Fixed points for discontinuous monotone operators

In this paper, we use the monotone iterative method to prove the existence of the minimal and maximal fixed points of a discontinuous strongly monotone operator on an order interval in an ordered normed linear space. As an example of the application of our results, we show the existence of extremal solutions to a class of discontinuous initial value problems.     

Existence of solutions to a nonvariational elliptic boundary value problem with parameter and discontinuous nonlinearity

We consider the question of the existence of the Dirichlet problem for second-order elliptic equations with spectral parameter and a nonlinearity discontinuous with respect to the phase variable. Here it is not assumed that the differential operator is formally selfadjoint. Using the method of upper and lower solutions, we establish results on the existence of nontrivial (positive and negative) solutions under positive values of the spectral parameter for the problems under study.     

序Banach空间中一类算子方程的可解性

利用改进的Mnch紧性条件,研究了一类算子方程Lx=Nx的解的存在性．本文结果并没有要求算子L和N的连续性．作为应用,讨论了右端项不连续的隐式椭圆偏微分方程的边值问题的解的存在性．     

间断系数的二阶椭圆型方程解的正则性

该文研究的问题源自于生物学与物理学中具有间断介电系数的静电场。作者以拟微分算子为主要工具讨论具间断系数的半线性二阶椭圆型方程解的存在性和正则性 。     

Existence of weak solutions of stochastic differential equations with standard and fractional Brownian motion,discontinuous coefficients,and a partly degenerate diffusion operator

We obtain estimates for functionals of solutions of stochastic differential equations with standard and fractional Brownian motion. We prove a theorem on the existence of weak solutions of stochastic differential equations with standard and fractional Brownian motion, discontinuous coefficients, and a partly degenerate diffusion operator.     

On a class of scalar variational inequalities with measure data

We consider a class of variational inequalities defining the so-called hysteresis play operator. We propose a new approach to discontinuous BV-solutions based on measure theoretical arguments, which enable us to infer the existence of solutions as a simple consequence of the classical theory. In this way, we generalize a recent result where only continuous BV-solutions were studied. We also provide a representation formula which allows to deduce the continuity of the play operator from general theorems on hysteresis operators.     

Existence of solutions of the nonautonomous abstract Cauchy problem of second order

In this paper the existence of solutions of a nonautonomous abstract Cauchy problem of second order is considered. Assuming appropriate conditions on the operator of the equation, we establish the existence of mild solutions and, in some cases, we construct an evolution operator associated to the homogeneous equation. Using this evolution operator we obtain existence of solutions for the inhomogeneous equation. Finally, we apply our results to study the existence of solutions of the nonautonomous wave equation.     

Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients

We deal with a Hamilton-Jacobi equation with a Hamiltonian that is discontinuous in the space variable. This is closely related to a conservation law with discontinuous flux. Recently, an entropy framework for single conservation laws with discontinuous flux has been developed which is based on the existence of infinitely many stable semigroups of entropy solutions based on an interface connection. In this paper, we characterize these infinite classes of solutions in terms of explicit Hopf-Lax type formulas which are obtained from the viscosity solutions of the corresponding Hamilton-Jacobi equation with discontinuous Hamiltonian. This also allows us to extend the framework of infinitely many classes of solutions to the Hamilton-Jacobi equation and obtain an alternative representation of the entropy solutions for the conservation law. We have considered the case where both the Hamiltonians are convex (concave). Furthermore, we also deal with the less explored case of sign changing coefficients in which one of the Hamiltonians is convex and the other concave. In fact in convex-concave case we cannot expect always an existence of a solution satisfying Rankine-Hugoniot condition across the interface. Therefore the concept of generalised Rankine-Hugoniot condition is introduced and prove existence and uniqueness.     

Existence results for operator equations in partially ordered sets and applications

In this paper, we prove existence results for operator equations in partially ordered sets, and apply the obtained results to operator equations in ordered Banach spaces and to semilinear functional parabolic and elliptic problems involving discontinuous nonlinearities.     

Existence and approximation of solutions for discontinuous functional differential equations

In this paper existence of solutions of initial value problems for discontinuous functional differential equations is investigated firstly. By applying the method of upper and lower solutions, which may be discontinuous, some existence results of extremal solutions are obtained. Furthermore, we also develop a monotone iterative technique for obtaining extremal solutions which are obtained as limits of monotone sequences.     

On solvability of variational inequalities with discontinuous semimonotone operators

By using the method of monotone operators, a theorem on the existence of the solution with a special property is obtained for an elliptic variational inequality with discontinuous semimonotone operator; this theorem is then used to prove the existence of a semicorrect solution of a variational inequality with a differential semilinear high-order operator of elliptic type with a nonsymmetric linear part and discontinuous nonlinearity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 443–447, March, 1993.     

