Controllability of a class of Volterra equations in Hilbert spaces with completely monotone kernel

We consider a parabolic Volterra integro-differential equation in Hilbert space with a completely monotone convolution kernel and a sectorial operator. Utilizing a state space setting, we show that for a large class of kernels the state cannot be controlled exactly to zero. On the other hand, equations of our type are always approximately controllable, provided the control operator has dense range.     

Integro-differential equations for the characteristic functional that are generated by a system of random integral equations

We consider a boundary value problem for a special system of integro-differential equations with variational derivatives. We establish the relationship between this problem and a system of integral equations with a power-law nonlinearity whose kernels and right-hand sides are random functions. We study the solvability of the boundary value problem. Special cases and examples are considered.     

An inverse spectral problem for integro-differential Dirac operators with general convolution kernels

ABSTRACT

An integro-differential Dirac system with a convolution kernel consisting of four independent functions is considered. We prove that the kernel is uniquely determined by specifying the spectra of two boundary value problems with one common boundary condition. The proof is based on the reduction of this nonlinear inverse problem to solving some nonlinear integral equation, which we solve globally. On this basis we also obtain a constructive procedure for solving the inverse problem along with necessary and sufficient conditions for its solvability in an appropriate class of kernels.     

On multi-dimensional integral equations of convolution type with shift

The criterion of invertibility or Fredholmness of some multi-dimensional integral equations with Carleman type shifts are given. The investigation is based on some Banach space approach to equations with an involutive operator. A modified version of this approach is also presented in the paper.This approach is applied to multi-dimensional convolution type equations when the kernels may be integrable or of singular Calderon-Zygmund-Mikhlin type and shift generated by a linear transformation in the Euclidean space satisfying the generalized Carleman condition. The convolution type equations are also specially considered in the two-dimensional case in a sector on the plane symmetric with respect to one of the axes and the corresponding reflection shift. Another application deals with multi-dimensional equations with homogeneous kernels and the shift .     

Dilation theorems for positive definite operator kernels having majorants

Dilation theorems for Banach space valued stochastic processes and operator valued positive definite kernels are considered. It is shown, e.g., that a Banach space valued stochastic process X can be dilated to another process Y, if and only if the covariance kernel of Y is a majorant of the covariance kernel of X. Positive definite operator kernels having majorants of certain special type are characterized.     

A continuous model of the dynamical systems capable to memorise multiple shapes

This paper proposes the novel approach to the mathematical synthesis of continuous self-organising systems capable to memorise and restore own multiple shapes defined by means of functions of single spatial variable or parametric models in two-dimensional space. The model is based on the certain universal form of the integral operator with the kernel representing the system memory. The technique for memorising shapes uses the composition of singular kernels of integral operators. The whole system is described by the potential function, whose minimisation leads to the non-linear dynamics of shape reconstruction by integro-differential non-linear equations with partial derivatives. The corresponding models are proposed and analysed for both parametric and non-parametric shape definitions. Main features of the proposed model are considered, and the results of numerical simulation are shown in case of three shapes memorising and retrieval. The proposed model can be used in theory of smart materials, artificial intelligence and some other branches of non-linear sciences where the effect of multiple shapes memorising and retrieval appears as the core feature.     

Asymptotic behaviour of solutions of linear integro-differential equations

Given the linear integro-differential equation (P_o) on a reflexive Banach space, we prove the existence of unbounded solutions with an exponential growth rate for a class of initial-value problems. Since the appearing kernel functions are of convolution type on a semi-axis, abstract Wiener-Hopf techniques, recently developed by Feldman [3,4,5], are used for the construction of the resolving operator associated with the problems under consideration. Applicability of the results is shown to initial boundary-value problems arising in the theory of generalized heat conduction in materials with memory and of viscoelasticity.     

Numerical solution of Volterra integral and integro-differential equations with rapidly vanishing convolution kernels

Variable stepsize algorithms for the numerical solution of nonlinear Volterra integral and integro-differential equations of convolution type are described. These algorithms are based on an embedded pair of Runge–Kutta methods of order p=5 and p=4 proposed by Dormand and Prince with interpolation of uniform order p=4. They require O(N) number of kernel evaluations, where N is the number of steps. The cost of the algorithms can be further reduced for equations that have rapidly vanishing convolution kernels, by using waveform relaxation iterations after computing the numerical approximation by variable stepsize algorithm on some initial interval. AMS subject classification (2000)  65R20, 45L10, 93C22     

Singularly Perturbed Nonlinear Integrodifferential Systems with Fast Varying Kernels

We consider nonlinear singularly perturbed integro-differential equations with fast varying kernels. It is assumed that the spectrum of the limiting operator lies in the closed left half-plane Re0. We derive an algorithm for obtaining regularized (in the sense of Lomov) asymptotic solutions in both the nonresonance and resonance cases. In deriving the algorithm, we essentially use the regularization apparatus for integral operators with fast varying kernels, developed earlier by the authors for linear integral and integro-differential systems. The algorithm is justified and the existence of a solution of the original nonlinear problem is proved by means of the Newton method for operator equations.     

