Dependence on parameters for a discrete Emden-Fowler equation

We investigate the dependence on parameters for the discrete boundary value problem connected with the Emden-Fowler equation. A variational method is used in order to obtain a general scheme allowing for investigating the dependence on parameters of discrete boundary value problems.     

Coupled intervals in the discrete calculus of variations: necessity and sufficiency

In this work we study nonnegativity and positivity of a discrete quadratic functional with separately varying endpoints. We introduce a notion of an interval coupled with 0, and hence, extend the notion of conjugate interval to 0 from the case of fixed to variable endpoint(s). We show that the nonnegativity of the discrete quadratic functional is equivalent to each of the following conditions: The nonexistence of intervals coupled with 0, the existence of a solution to Riccati matrix equation and its boundary conditions. Natural strengthening of each of these conditions yields a characterization of the positivity of the discrete quadratic functional. Since the quadratic functional under consideration could be a second variation of a discrete calculus of variations problem with varying endpoints, we apply our results to obtain necessary and sufficient optimality conditions for such problems. This paper generalizes our recent work in [R. Hilscher, V. Zeidan, Comput. Math. Appl., to appear], where the right endpoint is fixed.     

Mixed boundary value problems and parametrices in the edge calculus

Mixed elliptic boundary value problems are characterised by conditions which have a jump along an interface of codimension 1 on the boundary. We study such problems in weighted edge spaces and show the Fredholm property and the existence of parametrices under additional conditions of trace and potential type on the interface. We develop a new method for computing the interface conditions in terms of the index of boundary value problems in weighted spaces on infinite cones, combined with structures from the calculus of boundary value problems on a manifold with edges. This will be illustrated by the Zaremba problem and other mixed problems for the Laplace operator. The approach itself is completely general.     

A discrete multivariate mean value theorem with applications

We establish a discrete multivariate mean value theorem for the class of positive maximum component sign preserving functions. A constructive and combinatorial proof is given based upon a simplicial algorithm and vector labeling. Moreover, we apply this theorem to a discrete nonlinear complementarity problem and an economic equilibrium problem with indivisibilities and show the existence of solutions in both problems under certain mild conditions.     

On the discrete boundary value problem for anisotropic equation

In this paper we consider the discrete anisotropic boundary value problem using critical point theory. Firstly we apply the direct method of the calculus of variations and the mountain pass technique in order to reach the existence of at least one nontrivial solution. Secondly we derive some version of a discrete three critical point theorem which we apply in order to get the existence of at least two nontrivial solutions.     

Difference schemes on uniform grids performed by general discrete operators

Our aim is to set the foundations of a discrete vectorial calculus on uniform n-dimensional grids, that can be easily reformulated on general irregular grids. As a key tool we first introduce the notion of tangent space to any grid node. Then we define the concepts of vector field, field of matrices and inner products on the space of grid functions and on the space of vector fields, mimicking the continuous setting. This allows us to obtain the discrete analogous of the basic first order differential operators, gradient and divergence, whose composition define the fundamental second order difference operator. As an application, we show that all difference schemes, with constant coefficients, for first and second order differential operators with constant coefficients can be seen as difference operators of the form for suitable choices of q, and  . In addition, we characterize special properties of the difference scheme, such as consistency, symmetry and positivity in terms of q, and  .     

Connection between holomorphic vector bundles and cohomology on a Riemann surface and conjugation boundary value problems

This paper studies interconnections between holomorphic vector bundles on compact Riemann surfaces and the solution of the homogeneous conjugation boundary value problem for analytic functions on the one hand, and cohomology and the solution of the inhomogeneous problem on the other. We establish that constructing the general solution to the homogeneous problem with arbitrary coefficients in the boundary conditions is equivalent to classifying holomorphic vector bundles. Solving the inhomogeneous problem is equivalent to checking the solvability of 1-cocycles with coefficients in the sheaf of sections of a bundle; in particular, the solvability conditions in the inhomogeneous problem determine obstructions to the solvability of 1-cocycles, i.e. the first cohomology group. Using this connection, we can apply the methods of boundary value problems to vector bundles. The results enable us to elucidate the role of boundary value problems in the general theory of Riemann surfaces.     

