ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES

In this paper, we focus on anticipated backward stochastic Volterra integral equations(ABSVIEs) with jumps. We solve the problem of the well-posedness of so-called M-solutions to this class of equation, and analytically derive a comparison theorem for them and for the continuous equilibrium consumption process. These continuous equilibrium consumption processes can be described by the solutions to this class of ABSVIE with jumps.Motivated by this, a class of dynamic risk measures induced by ABSVIEs with jumps are discussed.

NEW NUMERICAL METHOD FOR VOLTERRA INTEGRAL EQUATION OF THE SECOND KIND IN PIEZOELASTIC DYNAMIC PROBLEMS

The elastodynamic problems of piezoelectric hollow cylinders and spheres under radial deformation can be transformed into a second kind Volterra integral equation about a function with respect to time, which greatly simplifies the solving procedure for such elastodynamic problems. Meanwhile, it becomes very important to find a way to solve the second kind Volterra integral equation effectively and quickly. By using an interpolation function to approximate the unknown function, two new recursive formulae were derived, based on which numerical solution can be obtained step by step. The present method can provide accurate numerical results efficiently. It is also very stable for long time calculating.

Backward stochastic Volterra integral equations——a brief survey

In this paper, we present a brief survey on the updated theory of backward stochas-tic Volterra integral equations （BSVIEs, for short）. BSVIEs are a natural generalization of backward stochastic diff erential equations （BSDEs, for short）. Some interesting motivations of studying BSVIEs are recalled. With proper solution concepts, it is possible to establish the corresponding well-posedness for BSVIEs. We also survey various comparison theorems for solutions to BSVIEs.

Numerical Solutions of Volterra Equations Using Galerkin Method with Certain Orthogonal Polynomials

This work is aim at providing a numerical technique for the Volterra integral equations using Galerkin method. For this purpose, an effective matrix formulation is proposed to solve linear Volterra integral equations of the first and second kind respectively using orthogonal polynomials as trial functions which are constructed in the interval [-1,1] with respect to the weight function w(x)=1+x2. The efficiency of the proposed method is tested on several numerical examples and compared with the analytic solutions available in the literature.

SINGULAR CONTROL OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS

This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations,where the solution X^(u,ξ)(t)=X(t)is given X(t)=φ(t)+∫_(0)^(t) b(t,s,X(s),u(s))ds+∫_(0)^(t)σ(t,s,X(s),u(s))dB(s)+∫_(0)^(t)h(t,s)dξ(s).by Here d B(s)denotes the Brownian motion It?type differential,ξdenotes the singular control(singular in time t with respect to Lebesgue measure)and u denotes the regular control(absolutely continuous with respect to Lebesgue measure).Such systems may for example be used to model harvesting of populations with memory,where X(t)represents the population density at time t,and the singular control processξrepresents the harvesting effort rate.The total income from the harvesting is represented by J(u, ξ) = E[∫_(0)^(t) f_(0)(t,X(t), u(t))dt + ∫_(0)^(t)f_(1)(t,X(t))dξ(t) + g(X(T))] for the given functions f0,f1 and g,where T>0 is a constant denoting the terminal time of the harvesting.Note that it is important to allow the controls to be singular,because in some cases the optimal controls are of this type.Using Hida-Malliavin calculus,we prove sufficient conditions and necessary conditions of optimality of controls.As a consequence,we obtain a new type of backward stochastic Volterra integral equations with singular drift.Finally,to illustrate our results,we apply them to discuss optimal harvesting problems with possibly density dependent prices.

GLOBAL SOLUTIONS OF SYSTEMS OF NONLINEAR IMPULSIVE VOLTERRA INTEGRAL EQUATIONS IN BANACH SPACES

The existence of solutions for systems of nonlinear impulsive Volterra integral equations on the infinite interval R+ with an infinite number of moments of impulse effect in Banach spaces is studied. Some existence theorems of extremal solutions are obtained, which extend the related results for this class of equations on a finite interval with a finite. number of moments of impulse effect. The results are demonstrated by means of an example of an infinite systems for impulsive integral equations.

