on the existence and multiplicity of positive periodic solutions for first order vector differential equation abstract in this paper we consider the existence and multiplicity of positive periodic solutions for first order vector differential equation x t f t x t 0 a e t 0 under the periodic boundary value condition x 0 x here is a positive constant and f 0 r n r n is a caratheodory function some existence and multiplicity results of positive periodic solutions are derived by using a fixed point theorem in cones

existence of solutions to an initial value problem for the differential equation f t x x 0 using barrier strips abstract under sign assumptions we investigate the global existence of solutions of the initial value problem f t x x x 0 a where the function f t x p may be singular at x a

remarks on first and second order periodic boundary value problems on etudie les problemes aux valeurs limites periodiques d ordre 1 et 2 suivants u f t u u 0 u 2 et u f t u u 0 u 2 u 0 u 2 on obtient l existence de solutions extremales comme limites d iterees monotones

analysis of the parareal time parallel time integration method the parareal algorithm is a method to solve time dependent problems parallel in time it approximates parts of the solution later in time simultaneously to parts of the solution earlier in time in this paper the relation of the parareal algorithm to space time multigrid and multiple shooting methods is first briefly discussed the focus of the paper is on new convergence results that show superlinear convergence of the algorithm when used on bounded time intervals and linear convergence for unbounded intervals

nonlinear boundary value problems for first order impulsive functional differential equations this paper is concerned with solutions of nonlinear boundary value problems for a class of first order functional differential equations we introduce new concept of lower and upper solutions and present that the method of lower and upper solutions coupled with monotone iterative technique is still valid obviously we extend previous results

on model complete differential fields we develop a geometric approach to definable sets in differentially closed fields with emphasis on the question of orthogonality to a given strongly minimal set equivalently within a family of ordinary differential equations we consider those equations that can be transformed by differential algebraic transformations so as to yield solutions of a given fixed first order ode x we show that this sub family is usually definable in particular if x lives on a curve of positive genus as a corollary we show the existence of many model complete superstable theories of differential fields

periodic boundary value problems for a class of functional differential equations in this paper we show that the method of upper and lower solutions coupled with the monotone iterative technique is valid to obtain constructive proofs of existence of solutions for nonlinear periodic boundary value problems of functional differential equations without assuming properties of monotonicity in the nonlinear part

conservative tightly coupled simulations of stochastic multiscale systems multiphysics problems often involve components whose macroscopic dynamics is driven by microscopic random fluctuations the fidelity of simulations of such systems depends on their ability to propagate these random fluctuations throughout a computational domain including subdomains represented by deterministic solvers when the constituent processes take place in nonoverlapping subdomains system behavior can be modeled via a domain decomposition approach that couples separate components at the interfaces between these subdomains its coupling algorithm has to maintain a stable and efficient numerical time integration even at high noise strength we propose a conservative domain decomposition algorithm in which tight coupling is achieved by employing either picard s or newton s iterative method coupled diffusion equations one of which has a gaussian white noise source term provide a computational testbed for analysis of these two coupling strategies fully converged implicit coupling with newton s method typically outperforms its picard counterpart especially at high noise levels this is because the number of newton iterations scales linearly with the amplitude of the gaussian noise while the number of picard iterations can scale superlinearly at large time intervals between two subsequent inter solver communications the solution error for single iteration explicit picard s coupling can be several orders of magnitude higher than that for implicit coupling increasing the explicit coupling s communication frequency reduces this difference but the resulting increase in computational cost can make it less efficient than implicit coupling at similar levels of solution error depending on the communication frequency of the latter and the noise strength this trend carries over into higher dimensions although at high noise strength explicit coupling may be the only computationally viable option

monotone method for second order delayed differential equations with boundary value conditions in this paper we investigate the existence of extremal solutions of boundary value problems for second order differential equations with delayed arguments we show that a lower and upper solution method has been applied successfully to such problems an one sided lipschitz condition with corresponding constants is important assumption in our research

on the solvability of a singular initial value problem x f t x x using barrier strips abstract in this paper conditions which guarantee the existence of solutions to x f t x x x 0 a are given

the method of lower and upper solutions for systems of boundary value problems the method of lower and upper solutions for second order differential system x f t x with dirichlet boundary condition x 0 x 1 0 is used to give sufficient conditions for the existence of at least one solution

ordinary differential equations with nonlinear boundary conditions of antiperiodic type abstract the quasilinearization method is successfully applied to differential problems with nonlinear boundary conditions of antiperiodic type given are sufficient conditions under which the monotone iterations converge quadratically to a unique solution of our problem the advantage of this approach lies in the fact that the approximate solutions are obtained iteratively as solutions of the above mentioned iterations

