of global bifurcation between two eigenvalues of a certain type of the laplacian is proved for a class of abstract inclusions with compact multivalued mappings in a hilbert space it is shown how the degree can be determined near the eigenvalues of a particular type of an associated linear single valued problem and the jump of the degree is proved as a consequence global bifurcation for such abstract inclusions is obtained the weak formulation of the boundary value problem mentioned is a particular case