systems of differential equations of alternately retarded and advanced type abstract the main purpose of this paper is to study oscillatory and stability properties of linear systems of differential equations with piecewise constant argument in which the argument has a fixed point in each interval of a certain length the results obtained solve a problem proposed by i gyori and g ladas

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

stability of runge kutta methods in the numerical solution of equation u t au t a 0 u t this paper is concerned with the stability analysis of the runge kutta methods for the equation u t au t a0u t the stability regions for the runge kutta methods are determined the conditions that the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

a glimm effros dichotomy for borel equivalence relations a basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by glimm g12 in the locally compact group action case and extended by effros e 1 e2 in the polish group action case when additionally the induced equivalence relation is f it is the purpose of this paper to extend the glimm effros dichotomy to the very general context of an arbitrary borel equivalence relation not even necessarily induced by a group action despite the totally classical descriptive set theoretic nature of our result our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability or recursion theory on the integers

existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations in connection with the maximizing problem for the functional r u u lq u lp in w01 p 0 we consider the equation div u p 2 u x u q 2 u x x 1 p resp q p e has no nontrivial resp nonnegative nontrivial solution belonging to the class p u w01 p lq xi u q 2u lp p 1 i 1 2 n w01 p provided that is star shaped the crucial point of the proof of our result is to obtain an l estimate of weak solutions and to verify a certain pohozaev type inequality for weak solutions belonging to p

positive solutions for a class of equations with a p laplace like operator and weights abstract in this paper we will study the problem of existence of positive solutions to the problem d a r u b r g u 0 a e in 0 r lim r 0 a r u r 0 u r 0 where is an odd increasing homeomorphism of r and g c r is such that g z 0 for all z 0 with g 0 0 the functions a and b that we will refer to as weight functions satisfy a r 0 b r 0 for all r 0 r and are such that a b c 1 0 r l 1 0 r if has the form z z m z and a r r n 1 a r b r r n 1 b r n 2 then solutions of problem d provide solutions with radial symmetry for the problem p div a x m u u b x g u 0 x u 0 x where b 0 r denotes the ball with center 0 and radius r 0 in r n

essential countability of treeable equivalence relations we establish a dichotomy theorem characterizing the circumstances under which a treeable borel equivalence relation e is essentially countable under additional topological assumptions on the treeing we in fact show that e is essentially countable if and only if there is no continuous embedding of e1 into e our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations and allow us to show that up to continuous kakutani embeddability there is a minimum borel function which is not essentially countable to one

existence of sign changing solutions for an equation with a weighted p laplace operator we consider radial solutions of a general elliptic equation involving a weighted p laplace operator with a subcritical nonlinearity by a shooting method we prove the existence of solutions with any prescribed number of nodes the method is based on a change of variables in the phase plane a very general computation of an angular velocity and new estimates for the decay of an energy associated with an asymptotic hamiltonian problem estimating the rate of decay for the energy requires a sub criticality condition the method covers the case of solutions which are not compactly supported or which have compact support in the last case we show that the size of the support increases with the number of nodes

stability of runge kutta methods for the alternately advanced and retarded differential equations with piecewise continuous arguments this paper deals with the numerical properties of runge kutta methods for the solution of u t au t a 0u t 12 it is shown that the runge kutta method can preserve the convergence order the necessary and sufficient conditions under which the analytical stability region is contained in the numerical stability region are obtained it is interesting that the q methods with 0 q 12 are asymptotically stable some numerical experiments are given

asymptotic stability of runge kutta methods for nonlinear differential equations with piecewise continuous arguments this paper deals with the asymptotic stability of numerical solutions for differential equations with piecewise continuous arguments epcas the necessary and sufficient condition is given for non confluent runge kutta methods to preserve the stability of nonlinear scalar epcas as for systems we prove that some algebraically stable methods can preserve the asymptotic stability

stability of θ methods for advanced differential equations with piecewise continuous arguments abstract this paper deals with the stability analysis of numerical methods for the solution of advanced differential equations with piecewise continuous arguments we focus on the behaviour of the one leg method and the linear method in the solution of the equation x t ax t a 0 x t a 1 x t 1 with real a a 0 a 1 and designates the greatest integer function the stability regions of two methods are determined the conditions under which the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

