Asymptotics of the solution to a stationary piecewise-smooth reaction-diffusion equation with a multiple root of the degenerate equation

A piecewise-smooth second-order singularly perturbed differential equation whose right-hand side is a nonlinear function with a discontinuity on some curve is investigated. This is a new class of problems in the case where the degenerate equation has a multiple root on the left-hand side of the curve which separates the domain and an isolated root on the right-hand side of that curve. The asymptotics of a solution with an internal layer near a point on the discontinuous curve and the transition point is constructed. The method to construct the internal layer function is proposed. The behavior of the solution in the internal layer consisting of four zones essentially differs from the case of isolated roots. For sufficiently small parameter values, the existence of a smooth solution with an internal layer from the multiple root of the degenerate equation to the isolated root in the neighborhood of a point on the discontinuous curve is proved. The method can be shown to be effective in the given example.     

Hyperbolic stefan problem in a curvilinear sector

The problem with unknown boundaries for a first-order semilinear hyperbolic system is studied in the case where the curve of definition of the initial conditions degenerates to a point. An existence and uniqueness theorem for a classical solution of the problem is proved for small t.     

Contrast Structures in Problems for a Stationary Equation of Reaction-Diffusion-Advection Type with Discontinuous Nonlinearity

A singularly perturbed boundary-value problem for a nonlinear stationary equation of reaction-diffusion-advection type is studied. A new class of problems with discontinuous advective and reactive terms is considered. The existence of contrast structures in problems of this type is proved, and an asymptotic approximation of the solution with an internal transition layer of arbitrary order of accuracy is obtained.     

Resonance elliptic variational inequalities with discontinuous nonlinearities of linear growth

We consider resonance elliptic variational inequalities with second-order differential operators and discontinuous nonlinearities of linear growth. The theorem on existence of a strong solution is proved. The initial-value problem is reduced to the problem of existence of a fixed point for a compact multivalued mapping and then the existence of this point is established by the Leray–Schauder method.     

不连续随机算子随机不动点定理及其应用

该文利用半序理论和随机压缩映象原理,得到了一类不连续随机增算子随机不动点的唯一存在定理.作为应用,考虑了R~n中含间断项的一阶随机微分积分方程初值问题.     

Solving neutral delay differential equations with state-dependent delays

In this paper we consider a class of neutral delay differential equations with state dependent delays. For such equations the possible discontinuity in the derivative of the solution at the initial point may propagate along the integration interval giving rise to subsequent points, called “breaking points”, where the solution derivative is still discontinuous. As a consequence, in a right neighbourhood of each such point we have to face a Cauchy problem where the equation has a discontinuous right-hand side. In this case the existence and the uniqueness of the solution is no longer guaranteed to the right of such points and hence the solution of the neutral equation may either cease to exist or bifurcate. After illustrating why uniqueness and existence of the solution is no longer guaranteed for general state-dependent problems and showing a possible way to detect these occurrences automatically, we explain how to generalize/regularize the problem in order to suitably extend the solution beyond the breaking point. This is important, for example, when exploring numerically the presence of possible periodic orbits.     

Mathematical analysis of a scalar multidimensional conservation law with discontinuous flux

We deal in this paper with a scalar conservation law, set in a bounded multidimensional domain, and such that the convective term is discontinuous with respect to the space variable. First, we introduce a weak entropy formulation for the homogeneous Dirichlet problem associated with the first-order reaction-convection equation that we consider. Then, we establish an existence and uniqueness property for the weak entropy solution. The method of doubling variables and a pointwise reasoning along the curve of discontinuity are used to state uniqueness. Finally, the vanishing viscosity method allows us to prove the existence result. Another method to obtain the existence of a solution, which relies on the regularization of the flux, is also detailled, at least for a particular case.     

Time-independent reaction–diffusion equation with a discontinuous reactive term

A two-dimensional singularly perturbed elliptic equation referred to in applications as the reaction–diffusion equation is considered. The nonlinearity describing the reaction is assumed to be discontinuous on a certain closed curve. On the basis of the generalized asymptotic comparison principle, the existence of smooth solution is proven and the accuracy of the asymptotic approximation is estimated.     

Maximum and minimum solutions for nonlinear parabolic problems with discontinuities

In this paper we examine nonlinear parabolic problems with a discontinuous right hand side. Assuming the existence of an upper solution φ and a lower solution ψ such that ψ ≤ φ, we establish the existence of a maximum and a minimum solution in the order interval [ψ, φ]. Our approach does not consider the multivalued interpretation of the problem, but a weak one side “Lipschitz” condition on the discontinuous term. By employing a fixed point theorem for nondecreasing maps, we prove the existence of extremal solutions in [ψ, φ for the original single valued version of the problem.     

Approximate Newton methods and homotopy for stationary operator equations

A quadratically convergent algorithm based on a Newton-type iteration is defined to approximate roots of operator equations in Banach spaces. Fréchet derivative operator invertibility is not required; approximate right inverses are used in a neighborhood of the root. This result, which requires an initially small residual, is sufficiently robust to yield existence; it may be viewed as a generalized version of the Kantorovich theorem. A second algorithm, based on continuation via single, Euler-predictor-Newton-corrector iterates, is also presented. It has the merit of controlling the residual until the homotopy terminates, at which point the first algorithm applies. This method is capable of yielding existence of a solution curve as well. An application is given for operators described by compact perturbations of the identity.     

