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This paper concerns the asymptotic behaviour of solutions of a linear convolution Volterra summation equation with an unbounded forcing term. In particular, we suppose the kernel is summable and ascribe growth bounds to the exogenous perturbation. If the forcing term grows at a geometric rate asymptotically or is bounded by a geometric sequence, then the solution (appropriately scaled) omits a convenient asymptotic representation. Moreover, this representation is used to show that additional growth properties of the perturbation are preserved in the solution. If the forcing term fluctuates asymptotically, we prove that fluctuations of the same magnitude will be present in the solution and we also connect the finiteness of time averages of the solution with those of the perturbation. Our results, and corollaries thereof, apply to stochastic as well as deterministic equations, and we demonstrate this by studying some representative classes of examples. Finally, we show that our theory can be extended to cover a class of nonlinear equations via a straightforward linearisation argument.

We develop precise bounds on the growth rates and fluctuation sizes of unbounded solutions of deterministic and stochastic nonlinear Volterra equations perturbed by external forces. The equation is sublinear for large values of the state, in the sense that the state dependence is negligible relative to linear functions. If an appropriate functional of the forcing term has a limit L at infinity, the solution of the differential equation behaves asymptotically like the underlying unforced equation when L is zero, like the forcing term when L is infinite and inherits properties of both the forcing term and underlying differential equation for finite positive values of L. Our approach carries over in a natural way to stochastic equations with additive noise, and we treat the illustrative cases of Brownian and Levy noise.

In this paper, stochastic Volterra equations driven by cylindrical Wiener process in Hilbert space are investigated. Sufficient conditions for existence of strong solutions are given. The key role is played by convergence of $\alpha$-times resolvent families.

Sufficient conditions for existence and uniqueness of the solution of the Volterra integral equations of the first kind with piecewise continuous kernels are derived in framework of Sobolev-Schwartz distribution theory. The asymptotic approximation of the parametric family of generalized solutions is constructed. The method for the solution's regular part refinement is proposed using the successive approximations method.

We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory called algebraic integration. In the Young case, that is for a driving signal with H\"older exponent greater than 1/2, we obtain a global solution, and are able to handle the case of a singular Volterra coefficient. In case of a driving signal with H\"older exponent in (1/3,1/2], we get a local existence and uniqueness theorem. The results are easily applied to the fractional Brownian motion with Hurst coefficient H>1/3.

‎‎ It is important to note that mixed systems of first and second-kind Volterra–Fredholm integral equations are ill-posed problems, so that solving discretized system of such problems has a lot of difficulties. We will apply the regularization method to convert this mixed system (ill-posed problem) to system of the second kind Volterra–Fredholm integral equations (well-posed problem). A numerical method based on Chebyshev wavelets is suggested for solving the obtained well-posed problem, and convergence analysis of the method is discussed. For showing efficiency of the method, some test problems, for which the exact solution is known, are considered.

An efficient discrete collocation method for solving Volterra type weakly singular integral equations with logarithmic kernels is investigated. One of features of these equations is that, in general the first erivative of solution behaves like as a logarithmic function, which is not continuous at the origin. In this paper, to make a compatible approximate solution with the exact ones, we introduce a new collocation approach, which applies the M¨untz logarithmic polynomials(Muntz polynomials with logarithmic terms) as basis functions. Moreover, since implementation of this technique leads to integrals with logarithmic singularities that are often difficult to solve numerically, we apply a suitable quadrature method that allows the exact evaluation of integrals of polynomials with logarithmic weights. To this end, we first remind the well-known Jacobi–Gauss quadrature and then extend it to integrals with logarithmic weights. Convergence analysis of the proposed scheme are presented, and some numerical results are illustrated to demonstrate the efficiency and accuracy of the proposed method.

