Approximations of Solutions to Abstract Neutral Functional Differential Equation

In the present work, we study the approximations of solutions to the abstract neutral functional differential equations with bounded delay. We consider an associated integral equation and a sequence of approximate integral equations. We establish the existence and uniqueness of the solutions to every approximate integral equation using the fixed point arguments. We then prove the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Next, we consider the Faedo–Galerkin approximations of the solutions and prove some convergence results. Finally, we demonstrate the application of the results established.     

Integral Equations in Measure Spaces

In this paper, we consider a class of integral equations in measure spaces, and the corresponding integral inequalities. Special cases are Volterra type integral equations and Gronwall type integral inequalities. We give different necessary and su.cient, and only su.cient conditions which together with the Lipschitz condition imply the existence and the uniqueness of solutions of the considered integral equations. We study the successive approximations for the considered integral equations. We derive estimates for the solutions of the studied integral equations and integral inequalities. Submitted: June 20, 2000?Revised: July 10, 2001     

Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities

We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global equations, and establish a sup ? inf type Harnack inequality of Schoen for integral equations.     

Stochastic PDIEs with nonlinear Neumann boundary conditions and generalized backward doubly stochastic differential equations driven by Lévy processes

In this paper, a new class of generalized backward doubly stochastic differential equations (GBDSDEs in short) driven by Teugels martingales associated with Lévy process and the integral with respect to an adapted continuous increasing process is investigated. We obtain the existence and uniqueness of solutions to these equations. A probabilistic interpretation for solutions to a class of stochastic partial differential integral equations (PDIEs in short) with a nonlinear Neumann boundary condition is given.     

Stationary Schr?dinger equation in nonrelativistic quantum mechanics and the functional integral

We formulate a method for representing solutions of homogeneous second-order equations in the form of a functional integral or path integral. As an example, we derive solutions of second-order equations with constant coefficients and a linear potential. The method can be used to find general solutions of the stationary Schr?dinger equation. We show how to find the spectrum and eigenfunctions of the quantum oscillator equation. We obtain a solution of the stationary Schr?dinger equation in the semiclassical approximation, without a singularity at the turning point. In that approximation, we find the coefficient of transmission through a potential barrier. We obtain a representation for the elastic potential scattering amplitude in the form of a functional integral.     

Investigation of solutions to one family of mathematical models of living systems

We consider a family of integral equations used as models of some living systems. We prove that an integral equation is reducible to the equivalent Cauchy problem for a non-autonomous differential equation with point or distributed delay dependently on the choice of the survival function of system elements. We also study the issues of the existence, uniqueness, nonnegativity, and continuability of solutions. We describe all stationary solutions and obtain sufficient conditions for their asymptotic stability. We have found sufficient conditions for the existence of a limit of solutions on infinity and present an example of equations where the rate of generation of elements of living systems is described by a unimodal function (namely, the Hill function).     

Faedo–Galerkin Approximation of Solution for a Nonlocal Neutral Fractional Differential Equation with Deviating Argument

In this work, we study a class of nonlocal neutral fractional differential equations with deviated argument in the separable Hilbert space. We obtain an associated integral equation and then, consider a sequence of approximate integral equations. We investigate the existence and uniqueness of the mild solution for every approximate integral equation by virtue of the theory of analytic semigroup theory via the technique of Banach fixed point theorem. Next we demonstrate the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. The Faedo–Galerkin approximation of the solution is studied and demonstrated some convergence results. Finally, we give an example.

     

On a Class of Systems of Linear and Nonlinear Fredholm Integral Equations of the Third Kind with Multipoint Singularities

Based on a new approach, we show that finding solutions for a class of systems of linear (respectively, nonlinear) Fredholm integral equations of the third kind with multipoint singularities is equivalent to finding solutions of systems of linear (respectively, nonlinear) Fredholm integral equations of the second kind with additional conditions. We study the existence, nonexistence, uniqueness, and nonuniqueness of solutions for this class of systems of Fredholm integral equations of the third kind with multipoint singularities.     

Closed-form solutions for some classes of singular integral equations with Cauchy kernel and infinite integration domain

We obtain closed-form solutions of singular integral equations with Cauchy kernels in the case of an infinite integration domain. Further, we find natural conditions for the uniqueness of solutions of these equations for the case in which the solution contains arbitrary functions or arbitrary constants.     

