on a class of singular elliptic systems we study the semilinear elliptic system u u p v q u 0in v u r v s v 0in u v 0on where rn n 1 is a smooth and bounded domain p q r s 0 under suitable ranges of exponents we obtain various results regarding the well posedness of our system

existence and comparison results for difference φ laplacian boundary value problems with lower and upper solutions in reverse order abstract this paper is devoted to the study of the existence and comparison results for nonlinear difference laplacian problems with mixed dirichlet neumann and periodic boundary value conditions we deduce existence of extremal solutions of periodic and neumann boundary value problems lying between a pair of lower and upper solutions given in reverse order we prove the optimality of some assumptions in

on the concept of general solution for impulsive differential equations of fractional order q 1 2 in this paper we provide the formulas of general solution for some impulsive differential equations of fractional order q 1 2

optimal feedback control for semilinear fractional evolution equations in banach spaces abstract in this paper we study optimal feedback controls of a system governed by semilinear fractional evolution equations via a compact semigroup in banach spaces by using the cesari property the fillippove theorem and extending the earlier work on fractional evolution equations we prove the existence of feasible pairs an existence result of optimal control pairs for the lagrange problem is presented

impulsive fractional functional differential equations in this paper a class of impulsive fractional functional differential equations is investigated the first purpose is introducing a natural formula of solutions for impulsive fractional functional differential equations the second purpose is establishing related new existence uniqueness and data dependence results examples are given to illustrate the results

existence and uniqueness of solutions for impulsive fractional differential equations motivated by some recent developments in the existence theory of impulsive fractional differential equations in this paper we present a general method for converting an impulsive fractional differential equation to an equivalent integral equation the applicability of the method is demonstrated by considering some boundary value problems for impulsive fractional differential equations

picard vessiot extensions of artinian simple module algebras abstract this paper pursues takeuchi s hopf algebraic approach m takeuchi a hopf algebraic approach to the picard vessiot theory j algebra 122 1989 481 509 to the picard vessiot pv theory for differential equations to involve the pv extensions of difference equations differential fields and total difference rings in the standard pv theory are unified here by artinian simple as module algebras over a cocommutative pointed smooth hopf algebra

a generalized gronwall inequality and its application to a fractional differential equation this paper presents a generalized gronwall inequality with singularity using the inequality we study the dependence of the solution on the order and the initial condition of a fractional differential equation

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

a note on extending actions of infinitesimal group schemes abstract we prove that iterative derivations on projective line cannot be expanded to iterative hasse schmidt derivations in the case when the iterativity rule is given by a non algebraic formal group

regularized solutions for abstract volterra equations abstract we introduce a two kernel dependent family of strong continuous operators defined in a banach space which allows us to consider in an unified treatment the notions of among others c 0 semigroups of operators cosine families n times integrated semigroups resolvent families and k generalized solutions the results are applied to the study of existence and uniqueness of solutions for the volterra equation of convolution type u f a au in the case a is not necessarily densely defined examples for equations defined in l p spaces are also given

steady state solutions for a general activator inhibitor model abstract we discuss existence results for a singular gierer meinhardt elliptic system with zero dirichlet boundary conditions which originally arose in studies of pattern formation in biology and has interesting and challenging mathematical properties the mathematical difficulties are that the system becomes singular near the boundary and it is non quasimonotone we show the existence of positive solutions for the general activator inhibitor model

a hopf algebraic approach to the picard vessiot theory abstract this paper shows what is really happening in picard vessiot theory our hopf algebraic approach simplifies and generalizes the theory among other things the generalization gives a picard vessiot theory for positive characteristic for fields with not necessarily commuting derivations and even for fields with a set of higher derivations

uniformly bounded weak and classical solutions to a singular parabolic system and applications abstract equations of both elliptic and parabolic type featuring singular nonlinearities have appeared in numerous works throughout the years in this work we consider a time dependent problem featuring nonlinearities of the form u p v q u r v s subject to homogeneous dirichlet boundary conditions and prove the existence of uniformly bounded weak and classical solutions under appropriate conditions on p q r s these results can in some ways be seen as a generalization of the results presented in 9 these results are obtained using a functional method motivated by works found in 8 22 6 etc and the boundary behaviour of a fundamental singular elliptic equation described in 16

