third order boundary value problems with nonlocal boundary conditions we treat the existence of multiple positive solutions for a third order boundary value problem with nonlocal boundary conditions in our work each of the three boundary conditions is a linear functional on c 0 1 that is given by a stieltjes integral but is not necessarily a positive functional

multiple solutions to a three point boundary value problem for higher order ordinary differential equations in this paper we provide sufficient conditions for the existence of at least three solutions to a three point boundary value problem for higher order ordinary differential equations the nonlinear term f in the differential equation under consideration may depend on higher order derivatives of arbitrary order and this is where the main novelty of this work lies by applying the two pairs of upper and lower solutions method of henderson and thompson as well as degree theory the existence of at least three solutions of the problem is given

positive solutions to differential inclusions with nonlocal nonlinear boundary conditions we demonstrate the existence of at least one positive solution to the differential inclusion u t f t u t email protected 0 1 equipped with the boundary conditions u 0 h f u and u 1 0 here f is linear functional realized as a lebesgue stieltjes integral and h is a nonlinear function consequently the boundary condition at t 0 can be both nonlocal and nonlinear by imposing an asymptotic condition on h we provide conditions under which the problem will have at least one positive solution

on boundary value problems of second order perturbed dynamic inclusions on time scales abstract in this paper an existence criterion is established for a class of second order perturbed dynamic inclusions on time scales with boundary conditions the main approach is a fixed point theorem for multi valued maps due to dhage

coupled mode equations and gap solitons in higher dimensions abstract we study wave packets in nonlinear periodic media in arbitrary d spatial dimension modeled by the cubic gross pitaevskii equation in the asymptotic setting of small and broad wave packets with n n carrier bloch waves the effective equations for the envelopes are first order coupled mode equations cmes we provide a rigorous justification of the effective equations the estimate of the asymptotic error is carried out in an l 1 norm in the bloch variables this translates to a supremum norm estimate in the physical variables in order to investigate the existence of gap solitons of the d dimensional cmes we discuss spectral gaps of the cmes for n 4 and d 2 a family of time harmonic gap solitons is constructed formally asymptotically and numerically moving gap solitons have not been found for d 1 and for the considered values of n due to the absence of a spectral gap in the standard moving frame variables

brownian motion in a poissonian potential we study here ad dimensional brownian motion in a random potentialv obtained as the sum of translations of a given fixed non negative shape function at the points of a poisson cloud of constant intensityv we are interested in the larget behavior for typical cloud configurations of the brownian path in timet under the influence of the natural feynman kac weight associated tov in particular we show that the location at timet of the process tends to be concentrated near points of suitably low local eigenvalue of 1 2 v which lie almost at distancet from the origin near these points one can find in the cloud a big hole or clearing of size const logt 1 d with volume like a ball of radiusr 0 d v logt 1 d

a note on the diffusion of directed polymers in a random environment a simple martingale argument is presented which proves that directed polymers in random environments satisfy a central limit theorem ford 3 and if the disorder is small enough this simplifies and extends an approach by j imbrie and t spencer

twin monotone positive solutions to a singular nonlinear third order differential equation abstract this paper discusses the existence of at least one or two nondecreasing positive solutions for the following singular nonlinear third order differential equation x t t f t x t 0 a t b x a x a x b 0 where 0 is a parameter c a b r f c a b 0 r t may be singular at t a b and f t s may be singular at s 0 green s function and the fixed point theorem of cone expansion and compression type are utilized to illustrate the results four nontrivial examples are included

exponential decay rate of survival probability in a disastrous random environment we study the exponential decay rate of the survival probability up to time t 0 of a random walker moving in zopf d in a temporally and spatially fluctuating random environment when the random walker has a speed parameter 0 we investigate the influence of on the exponential decay rate d in particular we prove that for any fixed d 1 d behaves like as log as 0

positive solutions to a system of second order nonlocal boundary value problems abstract in this paper we study the existence of positive solutions to a system of second order nonlocal boundary value problems by using fixed point index theory in a cone our hypotheses imposed on nonlinearities are those which characterize systems of nonlocal boundary value problems and our boundary value conditions are expressed in terms of possibly nonlinear functions of riemann stieltjes integrals thus generalizing and unifying the boundary value conditions in the literature therefore our results cannot be routinely deduced from the ones for single nonlocal problem in the literature

