On meromorphic solutions of linear partial differential equations of second order

This paper is concerned with entire and meromorphic solutions of linear partial differential equations of second order with polynomial coefficients. We will characterize entire solutions for a class of partial differential equations associated with the Jacobi differential equations, and give a uniqueness theorem for their meromorphic solutions in the sense of the value distribution theory, which also applies to general linear partial differential equations of second order. The results are complemented by various examples for completeness.     

Invariant imbedding and Riccati transformations

From its inception, the theory of invariant imbedding has been concerned with the study of various relations between the inputs and outputs of various physical processes. Where the processes could be modelled by differential or integro-differential equations, these ideas have led to the heuristic development of various functional relationships for the solutions of these equations. In this work, we show that for a general class of two point boundary value problems these relations can be obtained from mathematical arguments rather than physical ones. The principal result is the establishment of the equivalence of solving a family of two point boundary value problems and that of determining the existence of two transformations on the set of solutions of the given differential equations. We refer to these transformations as Riccati transformations. They are shown to be determined by a set of initial value problems which generalize the invariant imbedding equations obtained by previous authors. We work in the coordinate free setting of a Banach space. The usefulness of this approach is shown as we are able to readily extend our results to nonlocal and multipoint boundary conditions. An indication is made of how a similar theory applies to a class of problems for difference equations.     

Periodic nonautonomous differential equations with noninstantaneous impulsive effects

In this paper, we study a new class of periodic nonautonomous differential equations with periodic noninstantaneous impulsive effects. A concept of noninstantaneous impulsive Cauchy matrix is introduced, and some basic properties are considered. We give the representation of solutions to the homogeneous problem and nonhomogeneous problem by using noninstantaneous impulsive Cauchy matrix, and the variation of constants method, adjoint systems, and periodicity of solutions is verified under standard periodicity conditions. Further, we show the existence and uniqueness of solutions of semilinear problem and establish existence result for periodic solutions via Brouwer fixed point theorem and uniqueness and global asymptotic stability via Banach fixed point theorem.     

Existence and uniqueness of solutions for delay evolution equations of second order in time

We prove some results on the existence and uniqueness of solutions for a class of evolution equations of second order in time, containing some hereditary characteristics. Our theory is developed from a variational point of view, and in a general functional setting which permits us to deal with several kinds of delay terms. In particular, we can consider terms which contain spatial partial derivatives with deviating arguments.     

关于2n阶常微分方程两点边值问题解的存在性与唯一性

本文利用Leray－Schauder度理论建立了一类2n阶非线性常微分方程两点边值问题解的存在性与唯一性定理，以及利用Fredholm择一原理与Fourier展式，建立了一类2n阶线性常微分方程两点边值问题解的存在性唯一性定理．     

Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments

We consider nonlinear differential equations with piecewise constant arguments in the general case. This is based in the study of an equivalent integral equation, and in a solution of an integral inequality of Gronwall type. We establish the existence, uniqueness and the asymptotic behavior of the solutions of the equations. Equivalences, including unbounded solutions, with simpler equations are obtained.     

Stochastic partial differential equations with unbounded and degenerate coefficients

In this article, using DiPerna-Lions theory (DiPerna and Lions, 1989) [1], we investigate linear second order stochastic partial differential equations with unbounded and degenerate non-smooth coefficients, and obtain several conditions for existence and uniqueness. Moreover, we also prove the L¹-integrability and a general maximal principle for generalized solutions of SPDEs. As applications, we study nonlinear filtering problem and also obtain the existence and uniqueness of generalized solutions for a degenerate nonlinear SPDE.     