Solution and Analysis of the Fuzzy Volterra Integral Equations viaHomotopy Analysis Method

Homotopy Analysis Method(HAM)is semi-analytic method to solve the linear and nonlinear mathematical models which can be used to obtain the approximate solution.The HAM includes an auxiliary parameter,which is an efficient way to examine and analyze the accuracy of linear and nonlinear problems.The main aim of this work is to explore the approximate solutions of fuzzy Volterra integral equations(both linear and nonlinear)with a separable kernel via HAM.This method provides a reliable way to ensure the convergence of the approximation series.A new general form of HAM is presented and analyzed in the fuzzy domain.A qualitative convergence analysis based on the graphical method of a fuzzy HAM is discussed.The solutions sought by the proposed method show that the HAM is easy to implement and computationally quite attractive.Some solutions of fuzzy second kind Volterra integral equations are solved as numerical examples to show the potential of the method.The results also show that HAM provides an easy way to control and modify the convergence area in order to obtain accurate solutions.

L_（loc）~p-solutions of Nonlinear Impulsive Volterra Integral Equation in Abstract Space

In this paper,we discuss Llocp-solutions of a kind of nonlinear impulsive Volterra integral equation and present an existence theorem of solutions in Banach space.

Runge-Kutta Method and Bolck by Block Method to Solve Nonlinear Fredholm-Volterra Integral Equation with Continuous Kernel

In this paper, the existence and uniqueness of the solution of Fredholm-Volterra integral equation is considered (NF-VIE) with continuous kernel;then we used a numerical method to reduce this type of equations to a system of nonlinear Volterra integral equations. Runge-Kutta method (RKM) and Bolck by block method (BBM) are used to solve the system of nonlinear Volterra integral equations of the second kind (SNVIEs) with continuous kernel. The error in each case is calculated.

EXISTENCE OF SOLUTIONS FOR MIXED MONOTONE IMPULSIVE VOLTERRA INTEGRAL EQUATIONS IN BANACH SPACES

This paper studies the existence of solutions for mixed monotone impulsive Volterra integral equations on the infinite interval R+ with an infinite number of moments of impulse effect in Banach spaces. By using the mixed monotone iterative technique and Monch fixed point theorem, Some existence theorems of solutions and coupled minimal and maximal quasisolutions are obtained. Finally, an example is worked out.

EXTRAPOLATION FOR COLLOCATION METHOD OF THE FIRST KIND VOLTERRA INTEGRAL EQUATIONS

1. Introduction It is known that the following Cauchy problem for a parabolic partial differential equation (where the values at the right boundary, u.(1, t)=v(t) are unknown and sought for) is ill-posed: the solution (v) does not depend continuously on the data (g). In order to treat the ill-posedness and develop the numerical method, one reformulates the problem as a Volterra integral equation of the first kind wish a convolution type kernel (see Sneddon [1], Carslaw and Jaeger [2])

On Solving a System of Volterra Integral Equations with Relaxed Monte Carlo Method

A random simulation method was used for treatment of systems of Volterra integral equations of the second kind. Firstly, a linear algebra system was obtained by discretization using quadrature formula. Secondly, this algebra system was solved by using relaxed Monte Carlo method with importance sampling and numerical approximation solutions of the integral equations system were achieved. It is theoretically proved that the validity of relaxed Monte Carlo method is based on importance sampling to solve the integral equations system. Finally, some numerical examples from literatures are given to show the efficiency of the method.

