Mean-square stability of Milstein method for linear hybrid stochastic delay integro-differential equations

In this paper we study the mean-square (MS) stability of the Milstein method for linear stochastic delay integro-differential equations (SDIDE) with Markovian switching by extending the techniques of [Z. Wang, C. Zhang, An analysis of stability of Milstein method for stochastic differential equations with delay, Computers and Mathematics with Applications 51 (2006) 1445–1452; L. Ronghua, H. Yingmin, Convergence and stability of numerical solutions to SDDEs with Markovian switching, Applied Mathematics and Computation 175 (2006) 1080–1091]. It is established that the Milstein method is MS-stable for linear stochastic delay differential equations (Wang and Zhang (2006); in the above reference). Here we prove that it is MS-stable for linear SDIDE with Markovian switching also under suitable conditions on the integral term. A numerical example is provided to illustrate the theoretical results.     

On Estimates of Solutions to Systems of Nonlinear Differential Equations with Distributed Delay and Periodic Coefficients in the Linear Terms

Journal of Applied and Industrial Mathematics - We consider a system of nonlinear differential equations with distributed delay and periodic coefficients in the linear terms. Some sufficient...     

Existence and asymptotic behaviour of nonoscillatory solutions of second-order neutral differential equations with “maxima”

Bainov, D.D., V. Petrov and V. Proytcheva, Existence and asymptotic behaviour of nonoscillatory solutions of second order neutral differential equations with “maxima”, Journal of Computational and Applied Mathematics.In the paper, sufficient conditions for existence of positive and negative solutions of neutral equations with “maxima” are obtained and their asymptotic behaviour is investigated.     

On the boundedness of some rational difference equations

We study the boundedness character of solutions of some third order rational difference equations and we confirm some of the conjectures posed in Camouzis et al. [“Progress report on the boundedness character of third-order rational equations”, Journal of Difference Equations and Applications 11 (2005), 1029–1035] and [“On third order rational difference equations, part 6”, Journal of Difference Equations and Applications 11 (2005), 759–777].     

Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton–Jacobi equations

In this note, we reinterpret a discontinuous Galerkin method originally developed by Hu and Shu [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (1999) 666–690] (see also [O. Lepsky, C. Hu, C.-W. Shu, Analysis of the discontinuous Galerkin method for Hamilton–Jacobi equations, Applied Numerical Mathematics 33 (2000) 423–434]) for solving Hamilton–Jacobi equations. With this reinterpretation, numerical solutions will automatically satisfy the curl-free property of the exact solutions inside each element. This new reinterpretation allows a method of lines formulation, which renders a more natural framework for stability analysis. Moreover, this reinterpretation renders a significantly simplified implementation with reduced cost, as only a smaller subspace of the original solution space in [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (1999) 666–690; O. Lepsky, C. Hu, C.-W. Shu, Analysis of the discontinuous Galerkin method for Hamilton–Jacobi equations, Applied Numerical Mathematics 33 (2000) 423–434] is used and the least squares procedure used in [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (1999) 666–690; O. Lepsky, C. Hu, C.-W. Shu, Analysis of the discontinuous Galerkin method for Hamilton–Jacobi equations, Applied Numerical Mathematics 33 (2000) 423–434] is completely avoided.     

On <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">1</Emphasis></Superscript>-summability and asymptotic profiles for smooth solutions to Navier–Stokes equations in a 3D exterior domain

The exterior nonstationary problem is studied for the 3D Navier-Stokes equations. The L ¹ -summability is proved for smooth solutions which correspond to initial data satisfying certain symmetry and moment conditions. The result is then applied to show that such solutions decay in time more rapidly than observed in general. Furthermore, an asymptotic expansion is deduced and a lower bound estimate is given for the rates of decay in time. Mathematics Subject Classifications (1991): 35Q30, 76D05.On leave of absence from Institute of Applied Mathematics, Academy of Mathematics and System Sciences. Academia Sinica, Beijing 100080, Peoples Republic of China. Supported by JSPS     

On a Multivalued Version of the Sharkovskii Theorem and Its Application to Differential Inclusions, II

A multivalued version of the celebrated Sharkovskii theorem is established which is applicable to differential equations and inclusions for obtaining subharmonic periodic solutions. The results in our earlier paper (Set-Valued Anal. 10(1) (2002), 1–14) are completed to a sharp form. A multivalued analogue of the Levinson transformation theory (dissipativity implies the existence of harmonics) is stated.Mathematics Subject Classifications (2000) 34A60, 34C25, 47H04, 58C06.Jan Andres: Supported by the Council of Czech Goverment (J14/98:153100011) and by the grant No. 201-00-0768 of the Grant Agency of Czech Republic.     

Existence of periodic solutions to a system with functional response on time scales

This paper investigates the existence of periodic solutions of a three-species food-chain diffusive system with Beddington-DeAngelis functional responses and time delays in a two-patch environment on time scales. By using a continuation theorem based on coincidence degree theory, we obtain sufficient criteria for the existence of periodic solutions for the system. Moreover, when the time scale T is chosen as R or Z, the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the method is unified to provide the existence of the desired solutions for continuous differential equations and discrete difference equations.     

Second Order Scalar Functional Differential Equations with Piecewise Constant Arguments

The study of functional differential equations with piecewise constant arguments usually results in a study of certain related difference equations. In this paper we consider certain neutral functional differential equations of this type and the associated difference equations. We give conditions under which such equations with almost periodic time dependence will have unique almost periodic solutions, and for certain autonomous cases, we obtain certain stability results and also conditions for chaotic behavior of solutions. We are particularly concerned with such equations which are partially discretized versions of non-forced Duffing equations.     