periodic boundary value problem for first order impulsive functional differential equations this paper discusses the periodic boundary value problem for a class of first order impulsive functional differential equations we establish several existence results under weaker hypotheses by using the leray schauder alternative the lower and upper solution method and the monotone iterative technique the corresponding results in the literature are improved or extended some examples are also given to illustrate the advantage of the results

positive solution for system of nonlinear first order pbvps on time scales abstract in this paper we are concerned with the following system of nonlinear first order periodic boundary value problems on time scale t x i t f i t x 1 t x 2 t x n t 0 t 0 t x i 0 x i t i 1 2 n where f i 0 t 0 n r is continuous and there exists a constant m i 0 such that m i x i f i t x 1 x 2 x n 0 for x 1 x 2 x n 0 n t 0 t some existence criteria of positive solution are established by using a fixed point theorem for operators on cone

monotone method for first order functional differential equations we study periodic boundary value problems relative to a general class of first order functional differential equations for this class of problems we develop the monotone iterative technique our formulation is very general including delay differential equations functional differential equations with maxima and integro differential equations but the case where the operator defining the functional dependence is not necessarily lipschitzian is also considered

waveform relaxation methods for periodic differential functional systems the convergence of waveform relaxation methods applied to a periodic differential functional system is studied the continuous dependence is also discussed

verlet i r respa impulse is limited by nonlinear instabilities this paper shows that in molecular dynamics md when constant energy nve simulations of newton s equations of motion are attempted using the multiple time stepping mts integrator verlet i r respa impulse there are nonlinear instabilities when the longest step size is a third or possibly a fourth of the period s of the fastest motion s in the system this is demonstrated both through a thorough set of computer experiments and through the analysis of a nonlinear model problem the numerical experiments include not only the unconstrained dynamics simulation of a droplet of flexible water and a flexible protein but also the constrained dynamics simulation of a solvated protein representing a range of simulation protocols commonly in use by biomolecular modelers the observed and predicted instabilities match exactly previous work has identified and explained a linear instability for verlet i r respa impulse at around half the period of the fastest motion mandziuk and schlick discovered nonlinear resonances in single time stepping md integrators but unstable nonlinear resonances for mts integrators are reported here for the first time this paper also offers an explanation on the instability of mts constrained molecular dynamics simulations of explicitly solvated proteins more aggressive multiple step sizes are possible with mild langevin coupling or targeted langevin coupling and its combination with the mollified impulse method permits step sizes 3 to 4 times larger thanverlet i r respa impulse while still retaining some accuracy

proper generalized decomposition of time multiscale models summary models encountered in computational mechanics could involve many time scales when these time scales cannot be separated one must solve the evolution model in the entire time interval by using the finest time step that the model implies in some cases the solution procedure becomes cumbersome because of the extremely large number of time steps needed for integrating the evolution model in the whole time interval in this paper we considered an alternative approach that lies in separating the time axis one dimensional in nature in a multidimensional time space then for circumventing the resulting curse of dimensionality the proper generalized decomposition was applied allowing a fast solution with significant computing time savings with respect to a standard incremental integration copyright 2011 john wiley sons ltd

on discontinuous first order implicit boundary value problems t 0 1 2 under various boundary conditions cf e g 1 5 8 13 29 31 32 thereis little known about the existence of solutions of 1 2 even in the casewhen f is continuous unless it is possible to solve the equation 1 2 forthe highest derivative and then to apply standard existence theory such asleray schauder theory one approach taken for instance in 8 16 is toreduce equation 1 2 to a differential inclusion of the formu

a generalized upper and lower solutions method and multiplicity results for nonlinear first order ordinary differential equations abstract we introduce a generalized upper and lower solutions method for the solvability of first order ordinary differential equations u t t u t u 0 u 1 in order to cover the case when the function satisfies caratheodory conditions this method is then applied to get multiplicity results when the nonlinearity interacts with the real eigenvalue of the linearized problem our proofs are based on differential inequalities and classical leray schauder degree

periodic boundary value problem for first order impulsive delay differential equations in this paper by means of the method of upper and lower solutions and the monotone iterative technique the existence of maximal and minimal solutions of the periodic boundary value problem for first order impulsive delay differential equations is considered