impulsive fractional partial differential equations this paper deals with cauchy problem for a class of impulsive partial hyperbolic differential equations involving the caputo derivative our first purpose is to show that the formula of solutions in cited papers are incorrect next we reconsider a class of impulsive fractional partial hyperbolic differential equations and introduce a correct formula of solutions for cauchy problem in r n further some sufficient conditions for existence of the solutions are established by applying fixed point method at last we consider the cauchy problem in a banach space via the technique of measures of noncompactness and monch s fixed point theorem some examples are given to illustrate our results

upper and lower solution methods for fully nonlinear boundary value problems abstract sufficient conditions are given for the existence of a solution of a fourth order nonlinear boundary value problem with nonlinear boundary conditions the conditions assume the existence of a strong upper solution lower solution pair a concept that is defined in the paper the differential equation has nonlinear dependence on all lower order derivatives of the unknown in particular appropriate nagumo conditions are obtained and employed

existence and uniqueness of solutions of second order three point boundary value problems with upper and lower solutions in the reversed order abstract this paper studies the existence and uniqueness of solutions of second order three point boundary value problems with lower and upper solutions in the reversed order obtains the sufficient conditions for the existence and uniqueness of solutions by use of the monotone iterative method and gives the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution

weighted pseudo almost periodic functions and applications to evolution equations with delay in this paper we introduce a new class of vector valued weighted pseudo almost periodic functions and present some basic properties about these new functions moreover we obtain an new existence theorem of vector valued weighted pseudo almost periodic solution to the evolution equations x t ax t f t x t with delay in a banach space x

periodic boundary value problem for the second order impulsive functional differential equations this paper considers the existence of extreme solutions of the periodic boundary value problem for second order functional differential equations by using the method of lower and upper solutions coupled with monotone iterative technique

multipoint bvps for generalized impulsive fractional differential equations in this paper we study multipoint bvps for a class of generalized impulsive fractional order nonlinear differential equations we derive two new existence results under different conditions on the nonlinear terms via classical and generalized mittag leffler functions and fixed point methods finally four examples are given to illustrate our main results

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

the differential galois theory of strongly normal extensions differential galois theory the theory of strongly normal extensions has unfortunately languished this may be due to its reliance on kolchin s elegant but not widely adopted axiomatization of the theory of algebraic groups this paper attempts to revive the theory using a differential scheme in place of those axioms we also avoid using a universal differential field instead relying on a certain tensor product we identify automorphisms of a strongly normal extension with maximal differential ideals of this tensor product thus identifying the galois group with the closed points of an affine differential scheme moreover the tensor product has a natural coring structure which translates into the galois group operation composition of automorphisms this affine differential scheme splits i e is obtained by base extension from a not differential not necessarily affine group scheme as a consequence the galois group is canonically isomorphic to the closed or rational points of a group scheme defined over constants we obtain the fundamental theorem of differential galois theory giving a bijective correspondence between subgroup schemes and intermediate differential fields on the way to this result we study certain aspects of differential algebraic geometry e g closed immersions products local ringed space of constants and split differential schemes

differential arcs and regular types in differential fields we introduce differential arc spaces in analogy to the algebraic arc spaces and show that a differential variety is determined by its arcs at a point using differential arcs we show that if k 1 n is a differentially closed field of characteristic zero with n commuting derivations and p s k is a regular type over k then either p is locally modular or there is a definable subgroup g k of the additive group having a regular generic type that is nonorthogonal to p

impulsive problems for fractional differential equations with boundary value conditions we consider impulsive problems for a fractional differential equation with boundary value conditions natural formulae of a solution for our problem are introduced which can be regarded as a novelty item some related new existence results are established by applying fixed point methods via a generalized singular gronwall inequality examples are given to illustrate the results

on algebraic sigma groups we introduce the categories of algebraic varieties and groups over a difference field k under a linearly closed assumption on k a we prove an isotriviality theorem for groups this theorem immediately yields the key lemma in a proof of the manin mumford conjecture the present paper crucially uses ideas of pilay and ziegler 2003 but in a model theory free manner the applications to manin mumford are inspired by hrushovski s work 2001 and are also closely related to papers of pink and roessler 2002 and 2004

boundary value problems involving upper and lower solutions in reverse order in this paper we discuss the existence of extreme solutions of the boundary value problem for a class of first order functional equations with a nonlinear boundary condition in the presence of a lower solution a and an upper solution b with email protected email protected we establish existence results of extreme solutions by using the method of upper and lower solutions and a monotone iterative technique