In this paper we investigate two numerical schemes for the simulation of stochastic Volterra equations driven by space--time L\'evy noise of pure-jump type. The first one is based on truncating the small jumps of the noise, while the second one relies on series representation techniques for infinitely divisible random variables. Under reasonable assumptions, we prove for both methods $L^p$- and almost sure convergence of the approximations to the true solution of the Volterra equation. We give explicit convergence rates in terms of the Volterra kernel and the characteristics of the noise. A simulation study visualizes the most important path properties of the investigated processes.

We study uniqueness for a class of Volterra-type stochastic integral equations. We focus on the case of non-Lipschitz noise coefficients. The connection of these equations to certain degenerate stochastic partial differential equations plays a key role.

Optimal control problems of forward-backward stochastic Volterra integral equations (FBSVIEs, in short) with closed control regions are formulated and studied. Instead of using spike variation method as one may imagine, here we turn to treat the non-convexity of the control regions by borrowing some tools in set-valued analysis and adapting them into our stochastic control systems. A duality principle between linear backward stochastic Volterra integral equations and linear stochastic Fredholm-Volterra integral equations with conditional expectation are derived, which extends and improves the corresponding results in [25], [30]. Some first order necessary optimality conditions for optimal controls of FBSVIEs are established. In contrast with existed common routines to treat the non-convexity of stochastic control problems, here only one adjoint system and one-order differentiability requirements of the coefficients are needed.

In the paper regularity of solutions to stochastic Volterra equations in a separable Hilbert space is studied. Sufficient conditions for the temporal and spatial regularity of stochastic convolutions corresponding to the equations under consideration are provided. The results obtained generalize some well-known regularity results for solutions to stochastic differential equations. The paper is a continuation of previous author's papers concerning stochastic Volterra equations.

This study presents the process of using extrapolation methods to solve the nonlinear Volterra–Fredholm integral equations of the second kind. To do this, by approximating the integral terms contained in equations by a quadrature rule, the nonlinear Volterra–Fredholm integral equations of the second kind are reduced to a set of nonlinear algebraic equations. Then, the solution of the corresponding system of nonlinear equations is approxi-mated by an iterative method, and finally, these iterations are accelerated by an extrapolation method. We demonstrate the effectiveness of the pro-posed approach by solving some numerical examples.

In this paper, sinc-collocation method is discussed to solve Volterra func tional integral equations with delay function (t). Also the existence and uniqueness of numerical solutions for these equations are provided. This method improves conventional results and achieves exponential convergence. Numerical results are included to confirm the efficiency and accuracy of the method.

The purpose of this paper is to develop a general theory which gives sufficient conditions in terms of the kernel b, the operator A, and the forcing term f for the solution u of (V) to be bounded on t greater than or = 0 but less than infinity and which further assures that the solution u tends to a limit u sub infinity as t approaches infinity; under certain conditions u sub infinity = 0, under others u sub infinity is the unique solution of an appropriate 'limit equation' associated with (V). As one special case of this theory we give a complete analysis of the boundedness and asymptotic properties of the solution of the above heat flow problem, under physically reasonable assumptions concerning the relaxation functions, the nonlinear operator, the initial temperature distribution, and the external heat supply.

This report provides the theoretical background for an earlier report ('Frequency/Period Estimation and Adaptive Rejection of Sinusoidal Disturbances'). It also lays the foundation for a new type of transform analysis for linear recursion equations which appears to be quite useful in analyzing their spectral properties. Additional keywords: Floquet theory; difference equations; integral equations; linear control systems. (Author)

The nonlinear abstract Volterra equation of convolution type is considered. Boundedness and asymptotic properties of the solutions are established under the assumption that the kernel satisfies certain natural positivity conditions. An important property of linear and nonlinear diffusion equations is that the solutions of such equations obey a 'maximum principle'. In particular, if the initial data and the forcing terms are nonnegative, then the solution is nonnegative. The Volterra equation (V) is an abstraction of a mathematical model for nonlinear heat flow in a material with 'memory'. The kernel b in (V) can be expressed in terms of the physically meaningful heat flux and internal energy relaxation functions. In this paper equation (V) for a class of kernels b is considered which insure the positivity of the solution operator.