on algebraic relations between solutions of a generic painleve equation we prove that if y 00 f y y 0 t is a generic painlev e equation that is an equation in one of the families pi pv i but with the relevant complex parameters algebraically independent and if y1 yn are solutions such that y1 y 0 y2 y 0 yn y 0 n are algebraically dependent over c t then already for some 1 i j n yi y 0 yj y 0 are algebraically dependent over c t the proof combines results by the japanese school on irreducibility of the painlev e equations with the trichotomy theorem for strongly minimal sets in dierentially closed elds

periodic boundary value problems for impulsive integro differential equations of mixed type by developing a comparison result and using the monotone iterative technique we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for first order nonlinear impulsive integro differential equations of mixed type

existence and approximation of solutions for nonlinear functional differential equations with periodic boundary value conditions abstract this paper is concerned with the existence and approximation of solutions for a class of first order functional differential equations with periodic boundary conditions we present a new comparison result and extend previous results

finiteness theorems on hypersurfaces in partial differential algebraic geometry abstract hrushovski s generalization and application of jouanolou 1978 9 is here refined and extended to the partial differential setting with possibly nonconstant coefficient fields in particular it is shown that if x is a differential algebraic variety over a partial differential field f that is finitely generated over its constant field f 0 then there exists a dominant differential rational map from x to the constant points of an algebraic variety v over f 0 such that all but finitely many codimension one subvarieties of x over f arise as pull backs of algebraic subvarieties of v over f 0 as an application it is shown that the algebraic solutions to a first order algebraic differential equation over c t are of bounded height answering a question of eremenko two expected model theoretic applications to dcf 0 m are also given 1 lascar rank and morley rank agree in dimension two and 2 dimension one strongly minimal sets orthogonal to the constants are 0 categorical a detailed exposition of hrushovski s original unpublished theorem is included influenced by ghys 2000 5

singular and nonsingular first order initial value problems abstract under barrier strip type arguments we investigate the existence of global solutions to the initial value problem x f t x x x 0 a where the scalar function f t x p may be singular at x a

periodic boundary value problems for second order impulsive integro differential equations by developing a new comparison result and using the monotone iterative technique we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for second order nonlinear impulsive integro differential equations of mixed type

on the cauchy problem for first order discontinuous ordinary differential equations abstract we prove the existence of extremal caratheodory solutions for the initial value problem x t f t x t for a e t 0 1 x 0 0 under weak conditions over f 0 1 r r

nonlinear second order periodic boundary value problems abstract in this paper we study a periodic boundary value problem pbvp for brevity of second order it is well known that if and are upper and lower solutions respectively with then there exists a solution u such that u when nothing was known the main purpose of this paper is to consider this last case also we study the structure of the set of solutions of the pbvp in the sector when and we apply the results to the case

the monotone method for first order problems with linear and nonlinear boundary conditions in this paper we develop the monotone method for the first order problem u t f t u t for a e t a b when f is a caratheodory function and u w 1 1 a b we consider the nonlinear boundary conditions l u a u b 0 with l c r 2 r nondecreasing in x or nonincreasing in y and the linear boundary conditions a 0u 0 b 0u t l 0 with a 0 b 0 and l 0 r we prove the existence of solutions and the validity of the monotone method if there exists a lower solution a and an upper solution b with either a b for the linear conditions we obtain eight new concepts of lower and upper solutions which generalize previous known cases

remarks on periodic boundary value problems for functional differential equations we extend some results on existence and approximation of solution for a class of first order functional differential equations with periodic boundary conditions we show the validity of the monotone iterative technique under weaker hypotheses and present some examples

periodic boundary value problems for delay differential equations with impulses existence and approximation of solutions for an impulsive delay differential equation with periodic boundary value conditions are presented we use comparison principles and monotone iterative techniques

periodic boundary value problem for first order impulsive functional differential equations in this paper by means of the method of upper and lower solutions and the monotone iterative technique the existence of minimal and maximal solutions of the periodic boundary value problem for first order impulsive functional differential equations is considered

an iterative method for functional differential equations a suitable periodic boundary conditions for a functional differential equations email protected t f t x x t are conditions of the form x 0 q x 2 p q in this paper we use the notion of upper and lower solutions coupled with monotone iterative method to prove the existence of solutions of this periodic boundary value problem

existence and approximation of solutions for discontinuous functional differential equations in this paper existence of solutions of initial value problems for discontinuous functional differential equations is investigated firstly by applying the method of upper and lower solutions which may be discontinuous some existence results of extremal solutions are obtained furthermore we also develop a monotone iterative technique for obtaining extremal solutions which are obtained as limits of monotone sequences

an existence result on noncompact intervals for second order functional differential inclusions in this paper we investigate the existence of mild solutions on an infinite interval for second order functional differential inclusions in banach spaces we shall make use of a theorem of ma which is an extension to multivalued maps on locally convex topological spaces of schaefer s theorem