split step methods for the solution of the nonlinear schro dinger equation a split step method is used to discretize the time variable for the numerical solution of the nonlinear schrodinger equation the space variable is discretized by means of a finite difference and a fourier method these methods are analyzed with respect to various physical properties represented in the nls in particular it is shown how a conservation law dispersion and instability are reflected in the numerical schemes analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived these results are demonstrated by numerical examples

positive solutions of nonlinear singular third order two point boundary value problem abstract in this paper we are concerned with the existence of single and multiple positive solutions to the nonlinear singular third order two point boundary value problem u t a t f u t 0 0 t 1 u 0 u 0 u 1 0 where is a positive parameter under various assumptions on a and f we establish intervals of the parameter which yield the existence of at least one at least two and infinitely many positive solutions of the boundary value problem by using krasnoselskii s fixed point theorem of cone expansion compression type

justification of the coupled mode asymptotics for localized wavepackets in the periodic nonlinear schrodinger equation abstract we consider wavepackets composed of two modulated carrier bloch waves with opposite group velocities in the one dimensional periodic nonlinear schrodinger gross pitaevskii equation these can be approximated by first order coupled mode equations cmes for the two slowly varying envelopes under a suitably selected periodic perturbation of the periodic structure the cmes possess a spectral gap of the corresponding spatial operator and allow families of exponentially localized solitary waves parametrized by velocity this leads to a family of approximate solitary waves in the periodic nonlinear schrodinger equation besides a formal derivation of the cmes a rigorous justification of the approximation and an error estimate in the supremum norm are provided several numerical tests corroborate the analysis

brownian asymptotics in a poissonian environment we consider a brownian motion moving in a random potential obtained by translating a given fixed non negative shape function at the points of a poisson cloud we derive the almost sure principal long time behavior of the expectation of the natural feynman kac functional which is insensitive to the detail of the shape function we also study the situation of hard obstacles where brownian motion is killed once it comes within distancea of a point of the cloud the nature of the results then changes between the case whena is small or large in connection with the presence or absence of an infinite component in the complement of the obstacles

limiting results for the free energy of directed polymers in random environment with unbounded jumps we study asymptotic behaviors of the free energy for the directed polymer in random environment the polymer is allowed to make unbounded jumps and the environment is given by the bernoulli variables we first establish the existence and continuity including the negative infinity value of the coupling constant our proof of existence at differs from existing ones in that it avoids the direct use of subadditivity secondly we identify the asymptotics of the free energy at in the limit of the success probability of the bernoulli variables tending to one it is described by using the so called time constant of a certain directed first passage percolation our proof relies on a certain continuity property of the time constant which is of independent interest

on the stochastic heat equation with spatially colored random forcing we consider the stochastic heat equation of the following form t ut x lut x b ut x ut x t x for t 0 x r where l is the generator of a levy process and is a spatially colored temporally white gaussian noise we will be concerned mainly with the long term behavior of the mild solution to this stochastic pde for the most part we work under the assumptions that the initial data u0 is a bounded and measurable function and is nonconstant and lipschitz continuous in this case we find conditions under which the preceding stochastic pde admits a unique solution which is also weakly intermittent in addition we study the same equation in the case that lu is replaced by its massive dispersive analogue lu u where r and we describe accurately the effect of the parameter on the intermittence of the solution in the case that u is proportional to u the parabolic anderson model we also look at the linearized version of our stochastic pde that is the case when is identically equal to one any other constant works also in this case we study not only the existence and uniqueness of a solution but also the regularity of the solution when it exists and is unique

three solutions of an nth order three point focal type boundary value problem abstract we consider the following differential equation u n t f t u t t 0 1 together with the three point focal type boundary conditions u j 0 0 0 j n 3 u n 2 p 0 u n 1 1 0 where 1 2 p 1 by using two different fixed point theorems we offer criteria for the existence of three positive solutions of this problem examples are also included to illustrate the results obtained

positive solutions to boundary value problems with nonlinear boundary conditions abstract in this paper we consider the boundary value problem y t f t y t subject to the boundary conditions y a y and y 2 b 0 in this setting c rd a 2 b t r r is a continuous functional which represents a nonlinear nonlocal boundary condition by imposing sufficient structure on and the nonlinearity f we deduce the existence of at least one positive solution to this problem the novelty in our setting lies in the fact that may be strictly nonpositive for some y 0 our results are achieved by appealing to the krasnosel ski fixed point theorem we conclude with several examples illustrating our results and the generalizations that they afford