Analog of the cauchy function for a generalized equation with several deviations of the argument

We define a special multiplication of function series (skew multiplication) and a generalized Riemann-Stieltjes integral with function series as integration arguments. The generalized integrals and the skew multiplication are related by an integration by parts formula. The generalized integrals generate a family of linear generalized integral equations, which includes a family (represented in integral form via the Riemann-Stieltjes integral) of linear differential equations with several deviating arguments. A specific feature of these equations is that all deviating functions are defined on the same closed interval and map it into itself. This permits one to avoid specifying the initial functions and imposing any additional constraints on the deviating functions. We present a procedure for constructing the fundamental solution of a generalized integral equation. With respect to the skew multiplication, it is invertible and generates the product of the fundamental solution (a function of one variable) by its inverse function (a function of the second variable). Under certain conditions on the parameters of the equation, the product has all specific properties of the Cauchy function. We introduce the notion of adjoint generalized integral equation, obtain a representation of solutions of the original equation and the adjoint equation in generalized integral Cauchy form, and derive sufficient conditions for the convergence of solutions of a pair of adjoint equations.     

On the optimal control of systems governed by quasilinear integro-partial differential equations of parabolic type

Recently, M. N. O?uztöreli presented certain results on the existence and uniqueness of solutions of systems governed by a linear integro-partial differential equation of parabolic type with delayed arguments. Since his results admit only smooth coefficients, they could not be used directly in the study of the optimal control problems with bounded measurable control variables appearing in the coefficients of the system equations. In this paper, we consider a class of systems described by second-order quasilinear parabolic integro-partial differential equations with all but the second-order coefficients assumed bounded measurable. Our principal results are: Theorem 3.5, which establishes the existence and uniqueness of solutions of this class of systems (with controls in the coefficients), and Theorem 4.4, which gives a necessary condition for optimality for the corresponding controlled system.     

Teoremi di esistenza e di unicitá per le soluzioni deboli dei problemi al contorno per una classe di sistemi parabolici di equazioni del secondo ordine

In this paper we give some existence and uniqueness theorems for weak solutions of boundary value problems for a particular class of parabolic systems of linear partial differential equations of the second order with real coefficients. In particular some uniqueness theorems for equations with complex coefficients are deduced from the previous results.     

Backward stochastic partial differential equations with jumps and application to optimal control of random jump fields

We prove an existence and uniqueness result for a general class of backward stochastic partial differential equations (SPDE) with jumps. This is a type of equations, which appear as adjoint equations in the maximum principle approach to optimal control of systems described by SPDE driven by Lévy processes.     

The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications

This paper deals with a class of backward stochastic differential equations with Poisson jumps and with random terminal times. We prove the existence and uniqueness result of adapted solution for such a BSDE under the assumption of non-Lipschitzian coefficient. We also derive two comparison theorems by applying a general Girsanov theorem and the linearized technique on the coefficient. By these we first show the existence and uniqueness of minimal solution for one-dimensional BSDE with jumps when its coefficient is continuous and has a linear growth. Then we give a general Feynman-Kac formula for a class of parabolic types of second-order partial differential and integral equations (PDIEs) by using the solution of corresponding BSDE with jumps. Finally, we exploit above Feynman-Kac formula and related comparison theorem to provide a probabilistic formula for the viscosity solution of a quasi-linear PDIE of parabolic type.     

Parameter identification in nonlocal nonlinear evolution equations

We develop an approximation framework for identifying parameters in a general class of nonautonomous, nonlocal and nonlinear evolution equations. After establishing existence and uniqueness of solutions, we present a convergence theory for Galerkin approximations to inverse problems involving these equations. Our approach relies on the theory of maximal monotone operators in Banach spaces. An application to a nonautonomous nonlinear integral equation arising in heat flow is also discussed.     

