Existence and continuity with respect to parameter of solutions to stochastic Volterra equations in a plane

In this paper we study stochastic Volterra equations in a plane. These equations contain integrals with respect to fields of locally bounded variation and square-integrable strong martingales. We prove the existence and the uniqueness of solutions of such equations with locally integrable (in some measure) trajectories, assuming that the coefficients of equations possess the Lipschitz property with respect to the functional argument. We prove that a solution of a stochastic Volterra integral equation in a plane is continuous with respect to parameter.     

On Solutions of Backward Stochastic Volterra Integral Equations with Jumps in Hilbert Spaces

This paper studies the existence, uniqueness and stability of the adapted solutions to backward stochastic Volterra integral equations (BSVIEs) driven by a cylindrical Brownian motion on a separable Hilbert space and a Poisson random measure with non-Lipschitz coefficient. Moreover, a duality principle between the linear forward stochastic Volterra integral equations (FSVIEs) with jumps and the linear BSVIEs with jumps is established.     

Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator

This paper is devoted to proving the existence and uniqueness of solutions to Cauchy type problems for fractional differential equations with composite fractional derivative operator on a finite interval of the real axis in spaces of summable functions. An approach based on the equivalence of the nonlinear Cauchy type problem to a nonlinear Volterra integral equation of the second kind and applying a variant of the Banach’s fixed point theorem to prove uniqueness and existence of the solution is presented. The Cauchy type problems for integro-differential equations of Volterra type with composite fractional derivative operator, which contain the generalized Mittag-Leffler function in the kernel, are considered. Using the method of successive approximation, and the Laplace transform method, explicit solutions of the open problem proposed by Srivastava and Tomovski (2009) [11] are established in terms of the multinomial Mittag-Leffler function.     

Existence,uniqueness and stability of solutions for a class of nonlinear integral equations under generalized Lipschitz condition

In this paper, we prove the existence, uniqueness and the stability of solutions for some nonlinear functional-integral equations by using generalized Lipschitz condition. We prove a fixed point theorem to obtain the mentioned aims in Banach space X:= C([a, b],R). As application we study some Volterra integral equations with linear, nonlinear and singular kernel.     

Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation

This work is concerned with the extension of the Jacobi spectral Galerkin method to a class of nonlinear fractional pantograph differential equations. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with weakly singular kernel. Second, we analyze the existence and uniqueness of solutions for the obtained integral equation. Then, the Galerkin method is used for solving the equivalent integral equation. The error estimates for the proposed method are also investigated. Finally, illustrative examples are presented to confirm our theoretical analysis.     

On the existence of continuous solutions of nonlinear integral equations

In this paper, we study the existence of continuous solutions over compact intervals of some nonlinear integral equations. A special interest is devoted to the study of the existence as well as the uniqueness of nonlinear Volterra equations.     

非线性Volterra积分方程非平凡解的逼近方法及其应用

本文对一类具有卷积核的非线性Ｖｏｌｔｅｒｒａ型积分方程进行了讨论，给出了关于这类方程的非平凡解的存在性和解的逼近方法的一些结果。通过有关微分方程问题的实例，说明了所给结果的重要应用。     

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.     

Backward stochastic Volterra integral equations and some related problems

Backward stochastic Volterra integral equations (BSVIEs, for short) are introduced. The existence and uniqueness of adapted solutions are established. A duality principle between linear BSVIEs and (forward) stochastic Volterra integral equations is obtained. As applications of the duality principle, a comparison theorem is proved for the adapted solutions of BSVIEs, and a Pontryagin type maximum principle is established for an optimal control of stochastic integral equations.     

On some iterative numerical methods for a Volterra functional integral equation of the second kind

In this paper, we study an iterative numerical method for approximating solutions of a certain type of Volterra functional integral equations of the second kind (Volterra integral equations where both limits of integration are variables). The method uses the contraction principle and a suitable quadrature formula. Under certain conditions, we prove the existence and uniqueness of the solution and give error estimates for our approximations. We also included a numerical example which illustrates the fast approximations.     