boundary value problems for a class of impulsive functional equations this paper is related to the existence and approximation of solutions for impulsive functional differential equations with periodic boundary conditions we study the existence and approximation of extremal solutions to different types of functional differential equations with impulses at fixed times by the use of the monotone method some of the options included in this formulation are differential equations with maximum and integro differential equations in this paper we also prove that the lipschitzian character of the function which introduces the functional dependence in a differential equation is not a necessary condition for the development of the monotone iterative technique to obtain a solution and to approximate the extremal solutions to the equation in a given functional interval the corresponding results are established for the impulsive case the general formulation includes several types of functional dependence delay equations equations with maxima integro differential equations finally we consider the case of functional dependence which is given by nonincreasing and bounded functions

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

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

on a new class of impulsive fractional differential equations in this paper we consider fractional ordinary differential equations with not instantaneous impulses firstly we construct a uniform framework to derive a formula of solutions for impulsive fractional cauchy problem involving generalization of classical caputo derivative with the lower bound at zero in other words we mean a different solution keeping in each impulses the lower bound at zero which can better characterize the memory property of fractional derivative secondly we introduce a new concept of generalized ulam hyers rassias stability then we choose a fixed point theorem to derive a generalized ulam hyers rassias stability result for such new class of impulsive fractional differential equations finally an example is given to illustrate our main results

existence of solutions for nonlinear fractional stochastic differential equations abstract the fractional stochastic differential equations have wide applications in various fields of science and engineering this paper addresses the issue of existence of mild solutions for a class of fractional stochastic differential equations with impulses in hilbert spaces using fractional calculations fixed point technique stochastic analysis theory and methods adopted directly from deterministic fractional equations new set of sufficient conditions are formulated and proved for the existence of mild solutions for the fractional impulsive stochastic differential equation with infinite delay further we study the existence of solutions for fractional stochastic semilinear differential equations with nonlocal conditions examples are provided to illustrate the obtained theory

germs of arcs on singular algebraic varieties and motivic integration we study the scheme of formal arcs on a singular algebraic variety and its images under truncations we prove a rationality result for the poincare series of these images which is an analogue of the rationality of the poincare series associated to p adic points on a p adic variety the main tools which are used are semi algebraic geometry in spaces of power series and motivic integration a notion introduced by m kontsevich in particular we develop the theory of motivic integration for semi algebraic sets of formal arcs on singular algebraic varieties we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of oesterle

nonexistence of periodic solutions and s asymptotically periodic solutions in fractional difference equations nonexistence of periodic solution is proved for periodic right hand side existence of s asymptotically periodic solutions is discussed existence and uniqueness results are obtained for s asymptotically periodic solution nontrivial estimations of a ratio of gamma functions are used in this paper it is shown that a fractional difference equation with a periodic right hand side can not possess a periodic solution but it can have an s asymptotically periodic solution sufficient conditions are proved for the existence of a unique s asymptotically periodic solution an example is also given illustrating the obtained results

existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti periodic boundary conditions this paper describes the method of quasilinearization for first order nonlinear impulsive functional differential equations with anti periodic boundary conditions a monotone iterative technique coupled with lower and upper solutions is employed to obtain sequences of approximate solutions converging monotonically and quadratically to the unique solution of the problem at hand 2007 elsevier ltd all rights reserved

systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions abstract this paper is concerned with the existence of solutions for systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions by establishing new comparison results and applying the monotone iterative technique we obtain the sufficient conditions for the existence of extremal system of solutions

on a singular nonlinear semilinear elliptic problem where k x c2 b v9 a p 0 1 and l is a real parameter such singular elliptic problems arise in the contexts of chemical heterogeneous catalysts nonnewtonian fluids and also the theory of heat conduction in electrically conducting materials see 3 5 8 9 for a detailed discussion obviously 1 1 cannot have a solution u c2 v9 if k x is not vanishing near v however under various appropriate assumptions on k x we will obtain classical solutions of 1 1 for l belonging to a certain range and we will also obtain some uniqueness criteria here a classical solution is a solution u of 1 1 which belongs to c2 v mc v9 with u 0 in v we also study the boundary behaviour of solutions of 1 1 and we will show that the solution u of 1 1 lies in a certain holder class the special case when k x is negative and l 0 has been studied by several authors the existence and uniqueness of the solution were established by crandall rabinowitz and tartar 6 del pino 7 gomes 10 and lazer and mckenna