studies on the painlev equations iii second and fourth painlev equations p ii andp iv etude des transformations birationnelles canoniques du systeme de painleve associe a chacune des equations

monotone iterative techniques of v lakshmikantham for a boundary value problem for systems of impulsive differential difference equations by means of the monotone iterative techniques of v lakshmikantham a couple of minimal and maximal quasisolutions of a two point boundary value problem for a system of impulsive differential difference equations are constructed

periodic boundary value problem for non lipschitzian impulsive functional differential equations we consider a class of first order impulsive functional differential equations where the functional dependence is not necessarily a lipschitzian function the new maximum principle improves and extends previous results and uniqueness of solution between a lower and an upper solution for a particular nonlinear problem is presented we give conditions for existence of extremal solutions in an interval delimited by a lower and an upper solution

positive solutions for first order periodic boundary value problem this paper discusses the existence and multiplicity of positive solutions for the following first order periodic boundary value problemx t f t x t 0 0

periodic solutions of nonlinear boundary value problems one of the well known techniques in the theory of nonlinear boundary value problems bvp is the method of differential inequalities or the method of upper and lower solutions the method of alternative problems a global variant of the lyapunov schmidt method has been used in the study of problems at resonance the investigation of periodic bvp s forms an important subclass of problems at resonance our aim is to combine the two approaches to discuss nonlinear problems at resonance we restrict ourselves in this paper to the discussion of periodic boundary value problems in section 1 we shall indicate the method of upper and lower solutions the results for systems are given in a general way so as to include known results and also offer new directions section 2 deals with the abstract existence results at resonance in the desired framework in section 3 we consider the scalar periodic bvp s by combining the two techniques namely alternative methods and the method of differential inequalities

a monotone method for the equation uxyz f comparison theorems for the nonlinear boundary value problem u x y z f x y z u u x u y u z u x y u y z u z x with the boundary conditions u o y z a y z u x 0 z b z x u x y 0 c x y a y 0 c 0 y a 0 z b z 0 b 0 x c x 0 are established under certain conditions then it is shown that there exist sequences of lower and upper solutions which converge uniformly and monotonically to minimal and maximal solutions respectively

upper and lower solutions with jumps abstract we study a periodic boundary value problem for a first order differential equation from a new point of view which permits us to obtain an existence result involving upper and lower solutions with jump discontinuities we present some applications of our result with illustrative examples

discontinuous first order functional boundary value problems abstract in this paper we prove a new existence result for functional boundary value problems with first order functional differential equations under weak conditions on the nonlinear part and monotonicity but not continuity with respect to the functional variable discontinuities with respect to the unknown are allowed to occur over general classes of time dependent sets which are assumed to satisfy a kind of unviability or solubility condition

homogenization of hamiltonian systems with a strong constraining potential abstract the paper studies hamiltonian systems with a strong potential forcing the solutions to oscillate on a very small time scale in particular we are interested in the limit situation where the size of this small time scale tends to zero but the velocity components remain oscillating with an amplitude variation of the order o 1 the process of establishing an effective initial value problem for the limit positions will be called homogenization of the hamiltonian system this problem occurs in mechanics as the problem of realization of holonomic constraints as various singular limits in fluid flow problems in plasma physics as the problem of guiding center motion and in the simulation of biomolecules as the so called smoothing problem we suggest the systematic use of the notion of weak convergence in order to approach this problem this methodology helps to establish unified and short proofs of many known results which throw light on the inherent structure of the problem moreover we give a careful and critical review of the literature

convergent iterative schemes for time parallelization parallel methods are usually not applied to the time domain because of the inherit sequentialness of time evolution but for many evolutionary problems computer simulation can benefit substantially from time parallelization methods in this paper we present several such algorithms that actually exploit the sequential nature of time evolution through a predictor corrector procedure this sequentialness ensures convergence of a parallel predictor corrector scheme within a fixed number of iterations the performance of these novel algorithms which are derived from the classical alternating schwarz method are illustrated through several numerical examples using the reservoir simulator athena

existence of solutions of boundary value problems for differential equations with delayed arguments we study the existence and approximation of solution of boundary value problems for differential equations with delayed arguments sufficient conditions are established for the existence of a unique solution or extremal ones of the given problem a monotone iterative technique is applied

existence of periodic solutions for fully nonlinear first order differential equations in this paper we establish some existence results for periodic solutions of fully nonlinear first order scalar differential equation f t x x 0 the approach here uses the viscosity solutions method developed in recent decades and the classical upper lower solutions method

