anti periodic solutions to a class of nonlinear differential equations in hilbert space abstract a broad class of nonlinear non monotone anti periodic boundary value problems in a hilbert space is considered as a preliminary step the existence uniqueness and continuous dependence upon data of anti periodic solutions to some first and second order evolution equations associated to odd noncoercive monotone operators is established applications of the theory to nonlinear partial differential equations are also discussed

a note on existence of anti periodic and heteroclinic solutions for a class of second order odes abstract we discuss the existence of anti periodic solutions to the following second order differential equation q u t q by using fixed point theory together with the green s function for the anti periodic boundary value problem in the first part section 2 of the paper then in the next part section 3 we construct a kind of heteroclinic solution to some special cases of the equation above our method is variational in nature and is inspired by the ideas of rabinowitz and stredulinsky in p rabinowitz e stredulinsky on some results of moser and of bangert aihp anal nonlin 21 2004 673 688

nonlinear third order differential equations with anti periodic boundary conditions and some optimal control problems abstract existence uniqueness and continuous dependence theorems for a class of third order differential equations with anti periodic boundary conditions are given applications to optimal control of the deflection of a three layer beam are also given the methods used here are the leray schauder s continuation theorem the wirtinger poincare type inequalities the monotonicity method maximal monotone operators and the penalizations of convex functionals of barbu s type

imaging geometry through dynamics the observable representation for many stochastic processes there is an underlying coordinate space v with the process moving from point to point in v or on variables such as spin configurations defined with respect to v there is a matrix of transition probabilities whether between points in v or between variables defined on v and we focus on its slow eigenvectors those with eigenvalues closest to that of the stationary eigenvector these eigenvectors are the observables and can be used to recover geometrical features of v

almost invariant sets and invariant manifolds connecting probabilistic and geometric descriptions of coherent structures in flows abstract we study the transport and mixing properties of flows in a variety of settings connecting the classical geometrical approach via invariant manifolds with a probabilistic approach via transfer operators for non divergent fluid like flows we demonstrate that eigenvectors of numerical transfer operators efficiently decompose the domain into invariant regions for dissipative chaotic flows such a decomposition into invariant regions does not exist instead the transfer operator approach detects almost invariant sets we demonstrate numerically that the boundaries of these almost invariant regions are predominantly comprised of segments of co dimension 1 invariant manifolds for a mixing periodically driven fluid like flow we show that while sets bounded by stable and unstable manifolds are almost invariant the transfer operator approach can identify almost invariant sets with smaller mass leakage thus the transport mechanism of lobe dynamics need not correspond to minimal transport the transfer operator approach is purely probabilistic it directly determines those regions that minimally mix with their surroundings the almost invariant regions are identified via eigenvectors of a transfer operator and are ranked by the corresponding eigenvalues in the order of the sets invariance or leakiness while we demonstrate that the almost invariant sets are often bounded by segments of invariant manifolds without such a ranking it is not at all clear which intersections of invariant manifolds form the major barriers to mixing furthermore in some cases invariant manifolds do not bound sets of minimal leakage our transfer operator constructions are very simple and fast to implement they require a sample of short trajectories followed by eigenvector calculations of a sparse matrix

exact travelling wave solutions for the n 1 dimensional double sine and sinh gordon equations for the n 1 dimensional double sine and sinh gordon equations by using the approach of dynamical systems to the travelling wave solutions in different regions of the parameter space all possible bounded solutions solitary wave solutions kink and anti kink wave solutions and periodic wave solutions et al are obtained explicit exact parametric representations are given

anti periodic solutions for semilinear evolution equations in this paper we use the homotopy method to establish the existence and uniqueness of anti periodic solutions for the nonlinear anti periodic problem a t x bx f t a e t r x t t x t where a t x is a nonlinear map and b is a bounded linear operator from r n to r n sufficient conditions for the existence of the solution set are presented also we consider the nonlinear evolution problems with a perturbation term which is multivalued we show that for this problem the solution set is nonempty and weakly compact in w 1 2 i r n for the case of convex valued perturbation and prove the existence theorems of anti periodic solutions for the nonconvex case all illustrative examples are provided

exact explicit traveling wave solutions for the cdf equation abstract the traveling wave solutions of the calogero degasperis fokas cdf equation are studied by using the approach of dynamical systems and the theory of bifurcations with the aid of maple all bifurcations and phase portraits in the parametric space are obtained all possible explicit parametric representations of the bounded traveling wave solutions solitary wave solutions kink and anti kink wave solutions periodic wave solutions and breaking wave solutions are given moreover the reason for appearance of breaking waves and for persistence of smoothness of smooth traveling waves is explained