On closed-form solutions of triple series equations involving Laguerre polynomials

We consider some triple series equations involving generalized Laguerre polynomials. These equations are reduced to triple integral equations for Bessel functions. The closed-form solutions of the triple integral equations for Bessel functions are obtained and, finally, we get the closed-form solutions of triple series equations for Laguerre polynomials.     

一类二变量积分不等式及其在积分-微分方程中的应用

建立了一类二变量的积分不等式,该不等式包含了一个一重积分和两个二重积分.利用分析技巧,给出了积分不等式中未知函数的估计.这一结果可以作为研究积分-微分方程解的定性性质的工具.     

Numerical solution of multiple nonlinear Volterra integral equations

We analyze a discretization method for solving nonlinear integral equations that contain multiple integrals. These equations include integral equations with a Volterra series, instead of a single integral term, on one side of the equation. We prove existence and uniqueness of solutions, and convergence and estimates of the order of convergence for the numerical methods of solution.     

Some new integral inequalities and their applications in studying the stability of nonlinear integro-differential equations with time delay

In this paper, we generalize some integral inequalities to more general situations, and the inequalities of Pachpatte type are corollaries of our's. We establish bounds on the solutions, and we show the usefulness of our results in investigating the asymptotic behavior and the stability on the solutions of integral equations, differential equations and integro-differential equations with time delay.     

Regularity of Backward Stochastic Volterra Integral Equations in Hilbert Spaces

This article investigates backward stochastic Volterra integral equations in Hilbert spaces. The existence and uniqueness of their adapted solutions is reviewed. We establish the regularity of the adapted solutions to such equations by means of Malliavin calculus. For an application, we study an optimal control problem for a stochastic Volterra integral equation driven by a Hilbert space-valued fractional Brownian motion. A Pontryagin-type maximum principle is formulated for the problem and an example is presented.     

On the construction of a nonnegative solution for one class of Urysohn-type nonlinear integral equations on a semiaxis

We investigate one class of Urysohn-type nonlinear integral equations with noncompact operator. It is assumed that a Wiener–Hopf–Hankel-type linear integral operator is a local minorant for the initial Urysohn operator. We prove an alternative theorem on the existence of positive solutions and investigate the asymptotic behavior of the obtained solutions at infinity.     

Singularities of Solutions of One Class of Equations of Continuum Mechanics

We analyze one class of families of integral equations and describe the dependence of the singularities of solutions of integral equations on the dimensions of the families of kernels of equations. On the basis of these results, we propose procedures for the construction of approximate solutions for a small parameter.     

Stochastic PDIEs and backward doubly stochastic differential equations driven by Lévy processes

In this paper, a new class of backward doubly stochastic differential equations driven by Teugels martingales associated with a Lévy process satisfying some moment condition and an independent Brownian motion is investigated. We obtain the existence and uniqueness of solutions to these equations. A probabilistic interpretation for solutions to a class of stochastic partial differential integral equations is given.     

Some generalized integral inequalities and their applications

In this paper, we generalize some integral inequalities to more general situations. We establish bounds on the solutions and, by means of examples, we show the usefulness of our results in investigating the global existence of the solutions of integral equations and differential equations.     

On the Nyström discretization of integral equations on planar curves with corners

The Nyström method can produce ill-conditioned systems of linear equations when applied to integral equations on domains with corners. This defect can already be seen in the simple case of the integral equations arising from the Neumann problem for Laplace?s equation. We explain the origin of this instability and show that a straightforward modification to the Nyström scheme, which renders it mathematically equivalent to Galerkin discretization, corrects the difficulty without incurring the computational penalty associated with Galerkin methods. We also present the results of numerical experiments showing that highly-accurate solutions of integral equations on domains with corners can be obtained, irrespective of whether their solutions exhibit bounded or unbounded singularities, assuming that proper discretizations are used.     

Bifurcation of periodic solutions for wave equations with scalar nonlinearities

We treat wave equations with “scalar nonlinearities” and demonstrate the connection between the bifurcation theory of an associated system of Hammerstein integral equations and the existence of periodic solutions of the nonlinear wave equation.