Regularity of solutions to a reaction–diffusion equation on the sphere: the Legendre series approach

In the paper, we study some ‘a priori’ properties of mild solutions to a single reaction–diffusion equation with discontinuous nonlinear reaction term on the two‐dimensional sphere close to its poles. This equation is the counterpart of the well‐studied bistable reaction–diffusion equation on the Euclidean plane. The investigation of this equation on the sphere is mainly motivated by the phenomenon of the fertilization of oocytes or recent studies of wave propagation in a model of immune cells activation, in which the cell is modeled by a ball. Because of the discontinuous nature of reaction kinetics, the standard theory cannot guarantee the solution existence and its smoothness properties. Moreover, the singular nature of the diffusion operator near the north/south poles makes the analysis more involved. Unlike the case in the Euclidean plane, the (axially symmetric) Green's function for the heat operator on the sphere can only be represented by an infinite series of the Legendre polynomials. Our approach is to consider a formal series in Legendre polynomials obtained by assuming that the mild solution exists. We show that the solution to the equation subject to the Neumann boundary condition is C¹ smooth in the spatial variable up to the north/south poles and Hölder continuous with respect to the time variable. Our results provide also a sort of ‘a priori’ estimates, which can be used in the existence proofs of mild solutions, for example, by means of the iterative methods. Copyright © 2017 John Wiley & Sons, Ltd.     

Periodic solutions of a bang bang recurrence equation with least periods 1 through 9

In this paper, we study a very simple three term recurrence relation involving the discontinuous Heaviside step function. One reason for studying such an relation is that solutions of our recurrence relation are steady state distributions in some basic neural network models. Since analytic tools cannot be used to handle discontinuous models such as ours, existence of periodic solutions is investigated by combining combinatorial elimination technique as well as existence arguments for linear systems. By such means, we are able to obtain all periodic solutions with least periods 1 through 9. Some periodic solutions with periods 15, 18, 42 and 72 can also be found, but exhaustive results are not yet available.     

Periodic Solutions of Chebyshev Polynomials with Respect to the Discrete Variable

In this paper, we study a very simple three term recurrence relation involving the discontinuous Heaviside step function. One reason for studying such an relation is that solutions of our recurrence relation are steady state distributions in some basic neural network models. Since analytic tools cannot be used to handle discontinuous models such as ours, existence of periodic solutions is investigated by combining combinatorial elimination technique as well as existence arguments for linear systems. By such means, we are able to obtain all periodic solutions with least periods 1 through 8. Some periodic solutions with periods 12, 20 and 36 can also be found, but exhaustive results are not yet available.     

Existence of optimal controls for a class of systems governed by differential inclusions on a Banach space

Using Cesari's approach, we prove the existence of optimal controls for a class of systems governed by differential inclusions on a Banach space having the Radon-Nikodym property. Theorem 3.1 gives the existence result for optimal relaxed controls under fairly general assumptions on the system and the admissible controls. This result depends on a fundamental result (Theorem 2.1) that proves the existence of mild solutions of differential inclusions on a Banach space, which has also independent interest. Further, the preparatory results, such as Lemma 3.1 and Lemma 3.2, are also useful in the study of time-optimal and terminal control problems.For illustration of the results, we present two examples, one on distributed controls for a class of systems governed by nonlinear parabolic equations and the other on boundary controls with discontinuous boundary operator.This work is supported in part by the National Science and Engineering Council of Canada under Grant No. 7109.     

Existence results for degenerate p(x)-Laplace equations with Leray-Lions type operators

We show the existence and multiplicity of solutions to degenerate p(x)-Laplace equations with Leray-Lions type operators using direct methods and critical point theories in Calculus of Variations and prove the uniqueness and nonnegativeness of solutions when the principal operator is monotone and the nonlinearity is nonincreasing. Our operator is of the most general form containing all previous ones and we also weaken assumptions on the operator and the nonlinearity to get the above results. Moreover, we do not impose the restricted condition on p(x) and the uniform monotonicity of the operator to show the existence of three distinct solutions.     

6-Periodic travelling waves in an artificial neural network with bang–bang control

Doubly periodic travelling waves can be used to describe dynamic patterns of signals that govern movements of animals. In this paper, we study the existence of such waves in cellular networks involving the discontinuous Heaviside step function. This is done by finding ω-periodic solutions of an accompanying recurrence relation with a priori unknown parameters and the Heaviside function. Since analytic tools cannot be used to handle discontinuous models such as ours, existence of periodic solutions is investigated by means of symmetry, combinatorial techniques and accompanying linear systems. By such means, we are able to obtain all periodic solutions with least periods 1 through 6. Our techniques are new and good for other periodic solutions with relatively small periods.     

Qualitative Properties of Eigenvectors Related to Multivalued Operators and some Existence Results

In this work, we study a class of nonlinear eigenvalue problems related to fully discontinuous operators. In particular, we prove the existence of a critical point for two distinct problems. Connected with this problem, we also study a minimization problem with constraint, and we investigate the existence of solutions for a resonant case near zero. Moreover, we give some estimates and qualitative properties of solutions by using the relative rearrangement theory.     

On ℒ Characteristic of Some Potential Type Operators with Oscillating Symbols and Singularities of the Kernels on a Sphere

We obtain some new existence theorems for the maximal and minimal fixed points of discontinuous increasing operators. As applications, we consider the existence of the maximal and minimal solutions of nonlinear integro-differential equations with discontinuous terms in Banach spaces. Our results generalize and improve many well-known conclusions. This revised version was published online in June 2006 with corrections to the Cover Date.     

不连续泛函微分方程周期边值问题解的单调迭代技巧(英文)

文章通过运用上下解的方法证明了一类带有周期边值条件的不连续泛函微分方程解的存在性.并且采用不连续的上下解显著扩大了上下解的选择范围.进而,通过构造具有一致收敛性的上下解的单调迭代序列,得到了该周期边值问题的极值解,即最大、小解.