Cauchy problem with general discontinuous initial data along a smooth curve for 2-d Euler system

We study the Cauchy problems for the isentropic 2-d Euler system with discontinuous initial data along a smooth curve. All three singularities are present in the solution: shock wave, rarefaction wave and contact discontinuity. We show that the usual restrictive high order compatibility conditions for the initial data are automatically satisfied. The local existence of piecewise smooth solution containing all three waves is established.     

On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow

Summary. The aim of this work is to study a decoupled algorithm of a fixed point for solving a finite element (FE) problem for the approximation of viscoelastic fluid flow obeying an Oldroyd B differential model. The interest for this algorithm lies in its applications to numerical simulation and in the cost of computing. Furthermore it is easy to bring this algorithm into play. The unknowns are the viscoelastic part of the extra stress tensor, the velocity and the pressure. We suppose that the solution is sufficiently smooth and small. The approximation of stress, velocity and pressure are resp. discontinuous, continuous, continuous FE. Upwinding needed for convection of , is made by discontinuous FE. The method consists to solve alternatively a transport equation for the stress, and a Stokes like problem for velocity and pressure. Previously, results of existence of the solution for the approximate problem and error bounds have been obtained using fixed point techniques with coupled algorithm. In this paper we show that the mapping of the decoupled fixed point algorithm is locally (in a neighbourhood of ) contracting and we obtain existence, unicity (locally) of the solution of the approximate problem and error bounds. Received July 29, 1994 / Revised version received March 13, 1995     

Global exponential convergence in a delayed almost periodic Nicholson's blowflies model with discontinuous harvesting

This paper is concerned with a delayed Nicholson's blowflies model with discontinuous harvesting, which is described by an almost periodic nonsmooth dynamical system. Under some reasonable assumptions on the discontinuous harvesting function, by using the Filippov regulation techniques and the theory of dichotomy, together with the Halanay inequality, we establish some new criteria on the existence of positive almost periodic solution and its convergence. An example with numerical simulation is also presented to support the theoretical results.     

Existence and smoothness of continuous and discrete solutions of a two-dimensional shallow water problem over movable beds with nonlinear sediment transport relationship

A two-dimensional shallow water system over movable beds with nonlinear sediment transport relationship is considered in this paper. The existence of the solutions for the continuous system is proved here and their smoothness is investigated. A Galerkin method is employed to obtain a finite-dimensional problem. A Brouwer fixed point theorem is employed for this problem and it is shown that the model equations are satisfied by the limits of the resulting solution sequences.We also consider the discretized problem using a local discontinuous Galerkin scheme. We perform an error analysis and show that the method is convergent and the error is bounded according to a specific norm defined herein.     

On a class of Schrödinger systems with discontinuous nonlinearities

Using a non-smooth critical point theory for locally Lipschitz functionals, we investigate a class of stationary Schrödinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow up at infinity. The existence of nontrivial solution is obtained.     

Dynamical behavior for a class of predator–prey system with general functional response and discontinuous harvesting policy

In this paper, we introduce a class of predator–prey system with general functional response, whose harvesting policy is modeled by a discontinuous function. Based on the differential inclusions theory, topological degree theory in set‐valued analysis and generalized Lyapunov approach, we analyze the existence, uniqueness and global asymptotic stability of positive periodic solution. In particular, a series of useful criteria on existence, uniqueness and global asymptotic stability of the positive equilibrium point are established for the autonomous system corresponding to the non‐autonomous biological and mathematical model with a discontinuous right‐hand side. Moreover, some new sufficient conditions are provided to guarantee the global convergence in measure of harvesting solution and convergence in finite time of any positive solution for the autonomous discontinuous biological system. The obtained results, which improve and generalize previous works on dynamical behavior in the literature, are of interest for understanding and designing biological system with not only continuous or even Lipschitz continuous but also discontinuous harvesting function. Finally, we give three examples with numerical simulations to show the applicability and effectiveness of our main results. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses

In this paper, we present a general class of BAM neural networks with discontinuous neuron activations and impulses. By using the fixed point theorem in differential inclusions theory, we investigate the existence of periodic solution for this neural network. By constructing the suitable Lyapunov function, we give a sufficient condition which ensures the uniqueness and global exponential stability of the periodic solution. The results of this paper show that the Forti’s conjecture is true for BAM neural networks with discontinuous neuron activations and impulses. Further, a numerical example is given to demonstrate the effectiveness of the results obtained in this paper.     

Fixed point theorems for multivalued operators and application to discontinuous quasilinear BVP's

In this article, we prove an abstract fixed point theorem for increasing multivalued operators in ordered topological spaces, which provides an efficient tool to obtain qualitative results on the solution set of discontinuous quasilinear elliptic boundary value problems. In particular, we are able to show the existence of extremal solutions of such problems.     

Model oblique derivative problem for the heat equation with a discontinuous boundary function

The oblique derivative problem for the heat equation is considered in a model formulation with a boundary function that can be discontinuous and with the boundary condition understood as the limit in the normal direction almost everywhere on the lateral boundary of the domain. An example is given showing that the solution is not unique in this formulation. A solution is sought in the parabolic Zygmund space H ₁, which is an analogue of the parabolic Hölder space for an integer smoothness exponent. A subspace of H ₁ is introduced in which the existence and uniqueness of the solution is proved under suitable assumptions about the data of the problem.     

Nonlinear eigenvalue Neumann problems with discontinuities

In this paper we study nonlinear eigenvalue problems with Neumann boundary conditions and discontinuous terms. First we consider a nonlinear problem involving the p-Laplacian and we prove the existence of a solution for the multivalued approximation of it, then we pass to semilinear problems and we prove the existence of multiple solutions. The approach is based on the critical point theory for nonsmooth locally Lipschitz functionals.