extremal solutions of measure differential equations abstract we investigate the existence of the least and greatest solutions to measure differential equations as well as the relation between the extremal solutions and lower or upper solutions along the way we prove a fairly general local existence theorem and an analogue of peano s uniqueness theorem for measure differential equations finally we show that the general theory is applicable in the study of differential equations with impulses or dynamic equations on time scales

on monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations abstract in this paper we show that the method of monotone iterative technique is valid to obtain two monotone sequences that converge uniformly to extremal solutions of first and second order periodic boundary value problems and periodic solutions of functional differential equations we obtain some new results relative to the lower solution and upper solution with

monotone iterative method for first order functional difference equations with nonlinear boundary value conditions in this paper we show that the method of monotone iterative coupled with the upper and lower solutions is valid to obtain constructive proofs to the existence of solutions for first order functional difference equations with nonlinear boundary value conditions an example is given to illustrate the results obtained

time parallel multiscale multiphysics framework we introduce the time parallel compound wavelet matrix method tpcwm for modeling the temporal evolution of multiscale and multiphysics systems the method couples time parallel tp and cwm methods operating at different spatial and temporal scales we demonstrate the efficiency of our approach on two examples a chemical reaction kinetic system and a non linear predator prey system our results indicate that the tpcwm technique is capable of accelerating time to solution by 2 3 orders of magnitude and is amenable to efficient parallel implementation

approximation of solutions for nonlinear periodic boundary value problems with discontinuous upper and lower solutions we prove that it is possible to approximate the solutions of a nonlinear periodic boundary value problem via monotone sequences starting at piecewise continuous upper and lower solutions in fact we present here two different iterative methods explicit expression of each iteration is obtained by solving some linear initial value problems

discontinuous functional differential equations with delayed or advanced arguments we provide new results on the existence of extremal solutions for discontinuous differential equations with a deviated argument which can be either delayed or advanced the boundary condition is allowed to be discontinuous and to depend functionally on the unknown solution

extremality and comparison results for discontinuous third order functional initial boundary value problems extremality and comparison results are derived for third order functional initial boundary value problems differential equations of problems may involve discontinuous nonlinearities

advanced differential equations with nonlinear boundary conditions we investigate the existence of solutions for advanced differential equations with nonlinear boundary conditions sufficient conditions when the problem has extremal solutions or a unique solution are formulated linear advanced differential inequalities are also discussed

long time step methods for oscillatory differential equations considered are numerical integration schemes for nondissipative dynamical systems in which multiple time scales are present it is assumed that one can do an explicit separation of the rhs forces into fast forces and slow forces such that i the fast forces contain the high frequency part of the solution ii the fast forces are conservative and iii the reduced problem consisting only of the fast forces can be integrated much more cheaply than the full problem the fast forces are allowed to have low frequency components particular applications for which the schemes are intended include n body problems for which most of the forces are slow and nonlinear wave phenomena for which the fast forces can be propagated by spectral methods the assumption of cheap integration of fast forces implies that the overall cost of integration is primarily determined by the step size used to sample the slow forces a long time step method is one in which this step size exceeds half the period of the fastest normal mode present in the full system an existing method that comes close to qualifying is the impulse method also known as verlet i and r respa it is shown that it might fail though for a couple of reasons first it suffers a serious loss of accuracy if the step size is near a multiple of the period of a normal mode and second it is unstable if the step size is near a multiple of half the period of a normal mode proposed in this paper is a mollified impulse method having an error bound that is independent of the frequency of the fast forces it is also shown to possess superior stability properties theoretical results are supplemented by numerical experiments the method is efficient and reasonably easy to implement

extremality results for first order discontinuous functional differential equations abstract in this paper we derive extremality and comparison results for first order explicit and implicit functional differential equations equipped with functional initial conditions differential equations and initial conditions may involve discontinuities

continuous dependence of nonmonotonic discontinuous differential equations continuous dependence of solutions for a class of nonmonotonic discontinuous differential equations is studied first a local existence theorem due to z wu is extended to a larger class then a result concerning continuous dependence for this larger class is proven this employs a type of convergence similar to gihman s convergence criterion witch is defined to be for all a b and y formule the significance of gihman s convergence criterion is that for certain classes of differential equations it has been found to be necessary and sufficient for continuous dependence finally examples are presented to motivate and clarify this continuous dependence result

existence of solutions to initial value problems for first order differential equations under sign assumptions we investigate the global existence of solutions of the initial value problem f t x x 0 x a a where the function f t x p may be singular at t a