stability of runge kutta methods in the numerical solution of equation u t au t a0u t a1u t 1 this paper is concerned with the stability analysis of the runge kutta methods for the equation u t au t a 0u t a 1u t 1 the stability regions for the runge kutta methods are determined the conditions that the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

the classification of hypersmooth borel equivalence relations this paper is a contribution to the study of borel equivalence relations in standard borel spaces i e polish spaces equipped with their borel structure a class of such equivalence relations which has received particular attention is the class of hyperfinite borel equivalence relations these can be defined as the increasing unions of sequences of borel equivalence relations all of whose equivalence classes are finite or as it turns out equivalently those induced by the orbits of a single borel auto morphism hyperfinite equivalence relations have been classified in djk under two notions of equivalence borel bi reducibility and borel isomorphism

the improved linear multistep methods for differential equations with piecewise continuous arguments abstract this paper deals with the convergence of the linear multistep methods for the equation x t ax t a 0 x t numerical experiments demonstrate that the 2 step adams bashforth method is only of order p 0 when applied to the given equation an improved linear multistep methods is constructed it is proved that these methods preserve their original convergence order for ordinary differential equations odes and some numerical experiments are given

oscillations in systems of differential equations with piecewise constant argument abstract a linear system of differential equations with the argument t 1 2 is studied where denotes the greatest integer function of special interest are oscillatory and periodic properties of the solutions which depend on the eigenvalues of a certain matrix associated with the system therefore the results can be extended to more general dynamical systems

positive stationary solutions for p laplacian problems with nonpositive perturbation the paper is devoted to the existence of positive solutions of nonlinear elliptic equations with p laplacian we provide a general topological degree that detects solutions of the problem a u f u u m where a x d a x is a maximal monotone operator in a banach space x and f m x is a continuous mapping defined on a closed convex cone m x next we apply this general framework to a class of partial differential equations with p laplacian under dirichlet boundary conditions in the paper we employ general ideas from cwiszewski and kryszewski 2009 5 where a setting suitable for the one dimensional p laplacian was introduced

positive radial solutions for dirichlet problems with mean curvature operators in minkowski space abstract in this paper by using leray schauder degree arguments and critical point theory for convex lower semicontinuous perturbations of c 1 functionals we obtain existence of classical positive radial solutions for dirichlet problems of type div v 1 v 2 f x v 0 in b r v 0 on b r here b r x r n x r and f 0 r 0 r is a continuous function which is positive on 0 r 0

the structure of hyperfinite borel equivalence relations we study the structure of the equivalence relations induced by the orbits of a single borel automorphism on a standard borel space we show that any two such equivalence relations which are not smooth i e do not admit borel selectors are borel embeddable into each other this utilizes among other things work of effros and weiss using this and also results of dye varadarajan and recent work of nadkarni we show that the cardinality of the set of ergodic invariant measures is a complete invariant for borel isomorphism of aperiodic nonsmooth such equivalence relations in particular since the only possible such cardinalities are the finite ones countable infinity and the cardinality of the continuum there are exactly countably infinitely many isomorphism types canonical examples of each type are also discussed

dichotomy theorems for families of non cofinal essential complexity we prove that for every borel equivalence relation e either e is borel reducible to mathbb e 0 or the family of borel equivalence relations incompatible with e has cofinal essential complexity it follows that if f is a borel equivalence relation and cal f is a family of borel equivalence relations of non cofinal essential complexity which together satisfy the dichotomy that for every borel equivalence relation e either e in cal f or f is borel reducible to e then cal f consists solely of smooth equivalence relations thus the dichotomy is equivalent to a known theorem

differential equations alternately of retarded and advanced type we study differential equations with the argument m t k m where denotes the greatest integer function and k and m are positive integers such that k m the argument deviation t t m t k m is a function of period m and equals t for k t m k it changes sign in each interval mn k t m n 1 k vanishing at t mn

stability and oscillations of numerical solutions for differential equations with piecewise continuous arguments of alternately advanced and retarded type this paper is concerned with the numerical properties of q methods for the solution of alternately advanced and retarded differential equations with piecewise continuous arguments using two q methods namely the one leg q method and the linear q method the necessary and sufficient conditions under which the analytic stability region is contained in the numerical stability region are obtained and the conditions of oscillations for the q methods are also obtained it is proved that oscillations of the analytic solution are preserved by the q methods furthermore the relationships between stability and oscillations are revealed some numerical experiments are presented to illustrate our results