The unconditional stable and convergent difference methods with intrinsic parallelism for quasilinear parabolic systems

A kind of the general finite difference schemes with intrinsic parallelism forthe boundary value problem of the quasilinear parabolic system is studied without assum-ing heuristically that the original boundary value problem has the unique smooth vectorsolution. By the method of a priori estimation of the discrete solutions of the nonlineardifference systems, and the interpolation formulas of the various norms of the discretefunctions and the fixed-point technique in finite dimensional Euclidean space, the exis-tence and uniqueness of the discrete vector solutions of the nonlinear difference systemwith intrinsic parallelism are proved. Moreover the unconditional stability of the generalfinite difference schemes with intrinsic parallelism is justified in the sense of the continu-ous dependence of the discrete vector solution of the difference schemes on the discretedata of the original problems in the discrete w_2~(2,1) norms. Finally the convergence of thediscrete vector solutions of the certain differe     

THE UNCONDITIONAL STABLE DIFFERENCE METHODS WITH INTRINSIC PARALLELISM FOR TWO DIMENSIONAL SEMILINEAR PARABOLIC SYSTEMS

In this paper we are going to discuss the difference schemes with intrinsic parallelism for the boundary value problem of the two dimesional semilinear parabolic systems.The unconditional stability of the general finite difference schemes with intrinsic parallelism is justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete data of the original problems in the discrete W2^(2,1) norms.Then the uniqueness of the discrete vector solution of this difference scheme follows as the consequence of the stability.     

Elliptic boundary problems on manifolds with polycylindrical ends

We investigate general Shapiro-Lopatinsky elliptic boundary value problems on manifolds with polycylindrical ends. This is accomplished by compactifying such a manifold to a manifold with corners of in general higher codimension, and we then deal with boundary value problems for cusp differential operators. We introduce an adapted Boutet de Monvel's calculus of pseudodifferential boundary value problems, and construct parametrices for elliptic cusp operators within this calculus. Fredholm solvability and elliptic regularity up to the boundary and up to infinity for boundary value problems on manifolds with polycylindrical ends follows.     

On a class of semipositone discrete boundary value problems

In this paper, we deal with a class of semipositone discrete boundary value problems via critical point theory developed by Chang, and obtain nonexistence and multiplicity results on sublinear nonlinearities and an existence result on superlinear nonlinearities, respectively.     

Discrete non-linear inequalities and applications to boundary value problems

In this paper, we generalize some existing discrete Gronwall-Bellman-Ou-Iang-type inequalities to more general situations. These are in turn applied to study the boundedness, uniqueness, and continuous dependence of solutions of certain discrete boundary value problem for difference equations.     

Resolvent Estimates for Boundary Value Problems on Large Intervals Via the Theory of Discrete Approximations

In many applications such as the stability analysis of traveling waves, it is important to know the spectral properties of a linear differential operator on the whole real line. We investigate the approximation of this operator and its spectrum by finite interval boundary value problems from an abstract point of view. Under suitable assumptions on the boundary operators, we prove that the approximations converge regularly (in the sense of discrete approximations) to the all line problem, which has strong implications for the behavior of resolvents and spectra. As an application, we obtain resolvent estimates for abstract coupled hyperbolic–parabolic equations. Furthermore, we show that our results apply to the FitzHugh–Nagumo system.     