On the general linear coupled system for diffusion in media with two diffusivities

For the general linear coupled system of partial differential equations arising in the theory of diffusion in media with double diffusivity, simple uniqueness criteria, and a method of solution of boundary value problems are established. The equations studied retain the so-called cross terms which have been neglected in all previous investigations. Moreover, these equations arise as generalizations of a number of existing theories; for example, heat flow in heterogeneous multicomponent systems, flow of water in fissured rocks and a model of an arms race. The simple inequalities obtained on the various constants of the theory which guarantee uniqueness of solutions and existence of source solutions might serve as guidelines in an experimental determination of these constants. The solution procedure involves solving two boundary value problems for the classical diffusion equation and the formulae given mean that closed form expressions can be deduced for a number of commonly occurring boundary value problems. The paper emphasizes the general equations without special reference to particular physical applications or boundary value problems.     

Approximation of Solutions of Fractional-Order Delayed Cellular Neural Network on $$\varvec{[0,\infty )}$$

In this paper, we study a class of fractional-order cellular neural network containing delay. We prove the existence and uniqueness of the equilibrium solution followed by boundedness. Based on the theory of fractional calculus, we approximate the solution of the corresponding neural network model over the interval \([0,\infty )\) using discretization method with piecewise constant arguments and variation of constants formula for fractional differential equations. Furthermore, we conclude that the solution of the fractional-delayed system can be approximated for large t by the solution of the equation with piecewise constant arguments, if the corresponding linear system is exponentially stable. At the end, we give two numerical examples to validate our theoretical findings.     

Linear measure functional differential equations with infinite delay

We use the theory of generalized linear ordinary differential equations in Banach spaces to study linear measure functional differential equations with infinite delay. We obtain new results concerning the existence, uniqueness, and continuous dependence of solutions. Even for equations with a finite delay, our results are stronger than the existing ones. Finally, we present an application to functional differential equations with impulses.     

A general adjoint relation for functional differential and Volterra integral equations with application to control

A general adjoint relation is developed between solutions of linear functional differential equations and linear Volterra integral equations. Several useful representations for solutions of such equations arise as a consequence of the adjoint relationship. These representations are then used to obtain directly several results for controlling systems described by either linear functional differential equations or linear Volterra integral equations.This work was supported by the National Science Foundation under Grant No. GK-5798.     

Linear evolution equations in scales of Banach spaces

This work is devoted to the study of a class of linear time-inhomogeneous evolution equations in a scale of Banach spaces. Existence, uniqueness and stability for classical solutions is provided. We study also the associated dual Cauchy problem for which we prove uniqueness in the dual scale of Banach spaces. The results are applied to an infinite system of ordinary differential equations but also to the Fokker-Planck equation associated with the spatial logistic model in the continuum.     

Bivariate delta-evolution equations and convolution polynomials: Computing polynomial expansions of solutions

This paper describes an application of Rota and collaborator’s ideas, about the foundation on combinatorial theory, to the computing of solutions of some linear functional partial differential equations. We give a dynamical interpretation of the convolution families of polynomials. Concretely, we interpret them as entries in the matrix representation of the exponentials of certain contractive linear operators in the ring of formal power series. This is the starting point to get symbolic solutions for some functional-partial differential equations. We introduce the bivariate convolution product of convolution families to obtain symbolic solutions for natural extensions of functional-evolution equations related to delta-operators. We put some examples to show how these symbolic methods allow us to get closed formulas for solutions of genuine partial differential equations. We create an adequate framework to base theoretically some of the performed constructions and to get some existence and uniqueness results.     

Uniqueness of Kottler spacetime and the Besse conjecture

We establish a black hole uniqueness theorem for Schwarzschild–de Sitter spacetime, also called Kottler spacetime, which satisfies Einstein's field equations of general relativity with positive cosmological constant. Our result concerns the class of static vacuum spacetimes with compact spacelike slices and regular maximal level set of the lapse function. We provide a characterization of the interior domain of communication of the Kottler spacetime, which surrounds an inner horizon and is surrounded by a cosmological horizon. The proof combines arguments from the theory of partial differential equations and differential geometry, and is centered on a detailed study of a possibly singular foliation. We also apply our technique in the Riemannian setting, and establish the validity of the so-called Besse conjecture.