On an integro-differential equation with almost difference kernel on the half-line

We study the solvability of a class of integro-differential equations with almost difference kernel on the positive half-line. Using a special three-factor decomposition of the original integro-differential operator, we obtain sufficient conditions for the solvability of this equation in the class of tempered absolutely continuous functions. Under additional conditions on the kernel of the corresponding homogeneous equation with some value of the parameter occurring in it, we prove the existence of a nontrivial absolutely continuous solution, which, depending on the sign of the first moment of the kernel, is either a bounded function or has the asymptotics O(x), x → ∞.     

On an inverse spectral problem for first-order integro-differential operators with discontinuities

A convolution integro-differential operator of the first order with a finite number of discontinuities is considered. Properties of its spectrum are studied and a uniqueness theorem is proven for the inverse problem of recovering the convolution kernel along with the boundary condition from the spectrum.     

Well-posedness of a non-local abstract Cauchy problem with a singular integral

A non-local abstract Cauchy problem with a singular integral is studied, which is a closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, the well-posedness of the evolution system is proved under some boundedness and smoothness conditions on the coefficient functions. Furthermore, an isomorphism is established to extend the result to a partial integro-differential equation with a singular convolution kernel, which is a generalized form of the stationary Wigner equation. Our investigation considerably improves the understanding of the open problem concerning the well-posedness of the stationary Wigner equation with in ow boundary conditions.     

Solving integro-differential equations with Cauchy kernel

A simple method for solving the Fredholm singular integro-differential equations with Cauchy kernel is proposed based on a new reproducing kernel space. Using a transformation and modifying the traditional reproducing kernel method, the singular term is removed and the analytical representation of the exact solution is obtained in the form of series in the new reproducing kernel space. The advantage of the approach lies in the fact that, on the one hand, by improving the definition of traditional inner product, the representation of new reproducing kernel function becomes simple and requirement for image space of operator is weakened comparing with traditional reproducing kernel method; on the other hand, the approximate solution and its derivatives converge uniformly to the exact solution and its derivatives. Some examples are displayed to demonstrate the validity and applicability of the proposed method.     

Green and Generalized Green’s Functionals of Linear Local and Nonlocal Problems for Ordinary Integro-differential Equations

A linear, completely nonhomogeneous, generally nonlocal, multipoint problem is investigated for a second-order ordinary integro-differential equation with generally nonsmooth coefficients, satisfying some general conditions like p-integrability and boundedness. A system of three integro-algebraic equations named the adjoint system is introduced for the solution. The solvability conditions are found by the solutions of the homogeneous adjoint system in an “alternative theorem”. A version of a Green’s functional is introduced as a special solution of the adjoint system. For the problem with a nontrivial kernel also a notion of a generalized Green’s functional is introduced by a projection operator defined on the space of solutions. It is also shown that the classical Green and Cauchy type functions are special forms of the Green’s functional. The author passed away in 2006 prior to publication of the article.     

Dual equations of convolution type with kernels from different Banach algebras

We study dual integral equations of convolution type with kernels generated by functions from different Banach algebras of the type L₁(-, ) with weights, and defined by an operator equation. We establish theorems on solvability and Fredholmness, representations of solutions and of the resolvent kernel, and formulas for calculating the characteristic and the index of the corresponding operator.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 803–813, June, 1991.     

A derivative concept with respect to an arbitrary kernel and applications to fractional calculus

In this paper, we propose a new concept of derivative with respect to an arbitrary kernel function. Several properties related to this new operator, like inversion rules and integration by parts, are studied. In particular, we introduce the notion of conjugate kernels, which will be useful to guaranty that the proposed derivative operator admits a right inverse. The proposed concept includes as special cases Riemann‐Liouville fractional derivatives, Hadamard fractional derivatives, and many other fractional operators. Moreover, using our concept, new fractional operators involving certain special functions are introduced, and some of their properties are studied. Finally, an existence result for a boundary value problem involving the introduced derivative operator is proved.     

On the compactness of convolution-type operators in Morrey spaces

In a Morrey space, the product of the convolution operator with summable kernel and the operator of multiplication by an essentially bounded function is considered. Sufficient conditions for such a product to be compact are obtained. In addition, it is shown that the commutator of the convolution operator and the operator of multiplication by a function of weakly oscillating type is compact in a Morrey space.     

On Solvability of Integro-Differential Equations

A class of (possibly) degenerate integro-differential equations of parabolic type is considered, which includes the Kolmogorov equations for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces and in Sobolev-Slobodeckij spaces. Generalisations to stochastic integro-differential equations, arising in filtering theory of jump diffusions, will be given in a forthcoming paper.

     

Reconstruction of a Pair Integral Operator of the Convolution Type

For an arbitrary operator, we pose a general reconstruction problem inverse to the problem of finding solutions. For the pair operator considered, this problem is reduced to the equivalent problem of reconstruction of the kernels of the pair integral equation of the convolution type that generates this operator. In the cases investigated, we prove theorems that characterize the reconstruction of the corresponding kernels, which are constructed in terms of two functions from different Banach algebras of the type L ₁(–, ) with weight.     

Well-posed solvability of volterra integro-differential equations in Hilbert space

We study the well-posed solvability of initial value problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space. These equations are an abstract form of linear partial integro-differential equations that arise in the theory of viscoelasticity and have a series of other important applications. We obtain results on the wellposed solvability of the considered integro-differential equations in weighted Sobolev spaces of vector functions defined on the positive half-line and ranging in a Hilbert space.