TRANSIENT THERMAL STRESSES IN AN ORTHOTROPIC HOLLOW CYLINDER FOR AXISYMMETRIC PROBLEMS

For the thermoelastic dynamic axisymmetric problem of a finite orthotropic hollow cylinder,one comes closer to reality to involve the effect of axial strain than to consider the plane strain case only.However,additional mathematical difficulties should be encountered and a different solution procedure should be developed.By the separation of variables,the thermoelastic axisymmetric dynamic problem of an orthotropic hollow cylinder taking account of the axial strain is transformed to a Volterra integral equation of the second kind for a function of time,which can be solved efficiently and quickly by the interpolation method.The solutions of displacements and stresses are obtained. It is noted that the present method is suitable for an orthotropic hollow cylinder with an arbitrary thickness subjected to arbitrary axisymmetric thermal loads.Numerical comparison is made to show the effect of the axial strain on the displacements and stresses.

FIXED POINT THEOREMS FOR MEIR-KEELER CONDENSING OPERATORS VIA MEASURE OF NONCOMPACTNESS

In this article, we introduce the notion of Meir-Keleer condensing operator in a Banach space, a characterization using L-functions and provide a few generalization of Darbo fixed point theorem. Also, we introduce the concept of a bivariate Meir-Keleer condensing operator and prove some coupled fixed point theorems. As an application, we prove the existence of solutions for a large class of functional integral equations of Volterra type in two variables.

THIRD-ORDER NONLINEAR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEM

Third order singulary perturbed boundary value problem by means of differential inequality theories is studied. Based on the given results of second order nonlinear boundary value problem, the upper and lower solutions method of third order nonlinear boundary value problems by making use of Volterra type integral operator was established. Specific upper and lower solutions were constructed, and existence and asymptotic estimates of solutions under suitable conditions were obtained. The result shows that it seems to be new to apply these techniques to solving these kinds of third order singularly perturbed boundary value problem. An example is given to demonstrate the applications.

INVERSE PROBLEM FOR THE VISCOELASTIC MEDIUM WITH DISCONTINUOUS WAVE IMPEDANCE

In this paper, the inverse problem for the viscoelastic medium is investigated in the time domain, in which the wave impedance of the medium is discontinuous at the rear interface. The differentio-integral equations governing the behavior of the scattering and propagation operators are utilized to reconstruct the relaxation modulus of the viscoelastic medium. A new approach, in which only the one-side measurement reflection data for one round trip through the viscoelastic layer, is developed. The numerical examples are given at the end of the paper. Ir is shown that the curves of the reconstructed moduli coincide very well with the original relaxation moduli.

SOLUTIONS FOR A SYSTEM OF NONLINEAR RANDOM INTEGRAL AND DIFFERENTIAL EQUATIONS UNDER WEAK TOPOLOGY

In this paper, a Darbao type random fixed point theorem for a system of weak continuous random operators with random domain is first proved. When, by using the theorem, some existence criteria of random solutions for a systems of nonlinear random Volterra integral equations relative to the weak topology in Banach spaces are given. As applications, some existence theorems of weak random solutions for the random Cauchy problem of a system of nonlinear random differential equations are obtained, as well as the existence of extremal random solutions and random comparison results for these systems of random equations relative to weak topology in Banach spaces. The corresponding results of Szep, Mitchell-Smith, Cramer-Lakshmikantham, Lakshmikantham-Leela and Ding are improved and generalized by these theorems.

A COUPLED SYSTEM OF INTEGRAL EQUATIONS IN REFLEXIVE BANACH SPACES

We present an existence theorem for at least one weak solution for a coupled system of integral equations of Volterra type in a reflexive Banach spaces relative to the weak topology.

Existence of Solution for a Coupled System of Fractional Integro-Differential Equations on an Unbounded Domain

We present the existence of solution for a coupled system of fractional integro-differential equations. The differential operator is taken in the Caputo fractional sense. We combine the diagonalization method with Arzela-Ascoli theorem to show a fixed point theorem of Schauder.

Intuitionistic Menger inner product spaces and applications to integral equations

In this paper, first introduce and define an intuitionistic Menger inner product space, and then, obtain a new fixed point theorem in a complete intuitionistic Menger inner product space. As an application, the results are used to study the existence and uniqueness of the solution to a linear Volterra integral equation.