Volterra Operators Research Articles

Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: Analysis and numerical solution

A singular boundary value problem for a secondorder linear integrodifferential equation with Volterra and nonVolterra integral operators is formulated and analyzed. The equation is defined on +, has a weak singularity at zero and a strong singularity at infinity, and depends on several positive parameters. Under natural constraints on the coefficients of the equation, existence and uniqueness theorems for this problem with given boundary conditions at singular points are proved, asymptotic representations of the solution are given, and an algorithm for its numerical determination is described. Numerical computations are performed and their interpretation is given. The problem arises in the study of the survival probability of an insurance company over infinite time (as a function of its initial surplus) in a dynamic insurance model that is a modification of the classical Cramer-Lun� dberg model with a stochastic process rate of premium under a certain investment strategy in the financial market. A comparative analysis of the results with those produced by the model with deter� ministic premiums is given. DOI: 10.1134/S0965542512100077

Open loop control by equalization for an analog Duffing system

The inverse Volterra operators have been proposed as an open look control for sensors and actuators systems. Large mathematical development has been done in the area, however the implementation of a continuous time model of the inverse Volterra may be difficult. That is why in this work a discrete version of the Associated Linear Equations for the inverse operator is proposed. The control strategy is tested on an analog electronic Duffing oscillator system.

Long run behaviour of the autocovariance function of ARCH([formula omitted]) models

The asymptotic properties of the memory structure of ARCH(∞) equations are investigated. This asymptotic analysis is achieved by expressing the autocovariance function of ARCH(∞) equations as the solution of a linear Volterra summation equation and analysing the properties of an associated resolvent equation via the admissibility theory of linear Volterra operators. It is shown that the autocovariance function decays subexponentially (or geometrically) if and only if the kernel of the resolvent equation has the same decay property. It is also shown that upper subexponential bounds on the autocovariance function result if and only if similar bounds apply to the kernel.

On Orbits of Functions of the Volterra Operator

We study asymptotic properties of certain functions of the Volterra inte- gral operator V in L p (0,1) (1 ≤ p ≤∞ ). We also prove the Ritt property under minimal spectral assumptions for some functions of V in L 2 (0,1).

Rothe-Galerkin's method for a nonlinear integrodifferential equation

In this article we propose approximation schemes for solving nonlinear initial boundary value problem with Volterra operator. Existence, uniqueness of solution as well as some regularity results are obtained via Rothe-Galerkin method. Mathematics Subject Classification 2000: 35k55; 35A35; 65M20.

Optimal range theorems for operators with p-th power factorable adjoints

Consider an operator T : E ! X(µ) from a Banach space E to a Banach function space X(µ) over a finite measure µ such that its dual map is p-th power factorable. We compute the optimal range of T that is defined to be the smallest Banach function space such that the range of T lies in it and the restricted operator has p-th power factorable adjoint. For the case p = 1, the requirement on T is just continuity, so our results give in this case the optimal range for a continuous operator. We give examples from classical and harmonic analysis, as convolution operators, Hardy type operators and the Volterra operator.

EXISTENCE OF NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH ABSTRACT VOLTERRA OPERATORS

This paper deals with the existence of the solution process of a class of neutral stochastic functional differential equations with abstract Volterra operators defined on certain function spaces. Using fixed point techniques, existence and uniqueness of the solution process are proven. It is also noted that the method of successive approximations can be used to obtain some existence results.

Global dynamics of one class of nonlinear nonautonomous systems with time-varying delays

A class of nonautonomous systems of nonlinear delay differential equations was studied via construction of matrix inequalities and comparison techniques. The results for the nonautonomous systems with time-varying delays are novel, e.g., the global stability of differential equations with nonlinear (casual) Volterra operators is considered for the first time in the literature. Criteria obtained for permanence and global attractivity are explicit and hence are convenient for applying/verifying in practice. We illustrate applications of the results obtained to the nonautonomous and asymptotically autonomous Nicholson-type models.

Volterra operator, area integral and Carleson measures

We characterize the boundedness of Volterra operators from Bergman spaces to Hardy spaces. Area integral operators and Carleson measures are heavily involved.