bouncing solutions of an equation with attractive singularity for any n m 2 in it is proved the existence of 2m periodic solutions with n bouncings in each period for a second order forced equa tion with attractive singularity by using the approach of successor map and poincare birkho twist theorem

multiple phases in stochastic dynamics geometry and probabilities stochastic dynamics is generated by a matrix of transition probabilities certain eigenvectors of this matrix provide observables and when these are plotted in the appropriate multidimensional space the phases in the sense of phase transitions of the underlying system become manifest as extremal points this geometrical construction which we call an observable representation of state space can allow hierarchical structure to be observed it also provides a method for the calculation of the probability that an initial points ends in one or another asymptotic state

first order impulsive ordinary differential equations with anti periodic and nonlinear boundary conditions the monotone iterative technique is a powerful method that have been used to approximate the extremal solutions of several problems 1 2 on the other hand impulsive differential equations are a basic tool to study some problems of biology medicine engineering and physics 3 in this paper we deal with the following nonlinear boundary value problem for first order differential equation with impulses at fixed points

on the calogero degasperis fokas equation in 2 1 dimensions abstract in this paper we study a 2 1 dimensional integrable calogero degasperis fokas equation derivable by using a method proposed by calogero a catalogue of classical symmetry reductions are given these reductions to partial differential equations in 1 1 admit symmetries which lead to further reductions i e to second order ordinary differential equations these odes provide several classes of solutions all of them are expressible in terms of known functions some of them are expressible in terms of the second and third painleve trascendents the corresponding solutions of the 2 1 dimensional equation involve up to three arbitrary smooth functions consequently the solutions exhibit a rich variety of qualitative behaviour indeed by making appropriate choices for the arbitrary functions we exhibit solitary waves coherent structures and bound states

anti periodic oscillations of some second order differential equations and optimal control problems abstract this paper deals with periodic and anti periodic boundary conditions for wave and heat equations as well as for some second order ordinary differential equations optimal control problems associated with such equations are studied and some numerical experiments to determine the solution of the problem y t 3 sin t y t cos t 0 t y i 0 y i 0 i 0 1 are given some open problems related to the invertibility of i e ta are pointed out

an analytic framework for identifying finite time coherent sets in time dependent dynamical systems abstract the study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems barriers to transport which mitigate mixing are currently the subject of intense study in the autonomous setting the use of transfer operators perron frobenius operators to identify invariant and almost invariant sets has been particularly successful in the nonautonomous time dependent setting coherent sets a time parameterised family of minimally dispersive sets are a natural extension of almost invariant sets the present work introduces a new analytic transfer operator construction that enables the calculation of finite time coherent sets sets are that minimally dispersive over a finite time interval this new construction also elucidates the role of diffusion in the calculation and we show how properties such as the spectral gap and the regularity of singular vectors scale with noise amplitude the construction can also be applied to general markov processes on continuous state space

existence of an anti periodic solution for the quasilinear wave equation with viscosity abstract we prove the existence of a strong anti periodic solution for the quasilinear wave equation with viscosity u tt mathop rm div bigl lbrace sigma bigl abla u 2 bigr abla u bigr rbrace delta u t f x t qquad hbox in omega times r under the dirichlet boundary conditionu t 0 where is a bounded domain inrnwith the boundary and v2 is a function like 1 sqrt 1 v 2

anti periodic boundary value problems of second order impulsive differential equations this paper discusses anti periodic boundary value problems of second order impulsive differential equations by using the method of upper and lower solutions coupled with the monotone iterative technique new existence results of coupled solutions and uniqueness of problems are obtained

mixed states for an allen cahn type equation this paper continues the recent study of an allen cahn model pde 1 by eliminating a strong spatial reversibility condition and by weakening certain nondegeneracy conditions on families of basic heteroclinic solutions enabling us to obtain multibump solutions in a much more general setting as in 1 novel minimization arguments play a key role in finding solutions

reduction technique for matrix nonlinear evolution equations solvable by the spectral transform the main purpose of this paper is to describe a technique of reduction whereby from the class of evolution equations for matrices of order n solvable via the spectral transform associated to the matrix linear schrodinger eigenvalue problem one derives subclasses of nonlinear evolution equations involving less than n2 fields to illustrate the method from the equations for matrices of order 2 two subclasses of equations for 2 fields rather than 4 are obtained the first class coincides or rather includes that solvable via the spectral transform associated to the generalized zakharov shabat spectral problem further reduction to nonlinear evolution equations for a single field reproduces a number of well known equations but also yields a novel one highly nonlinear the second class also yields highly nonlinear equations some examples are given including another novel evolution equation for a single field