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     

Stochastic Volterra equations in Banach spaces and stochastic partial differential equation

In this paper, we study the existence-uniqueness and large deviation estimate for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then we apply them to a large class of semilinear stochastic partial differential equations (SPDE), and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Moreover, stochastic Navier-Stokes equations are also investigated.     

Improving the accuracy of solutions of the linear second kind volterra integral equations system by using the taylor expansion method

This paper presents a new and an efficient method for determining solutions of the linear second kind Volterra integral equations system. In this method, the linear Volterra integral equations system using the Taylor series expansion of the unknown functions transformed to a linear system of ordinary differential equations. For determining boundary conditions we use a new method. This method is effective to approximate solutions of integral equations system with a smooth kernel, and a convolution kernel. An error analysis for the proposed method is provided. And illustrative examples are given to represent the efficiency and the accuracy of the proposed method.     

Osgood type condition for the Volterra integral equations with bounded and nonincreasing kernels

This paper provides the necessary and sufficient Osgood type condition for the existence of blow-up solutions of Volterra equation with kernels being nonincreasing and bounded functions. Examples of such equations related to models of anomalous diffusion as well as some integral estimates of blow-up time are also presented.     

On a Perturbed System of Volterra Integral Equations

In this paper, we study the existence, uniqueness, and other properties of solutions of a system of Volterra integral equations under a general class of perturbations. The representation formula, a simple and classic application of the Leray–Schauder alternative, and a certain integral inequality with explicit estimate are used to establish the results.     

ON CERTAIN BOUNDARY VALUE PROBLEMS FOR NONLINEAR INTEGRODIFFERENTIAL EQUATIONS

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Noncompact‐type Krasnoselskii fixed‐point theorems and their applications

In this paper, we first establish some user‐friendly versions of fixed‐point theorems for the sum of two operators in the setting that the involved operators are not necessarily compact and continuous. These fixed‐point results generalize, encompass, and complement a number of previously known generalizations of the Krasnoselskii fixed‐point theorem. Next, with these obtained fixed‐point results, we study the existence of solutions for a class of transport equations, the existence of global solutions for a class of Darboux problems on the first quadrant, the existence and/or uniqueness of periodic solutions for a class of difference equations, and the existence and/or uniqueness of solutions for some kind of perturbed Volterra‐type integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

Generalized Jacobi Spectral-Galerkin Method for Nonlinear Volterra Integral Equations with Weakly Singular Kernels

We propose a generalized Jacobi spectral-Galerkin method for the nonlinear Volterra integral equations (VIEs) with weakly singular kernels. We establish the existence and uniqueness of the numerical solution, and characterize the convergence of the proposed method under reasonable assumptions on the nonlinearity. We also present numerical results which are consistent with the theoretical predictions.     

The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations

Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels.

     

Lévy-driven Volterra Equations in Space and Time

We investigate nonlinear stochastic Volterra equations in space and time that are driven by Lévy bases. Under a Lipschitz condition on the nonlinear term, we give existence and uniqueness criteria in weighted function spaces that depend on integrability properties of the kernel and the characteristics of the Lévy basis. Particular attention is devoted to equations with stationary solutions, or more generally, to equations with infinite memory, that is, where the time domain of integration starts at minus infinity. Here, in contrast to the case where time is positive, the usual integrability conditions on the kernel are no longer sufficient for the existence and uniqueness of solutions, but we have to impose additional size conditions on the kernel and the Lévy characteristics. Furthermore, once the existence of a solution is guaranteed, we analyze its asymptotic stability, that is, whether its moments remain bounded when time goes to infinity. Stability is proved whenever kernel and characteristics are small enough, or the nonlinearity of the equation exhibits a fractional growth of order strictly smaller than one. The results are applied to the stochastic heat equation for illustration.