In this work, we consider a general class of nonlinear Volterra integro-differential equations with Atangana–Baleanu derivative. We use the operational matrices based on the shifted Legendre polynomials to obtain numerical solution of the considered equations. By approximating the unknown function and its derivative in terms of the shifted Legendre polynomials and substituting these approximations into the original equation and using the collocation points, the original equation is reduced to a system of nonlinear algebraic equations. An error estimate of the numerical solution is proved. Finally, some examples are included to show the accuracy and validity of the proposed method.

We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS) list into the discrete Schr\"odinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to B\"acklund transformations for some particular cases of the discrete Krichever-Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate B\"acklund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.

We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS) list into the discrete Schr\"odinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to B\"acklund transformations for some particular cases of the discrete Krichever-Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate B\"acklund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.

Consider the nonlinear Volterra equation u(t) + (b*Au) not an element of f(t). This paper discusses existing and recent results for the following problems concerning this equation the global existence and uniqueness of solutions and their continuous dependence on the data; the boundedness and asymptotic behavior as t approaches infinity in th special cases when X = H is a real Hilbert space and A is either a maximal monotone operator on H or A is a subdifferential of a proper, convex, lower semicontinuous function; and the existence, boundedness, and asymptotic behavior of positive solutions in the general settting. The theory is used to study one possible model problem for heat flow in a material with 'memory' which can be transformed to the equivalent from of the equation under physically reasonable assumptions; the latter provide a motivation for the natural setting of much of the theory developed here.

The objective of the article is to treat the Schr\"{o}dinger equation in parallel with a standard treatment of the heat equation. In the mathematics literature, the heat equation initial value problem is converted into a Volterra integral equation of the second kind, and then the Picard algorithm is used to find the exact solution of the integral equation. The Poisson Integral theorem shows that the Poisson integral formula with the Schrodinger kernel holds in the Abel summable sense. Furthermore, the Source integral theorem provides the solution of the initial value problem for the nonhomogeneous Schrodinger equation. Folland's proof of the Generalized Young's inequality is used as a model for the proof of the L^p lemma. Basically the Generalized Young's theorem is in a more general form where the functions take values in an arbitrary Banach space. The L^1, L^p and the L^\infty lemmas are inductively applied to the proofs of their respective Volterra theorems in order to prove that the Neumman series converge with respect to the topology L^p(I;B}, where I is a finite time interval, B is an arbitrary Banach space, and 1 = < p = < \infty. The Picard method of successive approximations is to be used to construct an approximate solution which should approach the exact solution as n->\infty. To prove convergence, Volterra kernels are introduced in arbitrary Banach spaces. The Volterra theorems are proved and applied in order to show that the Neumann series for the Hilbert-Schmidt kernel and the unitary kernel converge to the exact Green function.

This paper formulates and studies a stochastic maximum principle for forward-backward stochastic Volterra integral equations (FBSVIEs in short), while the control area is assumed to be convex. Then a linear quadratic (LQ in short) problem for backward stochastic Volterra integral equations (BSVIEs in short) is present to illustrate the aforementioned optimal control problem. Motivated by the technical skills in solving above problem, a more convenient and briefer method for the unique solvability of M-solution for BSVIEs is proposed. At last, we will investigate a risk minimization problem by means of the maximum principle for FBSVIEs. Closed-form optimal portfolio is obtained in some special cases.

This paper formulates and studies a stochastic maximum principle for forward-backward stochastic Volterra integral equations (FBSVIEs in short), while the control area is assumed to be convex. Then a linear quadratic (LQ in short) problem for backward stochastic Volterra integral equations (BSVIEs in short) is present to illustrate the aforementioned optimal control problem. Motivated by the technical skills in solving above problem, a more convenient and briefer method for the unique solvability of M-solution for BSVIEs is proposed. At last, we will investigate a risk minimization problem by means of the maximum principle for FBSVIEs. Closed-form optimal portfolio is obtained in some special cases.