a note on multi point boundary value problems abstract multi point boundary value problems for a second order ordinary differential equation are considered in this note an existence result is obtained with the help of coincidence degree theory

positive solutions of a second order integral boundary value problem in this paper we study the existence of positive solutions of a second order integral boundary value problems for ordinary differential equations our results presented here unify generalize and substantially improve the existing results in the literature moreover it is worthwhile to point out that our method will dispense with constructing a new green function

lyapunov exponent for the parabolic anderson model with levy noise we consider the asymptotic almost sure behavior of the solution of the equation where y x x z d is a field of independent levy processes and is the discrete laplacian

comparison of partition functions in a space time random environment let z 1 and z 2 be partition functions in the random polymer model in the same environment but driven by different underlying random walks we give a comparison in concave stochastic order between z 1 and z 2 if one of the random walks has more randomness than the other we also treat some related models the parabolic anderson model with space time levy noise brownian motion among space time obstacles and branching random walks in space time random environments we also obtain a necessary and sufficient criterion for z 1 preceq cv z 2 if the lattice is replaced by a regular tree

on nonlinear boundary conditions satisfying certain asymptotic behavior abstract in this paper we consider a second order boundary value problem of the form y t a t g y t y 0 y y 1 0 we demonstrate that if the nonlinear functional y satisfies certain asymptotic behavior then the problem will possess at least one positive solution these results generalize and improve on some recent results in the literature on boundary value problems with nonlocal nonlinear boundary conditions and we provide some examples which illustrate these generalizations and improvements

on large deviations for the parabolic anderson model the focus of this article is on the different behavior of large deviations of random functionals associated with the parabolic anderson model above the mean versus large deviations below the mean the functionals we treat are the solution u x t to the spatially discrete parabolic anderson model and a functional an which is used in analyzing the a s lyapunov exponent for u x t both satisfy a law of large numbers with lim t to infty frac 1 t log u x t lambda kappa and lim n to infty frac a n n alpha we then think of n and t as being the mean of the respective quantities an and log u t x typically the large deviations for such functionals exhibits a strong asymmetry large deviations above the mean take on a different order of magnitude from large deviations below the mean we develop robust techniques to quantify and explain the differences

three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second order ordinary differential equations abstract in this paper we consider the existence of at least three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second order ordinary differential equations the associated green s function for the boundary value problems is first given the proofs of our main results are based upon the leggett williams fixed point theorem finally we give an example to demonstrate our result

interaction of modulated pulses in the nonlinear schrodinger equation with periodic potential abstract we consider a cubic nonlinear schrodinger equation with periodic potential in a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several bloch bands is studied the notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic interaction of these pulses

traveling solitary waves in the periodic nonlinear schrodinger equation with finite band potentials this paper studies asymptotics of moving gap solitons in nonlinear periodic structures of finite contrast deep grating within the one dimensional periodic nonlinear schrodinger equation pnls periodic structures described by a finite band potential feature transversal crossings of band functions in the linear band structure and a periodic perturbation of the potential yields new small gaps novel gap solitons with o 1 velocity despite the deep grating are presented in these gaps an approximation of gap solitons is given by slowly varying envelopes which satisfy a system of generalized coupled mode equations gcme and by bloch waves at the crossing point the eigenspace at the crossing point is two dimensional and it is necessary to select bloch waves belonging to the two band functions this is achieved by an optimization algorithm traveling solitary wave solutions of the gcme then result in nearly solitary wave solutions of the pnls moving at an o 1 velocity across the periodic structure

nonnegative solutions to nonlocal boundary value problems for systems of second order differential equations dependent on the first order derivatives abstract in this paper we consider nonlocal boundary value problems for systems of second order differential equations with dependence on the first order derivatives and deviating arguments by using a fixed point theorem due to avery and peterson sufficient conditions are obtained that guarantee the existence of at least three nonnegative solutions to such problems we investigate our problem both for delayed and advanced arguments i i and also for the case when i t i t t t 0 1 in all cases arguments i i can change the character on 0 1 so in some subinterval i of 0 1 they can be delayed in i and advanced in 0 1 i some remarks concern also the case when differential equations do not depend on the first order derivatives examples illustrate some results