The Cauchy problem for a singularly perturbed Volterra integro-differential equation

The Cauchy problem for a singularly perturbed Volterra integro-differential equation is examined. Two cases are considered: (1) the reduced equation has an isolated solution, and (2) the reduced equation has intersecting solutions (the so-called case of exchange of stabilities). An asymptotic expansion of the solution to the Cauchy problem is constructed by the method of boundary functions. The results are justified by using the asymptotic method of differential inequalities, which is extended to a new class of problems.     

On constrained Volterra cubic stochastic operators

We consider constrained Volterra cubic stochastic operators and construct several Lyapunov functions for the constrained Volterra cubic stochastic operators. We prove that such kind operators do not have periodic trajectories. Finally, we show that the set of all constrained Volterra cubic stochastic operators is a convex compact set and find the extreme points of this set.     

Admissibility for discrete Volterra equations

In this paper we develop the theory of admissibility for linear discrete Volterra operators and obtain several necessary and sufficient conditions for admissibility in various sequence spaces. Using the results obtained, we study the existence of solutions (such as bounded, exponential or convergent solutions), of linear or nonlinear discrete Volterra summation equations.     

Application of fuzzy differential transform method for solving fuzzy Volterra integral equations

This paper deals with the solutions of fuzzy Volterra integral equations with separable kernel by using fuzzy differential transform method (FDTM). If the equation considered has a solution in terms of the series expansion of known functions, this powerful method catches the exact solution. To this end, we have obtained several new results to solve mentioned problem when FDTM has been applied. In order to show this capability and robustness, some fuzzy Volterra integral equations are solved in detail as numerical examples.     

Hungry Lotka–Volterra lattice under nonzero boundaries,block-Hankel determinant solution,and biorthogonal polynomials

In this paper, we first extend the hungry Lotka–Volterra lattice to a case of nonzero boundary conditions and present its corresponding exact solution expressed in terms of a block-Hankel determinant. Then, we establish a connection between this hungry Lotka–Volterra lattice under nonzero boundary conditions and a set of biorthogonal polynomials. It turns out that the hungry Lotka–Volterra lattice under nonzero boundary conditions possesses a Lax pair expressed in terms of the biorthogonal polynomials. Moreover, we consider two special cases of the hungry Lotka–Volterra lattice. For the case $M=1$, it reduces to the Lotka–Volterra lattice under nonzero boundary condition, which has been discussed in literature. We also present the result for $M=2$ in detail, which extends a known result to a case of nonzero boundary functions. All these results are obtained by virtue of Hirota's bilinear method and determinant techniques.     

On a class of backward stochastic Volterra integral equations

In this paper, under some restrictions of the time interval, we compare a class of backward stochastic Volterra integral equations with the corresponding simpler one; to be precise, we give the relations between their solutions under global and local Lipschitz conditions on their generator functions. Using these relations, it could be easier to study solutions of more complex equations, where coefficients in backward integrals could be treated as perturbations.     

Stochastic Volterra equations driven by fractional Brownian motion

This paper is devoted to study a class of stochastic Volterra equations driven by fractional Brownian motion. We first prove the Driver type integration by parts formula and the shift Harnack type inequalities. As a direct application, we provide an alternative method to describe the regularities of the law of the solution. Secondly, by using the Malliavin calculus, the Bismut type derivative formula is established, which is then applied to the study of the gradient estimate and the strong Feller property. Finally, we establish the Talagrand type transportation cost inequalities for the law of the solution on the path space with respect to both the uniform metric and the L₂-metric.     

On infinite dimensional quadratic Volterra operators

In this paper we study a class of quadratic operators named by Volterra operators on infinite dimensional space. We prove that such operators have infinitely many fixed points and the set of Volterra operators forms a convex compact set. In addition, it is described its extreme points. Besides, we study certain limit behaviors of such operators and give some more examples of Volterra operators for which their trajectories do not converge. Finally, we define a compatible sequence of finite dimensional Volterra operators and prove that any power of this sequence converges in weak topology.     

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.     