On minkowski metric and weighted tchebyshev norm in vector optimization ∗

The typical approach in solving vector optimization problems is to scalarize the vector cost function into a single cost function by means of some utility or value function. A very large class of utility function is given by the Minkowski’s metric proposed by Charnes and Cooper in the context of goal programming. This includes the special case of linear scalarization and the weighted Tchebyshev norm. We shall furnish a rigorous justification that there is no equivalent relationship between the general vector optimization problem and scalarized optimization problems using any Minkowski’s metric utility function. Furthermore, we also show that the weighted Tchebyshev norm is, in some sense, the best amongst the class of Minkowski’s metric utility functions since it is the only scalarization method which yields an equivalence relation between the weak vector optimization problem and a set of scalar optimization problems, without any convexity assumption     

凹角域上Green函数有限元逼近的逐点估计及有限元外推

本文运用权范数方法证明了多角形区域上Green函数有限元逼近的逐点估计 |G_z(x)-G_z~h(x)|≤C(h~|lnh|~3/|x-z|~α),?x,z∈Ω,其中C为与x,z,h无关的常数θ<α<β_M,β_M=π/α_M,α_M为Ω的最大内角.由此可导出凹角域上有限元逼近的渐近展开. 考虑模型问题     

Designing robust liner shipping schedules: Optimizing recovery actions and buffer times

It is important for liner shipping companies to maintain cost efficient and robust liner shipping networks. Regularly, they set up pro-forma schedules, yet it is difficult to stay on time. We consider the problem of managing the delays. Therefore, we need to determine an optimal recovery policy and buffer time allocation to the ship route in order to minimize the total costs associated with delays and recovery actions, such as increasing sailing speed. We introduce a general framework consisting of a mixed integer programming formulation to solve discrete stochastic decision problems with short and long term decisions and apply this framework to the above described problem. Furthermore, we propose and test four heuristics for this problem. We compared the results of our method with an existing liner shipping route schedule and found a cost decrease of 28.9% after optimizing the buffer time distribution compared to the cost of sailing the current route schedule at constant speed.     

Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: survey

In this paper we provide a survey of characterizations of the nonnegativity and positivity of discrete quadratic functionals which arise as the second variation for nonlinear discrete calculus of variations problems. These characterizations are in terms of (i) (strict) conjugate and (strict) coupled intervals, (ii) the conjoined bases of the associated Jacobi difference equation, and (iii) the solution of the corresponding Riccati difference equation. The results depend on the form of the boundary conditions of the quadratic functional and, basically, we distinguish three types: (a) separable endpoints with zero right endpoint (this of course includes the simplest case of both zero endpoints), (b) separable endpoints, and (c) jointly varying endpoints.     

UNCONDITIONAL STABLE DIFFERENCE METHODS WITH INTRINSIC PARALLELISM FOR SEMILINEAR PARABOLIC SYSTEMS OF DIVERGENCE TYPE

The general finite difference schemes with intrinsic parallelism for the boundary value problem of the semilinear parabolic system of divergence type with bounded measurable coefficients is studied. By the approach of the discrete functional analysis, the existence and uniqueness of the discrete vector solutions of the nonlinear difference system with intrinsic parallelism are proved. Moreover the unconditional stability of the general difference schemes with intrinsic parallelism justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete initial data of the original problems in the discrete W_2~(2,1) (Q△) norms. Finally the convergence of the discrete vector solutions of the certain difference schemes with intrinsic parallelism to the unique generalized solution of the original semilinear parabolic problem is proved.     

On the numerical analysis of nonlinear twofold saddle point problems

We provide a general abstract theory for the solvability andGalerkin approximation of nonlinear twofold saddle point problems.In particular, a Strang error estimate containing the consistencyterms arising from the approximation of the continuous operatorsinvolved is deduced. Then we apply these results to analysea fully discrete Galerkin scheme for a twofold saddle pointformulation of a nonlinear elliptic boundary value problem indivergence form. Some numerical results are also presented.     

Scalar discrete nonlinear multipoint boundary value problems

In this paper we provide sufficient conditions for the existence of solutions to scalar discrete nonlinear multipoint boundary value problems. By allowing more general boundary conditions and by imposing less restrictions on the nonlinearities, we obtain results that extend previous work in the area of discrete boundary value problems [Debra L. Etheridge, Jesús Rodriguez, Periodic solutions of nonlinear discrete-time systems, Appl. Anal. 62 (1996) 119–137; Debra L. Etheridge, Jesús Rodriguez, Scalar discrete nonlinear two-point boundary value problems, J. Difference Equ. Appl. 4 (1998) 127–144].