Volterra Operators Research Articles

A Symbolic Method for Solving a Class of Convolution-Type Volterra–Fredholm–Hammerstein Integro-Differential Equations under Nonlocal Boundary Conditions

Integro-differential equations involving Volterra and Fredholm operators (VFIDEs) are used to model many phenomena in science and engineering. Nonlocal boundary conditions are more effective, and in some cases necessary, because they are more accurate measurements of the true state than classical (local) initial and boundary conditions. Closed-form solutions are always desirable, not only because they are more efficient, but also because they can be valuable benchmarks for validating approximate and numerical procedures. This paper presents a direct operator method for solving, in closed form, a class of Volterra–Fredholm–Hammerstein-type integro-differential equations under nonlocal boundary conditions when the inverse operator of the associated Volterra integro-differential operator exists and can be found explicitly. A technique for constructing inverse operators of convolution-type Volterra integro-differential operators (VIDEs) under multipoint and integral conditions is provided. The proposed methods are suitable for integration into any computer algebra system. Several linear and nonlinear examples are solved to demonstrate the effectiveness of the method.

The square roots of some classical operators

We give complete descriptions of the set of square roots of certain classical operators, often providing specific formulas. The classical operators included in this discussion are the square of the unilateral shift, the Volterra operator, certain compress

Note on self-commutators of Volterra operator and its square

Let V be the classical Volterra operator on L2(0, 1). The quadratic forms and their applications are used in many branches of mathematics. In this area of mathematics have studied many properties for concrete operator such as the numerical range. In recent year, the concept of Volterra operator has attached the serious attention of many researchers. In this paper, we compute the operator norm of self-commutators of V and its square. We also study the length of arc and the area of numerical range of V.

Equalization Control of Nonlinear Systems by Discrete Models of the Volterra Operators

The Associated linear equations (ALEs) are parametric models of the Volterra operators. Whit them, a Volterra inverse is constructed with the objective of been used as open loop control of continuous nonlinear systems. However, most of the actual control systems are of discrete nature, this work introduces the novel discrete version of the ALEs. This discrete version is a series of ARX models of the Volterra operators for both the direct and the inverse series. These discrete models of the ALEs are used for an equalization strategy in order to create an open loop control on a reported simulated Duffing Oscillator and a physical Duffing system constructed by analog circuits.

Controlling Problem within a Class of Two-Level Positive Maps

This paper aims to define the set of unital positive maps on M2(C) by means of quantum Lotka–Volterra operators which are quantum analogues of the classical Lotka–Volterra operators. Furthermore, a quantum control problem within the class of quantum Lotka–Volterra operators are studied. The proposed approach will lead to the understanding of the behavior of the classical Lotka–Volterra systems within a quantum framework.

Some direct and inverse source problems in nonlinear evolutionary PDEs with Volterra operators

This paper deals with direct and inverse source problems for parabolic or byperbolic PDEs containing nonlinear Volterra operators (including the variable order time-fractional derivatives). We show the well posedness (existence and uniqueness of a solution) of various settings. The proof-technique is based on the contraction mapping theorem and on a bootstrap argument, which ensures the existence of a global in time solution.

Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey

In this survey we report about recent work on weighted Banach spaces of analytic functions on the unit disc and on the whole complex plane defined with sup-norms and operators between them. Results about the solid hull and core of these spaces and distance formulas are reviewed. Differentiation and integration operators, Cesàro and Volterra operators, weighted composition and superposition operators and Toeplitz operators on these spaces are analyzed. Boundedness, compactness, the spectrum, hypercyclicity and (uniform) mean ergodicity of these operators are considered.

On a Class of Non-Ergodic Lotka–Volterra Operator

In present paper, we study dynamical systems generated by a class of Lotka–Volterra operators. It is proved that such class of operator has non-ergodic property.

Norms of polynomials of the Volterra operator

We compute the operator norm of real-quadratic polynomials of the Volterra operator. This is used to test whether the Crouzeix conjecture holds for the Volterra operator.

Quaternionic triangular linear operators

Triangular operators are an essential tool in the study of nonselfadjoint operators that appear in different fields with a wide range of applications. Although the development of a quaternionic counterpart for this theory started at the beginning of this century, the lack of a proper spectral theory combined with problems caused by the underlying noncommutative structure prevented its real development for a long time. In this paper, we give criteria for a quaternionic linear operator to have a triangular representation, namely, under which conditions such operators can be represented as a sum of a diagonal operator with a Volterra operator. To this effect, we investigate quaternionic Volterra operators based on the quaternionic spectral theory arising from the S‐spectrum. This allow us to obtain conditions when a non‐selfadjoint operator admits a triangular representation.