This paper formulates and studies a stochastic maximum principle for forward-backward stochastic Volterra integral equations (FBSVIEs in short), while the control area is assumed to be convex. Then a linear quadratic (LQ in short) problem for backward stochastic Volterra integral equations (BSVIEs in short) is present to illustrate the aforementioned optimal control problem. Motivated by the technical skills in solving above problem, a more convenient and briefer method for the unique solvability of M-solution for BSVIEs is proposed. At last, we will investigate a risk minimization problem by means of the maximum principle for FBSVIEs. Closed-form optimal portfolio is obtained in some special cases.

We consider the asymptotic behavior of solutions of the difference equations of the form $x(n+1)=Ax(n) + \sum_{k=0}^n B(n-k)x(k) + y(n)$ in a Banach space $\X$, where $n=0,1,2,...$; $A,B(n)$ are linear bounded operator in $\X$. Our method of study is based on the concept of spectrum of a unilateral sequence. The obtained results on asymptotic stability and almost periodicity are stated in terms of spectral properties of the equation and its solutions. To this end, a relation between the Z-transform and spectrum of a unilateral sequence is established. The main results extend previous ones.

The aim of this work is to present, in self-contained form, results concerning fundamental and the most important questions related to linear stochastic Volterra equations of convolution type. The paper is devoted to study the existence and some kind of regularity of solutions to stochastic Volterra equations in Hilbert space and the space of tempered distributions, as well. In recent years the theory of Volterra equations, particularly fractional ones, has undergone a big development. This is an emerging area of research with interesting mathematical questions and various important applications. The increasing interest in these equations comes from their applications to problems from physics and engeenering, particularly from viscoelasticity, heat conduction in materials with memory or electrodynamics with memory.

In this paper we study well posedness of a certain nonlinear Volterra-Fredholm dynamic integral and integro-dynamic equations on unbounded interval from arbitrary time scale. We derive the time scale analogue of certain integral inequalities of Pachpatte type and using them with Banach's fixed-point theorem to establish the results.

We introduce a new family of multivalue and multistage methods based on Hermite–Birkhoff interpolation for solving nonlinear Volterra integro differential equations. The proposed methods that have high order and ex tensive stability region, use the approximated values of the first derivative of the solution in the m collocation points and the approximated values of the solution as well as its first derivative in the r previous steps. Convergence order of the new methods is determined and their linear stability is analyzed. Efficiency of the methods is shown by some numerical experiments.

We study the correct solvability of an abstract integro-differential equations in Hilbert space generalizing integro-differential equations arising in the theory of viscoelastisity. The equations under considerations are the abstract hyperbolic equations perturbed by the terms containing Volterra integral operators. We establish the correct solvability in the weighted Sobolev spaces of vector-valued functions on the positive semiaxis.

Solutions of stochastic Volterra (integral) equations are not Markov processes, and therefore classical methods, like dynamic programming, cannot be used to study optimal control problems for such equations. However, we show that by using {\em Malliavin calculus} it is possible to formulate a modified functional type of {\em maximum principle} suitable for such systems. This principle also applies to situations where the controller has only partial information available to base her decisions upon. We present both a sufficient and a necessary maximum principle of this type, and then we use the results to study some specific examples. In particular, we solve an optimal portfolio problem in a financial market model with memory.

In this paper we consider unbounded solutions of perturbed convolution Volterra summation equations. The equations studied are asymptotically sublinear, in the sense that the state--dependence in the summation is of smaller than linear order for large absolute values of the state. When the perturbation term is unbounded, it is elementary to show that solutions are also. The main results of the paper are mostly of the following form: the solution has an additional unboundedness property $U$ if and only if the perturbation has property $U$. Examples of property $U$ include monotone growth, monotone growth with fluctuation, fluctuation on $\mathbb{R}$ without growth, existence of time averages. We also study the connection between the times at which the perturbation and solution reach their running maximum, and the connection between the size of signed and unsigned running maxima of the solution and forcing term.