three anti periodic solutions for second order impulsive differential inclusions via nonsmooth critical point theory abstract a nonsmooth version of a three critical point theorem of ricceri due to iannizzotto is used to obtain three anti periodic solutions for a second order impulsive differential inclusions with a perturbed nonlinearity and two parameters

singular gierer meinhardt systems of elliptic boundary value problems abstract we establish existence results for singular gierer meinhardt elliptic systems with zero dirichlet boundary conditions gierer meinhardt systems are model problems for pattern formations of spatial tissue structures of morphogenesis the mathematical difficulties are that the system becomes singular near the boundary and it is non quasimonotone we show the existence of positive solutions for the activator inhibitor model with common sources

global existence and nonexistence for some degenerate and quasilinear parabolic systems abstract the author discusses the degenerate and quasilinear parabolic system u t u v u a u p v q and v t u v v b u r v s with dirichlet boundary conditions in a bounded domain and shows that the global existence depends crucially on the sign of the difference q r 1 p 1 s and the domain

global stability of impulsive fractional differential equations abstract in the present paper the global stability of systems of impulsive differential equations of fractional order with impulse effect at fixed moments of time is considered using piecewise continuous functions of the type of lyapunov s functions and a new fractional comparison principle sufficient conditions for various types of global stability are proved

on recent developments in the theory of boundary value problems for impulsive fractional differential equations this paper is motivated from some recent papers treating the boundary value problems for impulsive fractional differential equations we first make a counterexample to show that the formula of solutions in cited papers are incorrect second we establish a general framework to find the solutions for impulsive fractional boundary value problems which will provide an effective way to deal with such problems third some sufficient conditions for the existence of the solutions are established by applying fixed point methods meanwhile data dependence is obtained by using a new generalized singular gronwall inequality finally three examples are given to illustrate the results

almost automorphic mild solutions to fractional differential equations abstract we introduce the concept of resolvent families to prove the existence of almost automorphic mild solutions to the differential equation d t u t a u t t n f t 1 2 n z considered in a banach space x where f r x is almost automorphic we also prove the existence and uniqueness of an almost automorphic mild solution of the semilinear equation d t u t a u t f t u t 1 2 assuming f t x is almost automorphic in t for each x x satisfies a global lipschitz condition and takes values on x finally we prove also the existence and uniqueness of an almost automorphic mild solution of the semilinear equation d t u t a u t f t u t u t 1 2 under analogous conditions as in the previous case

positive solutions to second order four point boundary value problems for impulsive differential equations abstract in this paper by using the leggett williams theorem we present sufficient conditions which guarantee the existence of three positive solutions to second order four point boundary value problems for impulsive differential equations

some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions this paper investigates the existence and uniqueness of solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order email protected 1 2 our results are based on some standard fixed point theorems some examples are presented to illustrate the main results

regularity of an elliptic problem with a singular nonlinearity we study a semilinear boundary value problem with the feature that the vanishing boundary value makes the equation singular we prove that the positive solution is in general holder continuous up to the boundary and has even better regularity in some special cases

controllability and observability of impulsive fractional linear time invariant system in this paper we deal with the controllability and observability of impulsive fractional linear time invariant if lti for short system our main purpose is to built some necessary and sufficient conditions of controllability and observability for the if lti system at the same time we establish some conclusions of controllability and observability for a continuous fractional lti system which is a special case of the if lti system examples are given to illustrate our results

the mordell lang conjecture for function fields we give a proof of the geometric mordell lang conjecture in any characteristic our method involves a model theoretic analysis of the kernel of manin s homomorphism and of a certain analog in characteristic p department of mathematics massachusetts institute of technology 2 277 cambridge massachusetts 02139 current address department of mathematics hebrew university jerusalem israel e mail address ehud math mit edu

regularity of mild solutions for a class of fractional order differential equations in this article we show sufficient conditions ensuring the existence and uniqueness of a mild solution to the equation d au t au t d a 1f t u t 1 a 2 email protected r in the same space where f belongs here a is a sectorial operator defined in a banach space x d a is the fractional derivative in the riemann liouville sense and f x email protected x for each email protected x where w x is a vector valued subspace of the space of continuous and bounded functions the subspaces w x that we will consider in this article are the space of periodic almost periodic almost automorphic and compact almost automorphic vector valued functions among others in particular we extend and unify recent results established for the equation in the papers agarwal et al 2010 cuevas et al 2010 and cuevas and lizama 2008