Linear Operators and Equations with Partial Integrals

We consider linear operators and equations with partial integrals in Banach ideal spaces, spaces of vector functions, and spaces of continuous functions. We study the action, regularity, duality, algebras, Fredholm properties, invertibility, and spectral properties of such operators. We describe principal properties of linear equations with partial integrals. We show that such equations are essentially different compared to usual integral equations. We obtain conditions for the Fredholm alternative, conditions for zero spectral radius of the Volterra operator with partial integrals, and construct resolvents of invertible equations. We discuss Volterra–Fredholm equations with partial integrals and consider problems leading to linear equations with partial integrals.

An Algorithm for the Closed-Form Solution of Certain Classes of Volterra–Fredholm Integral Equations of Convolution Type

In this paper, a direct operator method is presented for the exact closed-form solution of certain classes of linear and nonlinear integral Volterra–Fredholm equations of the second kind. The method is based on the existence of the inverse of the relevant linear Volterra operator. In the case of convolution kernels, the inverse is constructed using the Laplace transform method. For linear integral equations, results for the existence and uniqueness are given. The solution of nonlinear integral equations depends on the existence and type of solutions ofthe corresponding nonlinear algebraic system. A complete algorithm for symbolic computations in a computer algebra system is also provided. The method finds many applications in science and engineering.

Historical behavior for a class of Lotka–Volterra systems

It is well‐known that discrete‐time Kolmogorov systems play important role in studying predator‐prey models with interactions. A topic of this paper deals with stochastic discrete‐time Kolmogorov systems which are called Lotka–Volterra operators. In the present paper, we provide a new class of LV operators, and establish that each of operator from this class has historical behavior, that is, their time averages diverge.

Self-interaction in a cosmic dark fluid: The four-kernel rheological extension of the equations of state

We establish a new self-consistent model of coupling between the cosmic dark energy and dark matter in the framework of the rheological approach, which is based on the representation of the equations of state in terms of integral operators of the Volterra-type. We elaborate the so-called four-kernel model, in the framework of which both the dark energy and dark matter pressures are presented by two integrals containing the energy densities of the dark energy and dark matter. For the Volterra operators, the kernels of which are associated with the effects of fading memory, the corresponding isotropic homogeneous cosmological model is shown to be exactly integrable. We consider the classification of the model exact solutions, based on the analysis of roots of the characteristic polynomial associated with the key equation of the presented model. The scalars of the pressure and energy-density of the dark energy and dark matter, the Hubble function and acceleration parameter are presented explicitly as the functions of the dimensionless scale factor. The scale factor as the function of the cosmological time is found in quadratures and is described analytically, qualitatively and numerically. Asymptotic analysis allowed us to classify the models with respect to behavior typical for the Big Rip, Little Rip and Pseudo Rip (de Sitter type). Two intriguing exact cosmological solutions are discussed, which describe the super-exponential expansion and the symmetric bounce. New solutions are presented, which correspond to the quasi-periodic behavior of the state functions of the dark fluid and of the geometric characteristics of the Universe.

Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity

This article investigates an equation with a rapidly oscillating inhomogeneity and with a rapidly decreasing kernel of an integral operator of Fredholm type. Earlier, differential problems of this type were studied in which the integral term was either absent or had the form of a Volterra-type integral. The presence of an integral operator and its type significantly affect the development of an algorithm for asymptotic solutions, in the implementation of which it is necessary to take into account essential singularities generated by the rapidly decreasing kernel of the integral operator. It is shown in tise work that when passing the structure of essentially singular singularities changes from an integral operator of Volterra type to an operator of Fredholm type. If in the case of the Volterra operator they change with a change in the independent variable, then the singularities generated by the kernel of the integral Fredholm-type operators are constant and depend only on a small parameter. All these effects, as well as the effects introduced by the rapidly oscillating inhomogeneity, are necessary to take into account when developing an algorithm for constructing asymptotic solutions to the original problem, which is implemented in this work.