We study quadrature methods for solving Volterra integral equations of the first kind with smooth kernels under the presence of noise in the right-hand sides, with the quadrature methods being generated by linear multistep methods. The regularizing properties of an a priori choice of the step size are analyzed, with the smoothness of the involved functions carefully taken into consideration. The balancing principle as an adaptive choice of the step size is also studied. It is considered in a version which sometimes requires less amount of computational work than the standard version of this principle. Numerical results are included.

We use energy methods to study global existence, boundedness, and asymptotic behavior as t approaches infinity, of solutions of the two Cauchy problems (and related initial-boundary value problems).

We define a class of functions which have a known decay rate coupled with a periodic fluctuation. We identify conditions on the kernel of a linear summation convolution Volterra equation which give the equivalence of the kernel lying in this class of functions and the solution lying in this class of functions. Some specific examples are examined. In particular this theory is used to provide a counter--example to a result regarding the rate of decay of the auto--covariance function of an ARCH($\infty$) process.

Motivated by the potential applications to the fractional Brownianmotion, we study Volterra stochasticdifferential of the form~:\begin{equation}X\_t = x+ \int\_0^tK(t,s)b(s,X\_s)ds + \int\_0^tK(t,s) \sigma(s,X\_s)\,dB\_s ,\tag{E} \label{eq:sdefbm}\end{equation}where $(B\_s, \, s\in [0,1])$ is a one-dimensional standard Brownianmotion and $(K(t,s), \, t,s \in [0,1])$ is a deterministic kernelwhose properties will be precised below but for which we don't assumeany boundedness property.

In this paper we introduce a numerical method for solving nonlinear Volterra integro-differential equations. In the first step, we apply implicit trapezium rule to discretize the integral in given equation. Further, the Daftardar-Gejji and Jafari technique (DJM) is used to find the unknown term on the right side. We derive existence-uniqueness theorem for such equations by using Lipschitz condition. We further present the error, convergence, stability and bifurcation analysis of the proposed method. We solve various types of equations using this method and compare the error with other numerical methods. It is observed that our method is more efficient than other numerical methods.

In this paper, sufficient conditions are established for the existence results of fractional order semilinear Volterra integrodifferential equations in Banach spaces. The results are obtained by using the theory of fractional cosine families and fractional powers of operators.

In this paper, an efficient method is presented for solving three dimensional Volterra integral equations of the second kind with continuous kernel. Shifted Chebyshev polynomial is applied to approximate a solution for these integral equations. This method transforms the integral equation to algebraic equations with unknown Chebyshev coefficients. Numerical results are calculated and the estimated error in each example is computed using Maple 17.

Linear autonomous functional differential equations of neutral type include Volterra integral and integrodifferential equations as special cases. The paper considers numerical approximation of solutions to these equations by first converting the initial value problem to an abstract Cauchy problem in a product space (R superscript n x weighted L-squared-space) and then using abstract approximation results for C sub 0 semigroups combined with Galerkin type ideas. In order to obtain concrete schemes subspaces of Legendre and Laguerre polynomials are used. The convergence properties of the algorithms are demonstrated by several examples.

S. Elaydi obtained a characterization of the stability of the null solution of the Volterra difference equation $$ x_n=\sum_{i=0}^{n-1} a_{n-i} x_i\textrm{,}\quad n\geq 1\textrm{,} $$ by localizing the roots of its characteristic equation $$ 1-\sum_{n=1}^{\infty}a_nz^n=0\textrm{.} $$ The assumption that $(a_n)\in\ell^1$ was the single hypothesis considered for the validity of that characterization, which is an insufficient condition if the ratio $R$ of convergence of the power series of the previous equation equals one. In fact, when $R=1$, this characterization conflicts with a result obtained by Erd\"os, Feller and Pollard. Here, we analyze the $R=1$ case and show that some parts of that characterization still hold. Furthermore, studies on stability for the $R

We prove a weak error estimate for the approximation in space and time of a semilinear stochastic Volterra integro-differential equation driven by additive space-time Gaussian noise. We treat this equation in an abstract framework, in which parabolic stochastic partial differential equations are also included as a special case. The approximation in space is performed by a standard finite element method and in time by an implicit Euler method combined with a convolution quadrature. The weak rate of convergence is proved to be twice the strong rate, as expected. Our convergence result concerns not only functionals of the solution at a fixed time but also more complicated functionals of the entire path and includes convergence of covariances and higher order statistics. The proof does not rely on a Kolmogorov equation. Instead it is based on a duality argument from Malliavin calculus.