On Detecting Solutions of Polynomial Nonlinear Difference Equations

In this paper a method for discovering solutions of nonlinear polynomial difference equations is presented. It is based on the concepts of i -operator and star-product. These notions create a proper algebraic background by means of which we can find linear equations "included" into the original nonlinear one and to seek for solutions among them. A corresponding algorithm and some examples are also provided.     

Remarks on oscillation of second-order linear difference equation

In this remark, we shall show that the Lemma 1 and parts of the proof of Theorem 1 in the paper “Oscillation criteria for second-order linear difference equations” (Applied Mathematics and Computation 145 (2003) 591-601) are incorrect.     

Existence of periodic solutions of second order nonlinear p-Laplacian difference equations

By using the critical point theory, the existence of periodic solutions to second order nonlinear p-Laplacian difference equations is obtained. The main approach used is a variational technique and the saddle point theorem. The problem is to solve the existence of periodic solutions of second order nonlinear p-Laplacian difference equations.     

Obstacle problem for nonlinear integro-differential equations arising in option pricing

Abstract  We study the obstacle problem for a class of nonlinear integro-partial differential equations of second order, possibly degenerate, which includes the equation modeling American options in a jump-diffusion market with large investor. The viscosity solutions setting reveals appropriate, because of a monotonicity property with respect to the integral term. The same property allows to approximate the problem by penalization and to obtain the existence and uniqueness of solutions via a comparison principle. We also give uniform estimates of the solutions of the penalized problems which allow to prove further regularity. Keywords: Integro-differential equations, Obstacle problem, Viscosity solutions, American options Mathematics Subject Classification (2000): 45K05, 35K85, 49L25, 91B24     

A one-model approach based on relaxed combinations of inputs for evaluating input congestion in DEA

This paper provides a one-model approach of input congestion based on input relaxation model developed in data envelopment analysis (e.g. [G.R. Jahanshahloo, M. Khodabakhshi, Suitable combination of inputs for improving outputs in DEA with determining input congestion — Considering textile industry of China, Applied Mathematics and Computation (1) (2004) 263–273; G.R. Jahanshahloo, M. Khodabakhshi, Determining assurance interval for non-Archimedean ele improving outputs model in DEA, Applied Mathematics and Computation 151 (2) (2004) 501–506; M. Khodabakhshi, A super-efficiency model based on improved outputs in data envelopment analysis, Applied Mathematics and Computation 184 (2) (2007) 695–703; M. Khodabakhshi, M. Asgharian, An input relaxation measure of efficiency in stochastic data analysis, Applied Mathematical Modelling 33 (2009) 2010–2023]. This approach reduces solving three problems with the two-model approach introduced in the first of the above-mentioned reference to two problems which is certainly important from computational point of view. The model is applied to a set of data extracted from ISI database to estimate input congestion of 12 Canadian business schools.     

Bifurcations from infinity in asymmetric nonlinear oscillators

We study the existence of large-amplitude periodic or almost periodic solutions of second order differential equations with asymmetric nonlinearities, when the system is close to "nonlinear resonance". Received September 1998     

Periodic solutions of certain hybrid delay-differential equations and their corresponding difference equations

In this paper we use a method due to Carvalho (A method to investigate bifurcation of periodic solution in retarded differential equations, J. Differ. Equ. Appl. 4 (1998), pp. 17–27) to obtain conditions for the existence of nonconstant periodic solutions of certain systems of hybrid delay-differential equations. We first deal with a scalar equation of Lotka–Valterra type; then a system of two equations in two unknowns that could model the interactions of two identical neurons. It will be seen that such solutions are determined by solutions of corresponding difference equations. Another paper in which this method is used is by Cooke and Ladeira (Applying Carvalho's method to find periodic solutions of difference equations, J. Differ. Equ. Appl. 2 (1996), pp. 105–115).

We first state Carvalho's result.     

Heteroclinic Cycles in Coupled Systems of Difference Equations

Cycling behavior, in which solution trajectories linger around steady-states and periodic solutions, is known to be a generic feature of coupled cell systems of differential equations. In this type of systems, cycling behavior can even occur independently of the internal dynamics of each cell. This conclusion has lead to the discovery of "cycling chaos", in which solution trajectories cycle around symmetrically related chaotic sets. In this work, we demonstrate that cycling behavior also occurs in coupled systems of difference equations. More specifically, we prove the existence of structurally stable cycles between fixed points, and use numerical simulations to illustrate that the resulting cycles can also persist independently of the internal dynamics of each cell. Consequently, we demonstrate that cycles involving periodic orbits as well as cycling chaos also occur in systems of difference equations.     

On the existence of periodic solutions in the thermistor problem with degenerate thermal conductivity

Abstract This paper deals with the existence of weak periodic solutions for a parabolic-elliptic system proposed as a model for a time dependent thermistor with degenerate thermal conductivity. Applying the maximal monotone mappings theory, we prove an existence result for weak periodic solutions. Keywords: Nonlinear parabolic-elliptic system of degenerate type, Periodic solutions, Thermistor problem Mathematics Subject Classification (2000): 35B10, 35J60, 35K65     

Strong NP-hardness of scheduling problems with learning or aging effect

Proofs of strong NP-hardness of single machine and two-machine flowshop scheduling problems with learning or aging effect given in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c; Applied Mathematical Modelling 37:1523–1536, 2013) contain a common mistake that make them incomplete. We reveal the mistake and provide necessary corrections for the problems in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; Applied Mathematical Modelling 37:1523–1536, 2013). NP-hardness of problems in Rudek (International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c) remains unknown because of another mistake which we are unable to correct.