About nonlinear integro-differential Volterra and Fredholm equations

Two nonlinear problems in terms of abstract operator equations of the form Bx = f are investigated in this paper. In the first problem the operator B contains a linear differential operator A , the Volterra operator K with kernel of convolution type and the inner product of vectors g ( x )Ф( u ) with nonlinear bounded functionals Φ. The first problem is given by equation Bu ( x ) = Аu ( x ) - Ku ( x ) - g ( x )Ф( u ) = f ( x ) with boundary condition D ( B ) = D ( A ). In the second problem the operator B contains a linear differential operator A and the inner product of vectors g ( x )F( Аu ) with nonlinear bounded on C [ a , b ] functionals F , where F ( Аu ) denotes the nonlinear Fredholm integral. The second problem is presented by equation Bu = Аu - gF ( Au ) = f with boundary condition D ( B ) = D ( A ). A direct method for exact solutions of nonlinear integro-differential Volterra and Fredholm equations is proposed. Specifically, the three theorems about existing exact solutions are proved in this paper. The first theorem is mean that for nonzero constant α0 Volterra integro-differential equation Аu ( x ) - Ku ( x ) = 0 is reducing to Volterra integral equation and has a unique zero solution. During it the operator A - K is closed and continuously invertible. Also, if the functions u ( t ), g ( t ) and f ( t ) are of exponential order α then nonhomogeneous equation Аu ( x ) - Ku ( x ) = f ( x ) for each f ( x ) has a unique solution, shown in this paper. The second theorem is mean that for the first investigated problem with an injective operator A - K , for f ( x ) and g ( x ) from C [ a , b ], the exact solution is given by equation u = ( A - K )- 1 f +( A - K )- 1 g b* for every vector b* = Ф( u ) that solves nonlinear algebraic (transcendental) system of n equations b = Ф(( A - K )- 1 f +( A - K )- 1 g b). And if the last algebraic system has no solution, then investigated problem also has no solution. The third theorem is means that exact solution of the second investigated problem is given by u = A - 1( f+g d*) for every vector d* = F ( Au ) that solves nonlinear algebraic (transcendental) system of n equations d = F ( f + g d). In this case we have same property - if the last algebraic system has no solution, then investigated problem also has no solution. Two particular examples for each considered problem are shown for illustration of exact solutions giving by perform the suggested in this paper methods. In the first example was considered integro-differential Volterra and Fredholm equation and in the second case was considered equation with nonlinear Fredholm integral.

On dynamics of Volterra and non-Volterra cubic stochastic operators

We consider Volterra and non-Volterra cubic stochastic operators. For a Volterra cubic stochastic operator defined on the two-dimensional simplex, it is shown that the vertices and the centre of the simplex are fixed points. The trajectory of such an operator starting from any point from the boundary of the simplex is convergent, and the trajectory of such an operator starting from any point from the interior of the simplex except the centre does not converge. Moreover, therein proven the mean of any trajectory does not converge. For a non-Volterra cubic stochastic operator defined on the two-dimensional simplex, it is proved that the uniqueness of a fixed point, which is repelling and any trajectory starting from the boundary of the simplex converges to a periodic trajectory which consists of three vertices of the simplex. The set of limit points of the trajectory starting from the interior of the simplex except the centre is an infinite subset of the boundary of the simplex.

Fixed point equations with abstract Volterra operators on spaces of functions of several variables

Several notions of abstract Volterra operators on spaces of functions of one variable are well known. In this paper, we introduce various notions of abstract Volterra operators in spaces of functions of several variables. Some fixed point equations with such abstract Volterra operators are also studied. The basic ingredient in the theory of step by step contraction is the notion of G-contraction. The relevance of step by step contraction principle is illustrated by applications in the theory of Darboux-Ionescu problem.

Volterra-type Quadratic Stochastic Operators with a Homogeneous Tournament

As is known [1], each quadratic stochastic operator of Volterra type acting on a finitedimensional simplex defines a certain tournament, the properties of which make it possible to study the asymptotic behavior of the trajectories of this Volterra operator. In this paper, we introduce the concept of a homogeneous tournament and study the dynamic properties of Volterra operators corresponding to homogeneous tournaments in the simplex S4.

Macroscale behavior of random lower triangular matrices

We analyze the macroscale behavior of random lower (and therefore upper) triangular matrices with entries drawn i.i.d. from a distribution with nonzero mean and finite variance. We show that such a matrix behaves like a probabilistic version of a Riemann sum and therefore in the limit behaves like the Volterra operator. Specifically, we analyze certain SOT-like and WOT-like modes of convergence for random lower triangular matrices to a scaled Volterra operator. We close with a brief discussion of moments.