Book Source: Digital Library of India Item 2015.198748 dc.contributor.author: Pyda Srinivas dc.date.accessioned: 2015-07-08T12:56:12Z dc.date.available: 2015-07-08T12:56:12Z dc.date.digitalpublicationdate: 0000-00-00 dc.identifier.barcode: 5990010100531 dc.identifier.origpath: /rawdataupload/upload/0100/533 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/198748 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 94 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher: Iit Kanpur dc.rights: Out_of_copyright dc.source.library: I I T Kanpur dc.subject.classification: Mathematics dc.title: Quality Properties Of Soluctions Of Volterra Integro-differential Equations

Aim of this work is to extend the results of Cl\'ement, Da Prato & Pr\"uss on the fractional white noise perturbation with Hurst parameter 0

Aim of this work is to extend the results of Cl\'ement, Da Prato & Pr\"uss on the fractional white noise perturbation with Hurst parameter 0

In this paper we make a subtle use of operator theory techniques and the well-known Schauder fixed-point principle to establish the existence of pseudo-almost automorphic solutions to some second-order damped integro-differential equations with pseudo-almost automorphic coefficients. In order to illustrate our main results, we will study the existence of pseudo-almost automorphic solutions to a structurally damped plate-like boundary value problem.

In this paper we make a subtle use of operator theory techniques and the well-known Schauder fixed-point principle to establish the existence of pseudo-almost automorphic solutions to some second-order damped integro-differential equations with pseudo-almost automorphic coefficients. In order to illustrate our main results, we will study the existence of pseudo-almost automorphic solutions to a structurally damped plate-like boundary value problem.

The research is devoted to a numerical solution of the Volterra equations of the first kind that were obtained using the Laplace integral transforms for solving the equation of heat conduction. The paper consists of an introduction and two sections. The first section deals with the calculation of kernels from the respective integral equations at a fixed length of the significand in the floating point representation of a real number. The PASCAL language was used to develop the software for the calculation of kernels, which implements the function of tracking the valid digits of the significand. The test examples illustrate the typical cases of systematic error accumulation. The second section presents the results obtained from the computational algorithms which are based on the product integration method and the midpoint rule. The results of test calculations are presented to demonstrate the performance of the difference methods.

This paper is devoted to the stochastic optimal control problems for systems governed by forward-backward stochastic Volterra integral equations (FBSVIEs, for short) with state constraints. Using Ekeland's variational principle, we obtain one kind of variational inequality. Then, by dual method, we derive a stochastic maximum principle which gives the necessary conditions for the optimal controls.

This paper studies existence and uniqueness of solutions to generalized Volterra integral equations. Since our proof for existence and uniqueness does not make use of Banach fixed point theorem unlike the previous papers focused on this subject, we can replace continuity property of the kernel function with the weaker one rd-continuity. The paper also covers results concerning the following concepts: The notion of resolvent kernel, and its role in formulation of the solution, the reciprocity property of kernels, Piccard iterates, relation between linear dynamic equations and Volterra integral equations, some special type of kernels together with several illustrative examples.

The aim of the paper is to study an optimal control problem on infinite horizon for an infinite dimensional integro-differential equation with completely monotone kernelskernels, where we assume that the noise enters the system when we introduce a control. We start by reformulating the state equation into a semilinear evolution equation which can be treated by semigroup methods. The application to optimal control provide other interesting result and require a precise descriprion of the properties of the generated semigroup. The main tools consist in studying the differentiability of the forward-backward system with infinite horizon corresponding with the reformulated problem and the proof of existence and uniqueness of of mild solutions to the corresponding HJB equation.