generalized differential galois theory a galois theory of differential fields with parameters is developed in a manner that generalizes kolchin s theory it is shown that all connected differential algebraic groups are galois groups of some appropriate differential field extension

nonlinear impulsive problems for fractional differential equations and ulam stability in this paper the first purpose is treating cauchy problems and boundary value problems for nonlinear impulsive differential equations with caputo fractional derivative we introduce the concept of piecewise continuous solutions for impulsive cauchy problems and impulsive boundary value problems respectively by using a new fixed point theorem we obtain many new existence uniqueness and data dependence results of solutions via some generalized singular gronwall inequalities the second purpose is discussing ulam stability for impulsive fractional differential equations some new concepts in stability of impulsive fractional differential equations are offered from different perspectives some applications of our results are also provided

global existence for a singular gierer meinhardt system abstract this paper is concerned with existence results for a singular gierer meinhardt system subject to zero dirichlet boundary conditions which originally arose in studies of pattern formation in biology the mathematical difficulties are that the system becomes singular near the boundary and it lacks a variational structure we use a functional method to obtain both upper and lower bounds for the perturbed system and then use sobolev embedding theorem to prove the existence of a pair of positive solutions under suitable conditions this method is first used in a singular parabolic system and is completely different than the traditional methods of sub and super solutions

definable sets in algebraically closed valued fields elimination of imaginaries it is shown that if k is an algebraically closed valued field with valuation ring r then th k has elimination of imaginaries if sorts are added whose elements are certain cosets in k n of certain definable r submodules of k n for all n 1 the proof involves the development of a theory of independence for unary types which play the role of 1 types followed by an analysis of germs of definable functions from unary sets to the sorts contents

fractional differential equation models with pulses and criterion for pest management in this paper we propose and analyze a new class of fractional order nonlinear differential equations with impulsive terms and criterion which provide a more holistic and less toxic approach to pest management sufficient conditions on the existence of solutions are investigated by using fixed point methods corresponding to compact or noncompact conditions respectively

extremal solutions for third order nonlinear problems with upper and lower solutions in reversed order abstract this paper deals with the existence of extremal solutions for the third order nonlinear boundary value problem u t f t u t t a b u a a u a b u b c in the presence of a pair of lower and upper solutions in reversed order here r r is an increasing homeomorphism f a b r r is a carathedory function and a b c r the proof follows from monotone iterative techniques which are based on suitable anti maximum principles for adequate operators to deduce such results we study some related problems coupled with boundary value conditions of the form p 0 u a q 0 u a a p 1 u b q 1 u b b u a c and u a a u a b u b c

bounded solutions to a singular parabolic system abstract in this paper we are concerned with the singular parabolic system u t u f x v p v t v g x u q in a smooth bounded domain r n subject to zero dirichlet conditions with initial conditions u 0 x v 0 x 0 this problem is of interest as it is related to some problems in biology and physics under suitable assumptions on p q and f x g x some existence results of weak and classical solutions are obtained using a functional method this method is motivated by such results found in 4 and 5 when dealing with singular parabolic systems and the related references within

difference galois theory of linear differential equations abstract we develop a galois theory for linear differential equations equipped with the action of an endomorphism this theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation the galois groups here are linear difference algebraic groups i e matrix groups defined by algebraic difference equations

the monotone method for neumann functional differential equations with upper and lower solutions in the reverse order abstract in this paper we show that the monotone technique produces two monotone sequences that converge uniformly to extremal solutions of second order functional differential equations and laplacian equations with neumann boundary value conditions moreover we obtain existence results assuming upper and lower solutions in the reverse order

existence of mild solutions for fractional integrodifferential equations of sobolev type with nonlocal conditions abstract this paper is concerned with the fractional integrodifferential equations of sobolev type with nonlocal condition in a separable banach space with the help of the theory of propagation family as well as the theory of the measures of noncompactness and the condensing maps we obtain the existence result of mild solutions for above equations two examples are given to illustrate the existence theorem

new comparison results for impulsive integro differential equations and applications we prove some new maximum principles for ordinary integro differential equations this allows us to introduce a new definition of lower and upper solutions which leads to the development of the monotone iterative technique for a periodic boundary value problem related to a nonlinear first order impulsive integro differential equation