Optimal estimates for the semigroup generated by the classical Volterra operator on L p -spaces

Optimal upper bounds are given for the norm of the semigroup (e −tV )t≥0, where V is the classical Volterra operator acting on L p [0,1], 1≤p≤∞. In particular, for every p∈[1,∞] we prove that $$\mathop{\overline{\lim}}_{t\to+\infty}\,\left(t^{-|1/4-1/(2p)|}\|e^{-tV}\|_{L_p}\right)>0.$$

A criterion for the unconditional basis property of eigenvectors for finite-rank perturbations of Volterra operators

A criterion for the unconditional basis property of eigenvectors for finite-rank perturbations of Volterra operators is given. Considerations are based on functional models for non-self-adjoint operators and on the technique of the Muckenhoupt matrix weights.

Trajectory attractors of energy balance climate models with bio-feedback

Motivated by coupling an energy balance climate model and a two-species competition model for the bio-sphere, one is led to the study of functional reaction-diffusion equations with memory and a nonlocal Volterra operator. The existence of a trajectory attractor is established. The work is motivated by similar studies in (12) for a energy balance model with latent heat flux and uses techniques developed in (11) and (12). It is a continuation of (18), where an abstract global existence and boundedness result was established.

Numerical solution of a singular integral equation with Cauchy kernel in the plane contact problem

This paper describes a collocation method for solving numerically a singular integral equation with Cauchy and Volterra operators, associated with a proper constraint condition. The numerical method is based on the transformation of the given integral problem into a hypersingular integral equation and then applying a collocation method to solve the latter equation. Convergence of the resulting method is then discussed, and optimal convergence rates for the collocation and discrete collocation methods are given in suitable weighted Sobolev spaces. Numerical examples are solved using the proposed numerical technique.

On admissibility of the resolvent of discrete Volterra equations

Suppose that a pair of sequence spaces is admissible with respect to a discrete linear Volterra operator. This paper gives sufficient conditions for the same pair of spaces to be admissible with respect to the associated resolvent operator. The spaces considered include spaces of weighted bounded and weighted convergent sequences. Classes of discrete kernels are discussed for which the appropriate weight sequences are not purely exponential, but the product of an exponential and a slowly decaying sequence.

Pointwise estimation of the difference of the solutions of a controlled functional operator equation in Lebesgue spaces

For a functional operator equation in Lebesgue space, we prove a statement on the pointwise estimate of the modulus of the increment of its global (on a fixed set Π ⊂ ℝn) solution under the variation of the control function appearing in this equation. As an auxiliary statement, we prove a generalization of Gronwall’s lemma to the case of a nonlinear operator acting in Lebesgue space. The approach used here is based onmethods from the theory of stability of existence of global solutions to Volterra operator equations.

Products of Volterra type operators and composition operators between weighted Bergman spaces of infinite order and weighted Bloch type spaces

Abstract We study the boundedness and compactness of products of Volterra operators and composition operators acting between weighted Bergman spaces of infinite order and weighted Bloch type spaces Bw .

Existence of heteroclinic orbits in simplified longitudinal flight dynamics

The existence of an infinite chain of heteroclinic orbits connecting saddle-nodes is proved, in the case of a simplified system of differential equations governing the longitudinal flight of an aircraft, using fixed point theory for nonlinear Volterra operators with convolution kernels. Numerical simulations are given and the importance of the obtained results is discussed from the point of view of the appearance of oscillatory behavior in longitudinal flight.

ON DYNAMICS OF ℓ-VOLTERRA QUADRATIC STOCHASTIC OPERATORS

We introduce a notion of ℓ-Volterra quadratic stochastic operator defined on (m - 1)-dimensional simplex, where ℓ ∈ {0,1,…, m}. The ℓ-Volterra operator is a Volterra operator if and only if ℓ = m. We study structure of the set of all ℓ-Volterra operators and describe their several fixed and periodic points. For m = 2 and 3, we describe behavior of trajectories of (m - 1)-Volterra operators. The paper also contains many remarks with comparisons of ℓ-Volterra operators and Volterra ones.

A new method for the generalized Hyers-Ulam-Rassias stability

We propose a new method, called the textit{the weighted space method}, for the study of the generalized Hyers-Ulam-Rassias stability. We use this method for a nonlinear functional equation, for Volterra and Fredholm integral operators.

Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator S on ℓ 2 is skew. We show its domain is a proper subset of the domain of its adjoint S ∗ , and − S ∗ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator T on L 2 [ 0 , 1 ] . We compare the domain of T with the domain of its adjoint T ∗ and show that the skew part of T admits two distinct linear maximal monotone skew extensions. These unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators cannot be significantly weakened, and they are simpler than the example given by Phelps–Simons. Interesting consequences on Fitzpatrick functions for sums of two maximal monotone operators are also given.