Genetic Volterra algebras and their derivations

The present paper is devoted to genetic Volterra algebras. We first study characters of such algebras. We fully describe associative genetic Volterra algebras, in this case all derivations are trivial. In general setting, i.e., when the algebra is not associative, we provide a su?cient condition to get trivial derivation on generic Volterra algebras. Furthermore, we describe all derivations of three dimensional generic Volterra algebras, which allowed us to prove that any local derivation is a derivation of the algebra.     

Abstract Volterra operators

We propose a new general definition of Volterra operators. Several types of evolutionary operators, including Volterra ones in the sense of A.N. Tikhonov, satisfy this definition. For equations with generalized Volterra operators we introduce the notions of local, global, and maximally extended solutions. For solutions to nonlinear equations we formulate the existence, uniqueness, and extendability conditions. The theorems proved in this paper imply both known and new results on the solvability of concrete equations. We adduce an example of the application of obtained results to the study of the Cauchy problem for functional differential equations.     

A novel numerical method and its convergence for nonlinear delay Volterra integro-differential equations

In this paper, we suggest a convergent numerical method for solving nonlinear delay Volterra integro-differential equations. First, we convert the problem into a continuous-time optimization problem and then use a shifted pseudospectral method to discrete the problem. Having solved the last problem, we can achieve the pointwise and continuous approximate solutions for the main delay Volterra integro-differential equations. Here, we analyze the convergence of the method and solve some numerical examples to show the efficiency of the method.     

Large Deviations for Stochastic Volterra Equations in the Plane

We prove a Large Deviation Principle for the family of solutions of Volterra equations in the plane obtained by perturbation of the driving white noise. One of the motivations for the study of such class of equations is provided by non-linear hyperbolic stochastic partial differential equations appearing in the construction of some path-valued processes on manifolds. The proof uses the method developped by Azencott for diffusion processes. The main ingredients are exponential inequalities for different classes of two-parameter stochastic integrals; these integrals are related to the representation of the stochastic term in the differential equation as a representable semimatringale.     

Rationalized Haar functions method for solving Fredholm and Volterra integral equations

This paper presents a computational technique for Fredholm integral equation of the second kind and Volterra integral equation of the second kind. The method is based upon Haar functions approximation. Properties of Rationalized Haar functions are first presented, the operational matrix of integration together with product operational matrix and Newton–Cotes nodes are utilized to reduce the computation of integral equations into some algebraic equations. The method is computationally attractive and applications are demonstrated through illustrative examples.     

Weighted integral and integro-differential inequalities

A general weighted integral inequality for two continuous functions on an interval [a,b] is presented. The equality conditions are given. This result implies the new inequalities for the incomplete beta and gamma functions as well as the related estimates for the confluent hypergeometric function, error function, and Dawson's integral. Also it implies various weighted integro-differential inequalities, those of the Opial type included, and some inequalities which involve the Erdélyi–Kober and Riemann–Liouville fractional integrals.     

Boundedness and compactness of Volterra type integral operators

We introduce some nested classes of Volterra type integral operators. For the operators of these classes we establish criteria for boundedness and compactness in Lebesgue spaces.     

Strong solutions to stochastic Volterra equations

In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties for stochastic convolutions are studied. Our main results provide sufficient conditions for strong solutions to stochastic Volterra equations.     

Nonlinear Volterra integral equations and the Schröder functional equation

We show an interesting connection between a special class of Volterra integral equations and the famous Schröder equation. The basic results provide criteria for the existence of nontrivial as well as blow-up solutions of the Volterra equation, expressed in terms of the convergence of some integrals. Examples related to Volterra equations with power and exponential nonlinearities are presented.     

Regularity Properties of Some Stochastic Volterra Integrals with Singular Kernel

We prove the Hölder continuity of some stochastic Volterra integrals, with singular kernels, under integrability assumptions on the integrand. Some applications to processes arising in the analysis of the fractional Brownian motion are given. The main tool is the embedding of some Besov spaces into some sets of Hölder continuous functions.     

On a class of Volterra nonlinear equations of parabolic type

In this paper, we show the backward uniqueness in time of solution for a class of Volterra nonlinear equations of parabolic type. We also give reasonable physical interpretation for our conclusion.