approximate mild solutions of fractional stochastic evolution equations in hilbert spaces in this paper we study the approximate mild solutions of a class of fractional stochastic evolution equations in hilbert spaces by constructing picard type approximate sequences we present the approximate results under the non lipschitz and linear growth conditions

existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay we deal in this paper with the mild solution for fractional semilinear differential equations with infinite delay d ax t ax t f t x t bx t email protected 0 t x t email protected t email protected 0 with t 0 and 0 0 on a complex banach space x by means of classical fixed points methods

parameterized picard vessiot extensions and atiyah extensions abstract generalizing atiyah extensions we introduce and study differential abelian tensor categories over differential rings by a differential ring we mean a commutative ring with an action of a lie ring by derivations in particular these derivations act on a differential category a differential tannakian theory is developed the main application is to the galois theory of linear differential equations with parameters namely we show the existence of a parameterized picard vessiot extension and therefore the galois correspondence for many differential fields with possibly non differentially closed fields of constants that is fields of functions of parameters other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic which will appear in a separate paper this application is based on differential categories developed in the present paper and not just differential algebraic groups and their representations

generalized hasse schmidt varieties and their jet spaces building on the abstract notion of prolongation developed in 10 the theory of iterative hasse schmidt rings and schemes is introduced simultaneously generalising difference and hasse schmidt differential rings and schemes this work provides a unified formalism for studying difference and differential algebraic geometry as well as other related geometries as an application hasse schmidt jet spaces are constructed generally allowing the development of the theory for arbitrary systems of algebraic partial difference differential equations where constructions by earlier authors applied only to the finite dimensional case in particular it is shown that under appropriate separability assumptions a hasse schmidt variety is determined by its jet spaces at a point

global and blow up solutions for compressible euler equations with time dependent damping abstract this paper deals with the cauchy problem for the compressible euler equations with time dependent damping where the time vanishing damping in the form of 1 t makes some fantastic variety of the dynamic system for 0 1 and 0 or 1 but 2 the solutions are proved to exist globally in time when the derivatives of the initial data are small but the initial data themselves can be arbitrarily large this is the so called challenging case of global solutions with large initial data while when the initial riemann invariants are monotonic and their derivatives with absolute value are large at least at one point then the solutions are still bounded but their derivatives will blow up at finite time somewhat like the singularity formed by shock waves for 1 and 0 or 1 but 0 1 the derivatives of solutions will blow up even for all initial data including the interesting case of blow up solutions with small initial data here the initial riemann invariants are monotonic in fact such a blow up phenomenon is determined by the mechanism of the dynamic system itself in order to prove the global existence of solutions with large initial data we introduce a new energy functional related to the riemann invariants which crucially enables us to build up the maximum principle for the corresponding riemann invariants and the uniform boundedness for the local solutions finally numerical simulations in different cases are carried out which further confirm our theoretical results

on the concept of general solution for impulsive differential equations of fractional order q 0 1 in this paper for impulsive differential equations with fractional order q 0 1 we show that the formula of solutions in cited papers are incorrect secondly we find out a formula of the general solution for impulsive cauchy problem with caputo fractional derivative q 0 1 further for a kind of impulsive fractional differential equations system with special initial value we come to an existence result for it by applying fixed point methods

positive solutions to singular boundary value problems for fractional functional differential equations with changing sign nonlinearity in this work a boundary value problem is discussed for a singular delay differential equation of fractional order the nonlinearity may change sign by means of the krasnoselskii fixed point theorem existence results for positive solutions are obtained examples illustrating our main results are also given

completeness in partial differential algebraic geometry abstract this paper is part of the model theory of fields of characteristic 0 equipped with m commuting derivation operators dcf 0 m it continues to partial differential fields work begun by wai yan pong who treated the case m 1 we study the concept of completeness in differential algebraic geometry applying methods of model theory and differential algebra our central tool in applying the valuative criterion developed in differential algebra by e r kolchin peter bloom and sally morrison is a fundamental theorem in classical elimination theory due to the model theorist lou van den dries we use this valuative criterion to give a new family of complete differential algebraic varieties in addition to completeness we prove some embedding theorems for differential algebraic varieties of arbitrary differential transcendence degree as a special case we show that every differential algebraic subvariety of the projective line which has lascar rank less than m can be embedded in the affine line

periodic boundary value problem for the first order impulsive functional differential equations this paper is concerned with the existence of extreme solutions of the periodic boundary value problem for a class of first order impulsive 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 meanwhile we extend previous results