Volterra Property Research Articles

Spectral properties of local and nonlocal problems for the diffusion-wave equation of fractional order

The paper investigates the issues of solvability and spectral properties of local and nonlocal problems for the fractional order diffusion-wave equation. The regular and strong solvability to problems stated in the domains, both with characteristic and non-characteristic boundaries are proved. Unambiguous solvability is established and theorems on the existence of eigenvalues or the Volterra property of the problems under consideration are proved.

Solvability and Volterra property of nonlocal problems for mixed fractional-order diffusion-wave equation

The paper is devoted to the study of one class of problems with nonlocal conditions for a mixed diffusion-wave equation with two independent variables. The main results of the work are the proof of regular and strong solvability, as well as the Volterra property of three problems with conditions pointwise connecting the values of the tangent derivative of the desired solution on one of the characteristics with derivatives in various directions of the solution on an arbitrary curve lying inside the characteristic triangle for a fractional-order diffusion-hyperbolic equation.

On the Adjoint Problem in a Domain with Deviation Out from the Characteristic for the Mixed Parabolic-Hyperbolic Equation with the Fractional Order Operator

В настоящей статье доказана классическая, сильная разрешимость и вольтерровость сопряженной задачи c отходом от характеристики для уравнения смешанного параболо-гиперболического типа с оператором дробного порядка в смысле Герасимова-Капуто. Целью исследования является решение сопряженной задачи для уравнения смешанного параболо-гиперболического типа дробного порядка. Учитывая, свойств операторов дробного порядка найдены сопряженный оператор и применены постановки сопряженной задачи. Для исследования поставленной задачи в параболической частью смешанной области решается первой краевой задачи для уравнения параболического типа дробного порядка в смысле Герасимова-Капуто. Используя, свойств функции Райта получено функциональное соотношение на линии перехода. Точно также решая, задачи Коши гиперболической частью смешанной области находим функциональное соотношение. Следовательно, поставленная задача эквивалентным образом сводится к интегральному уравнению Вольтерра второго рода со слабой особенностью. Согласно теории интегральных уравнений Вольтерра второго рода доказывается однозначной разрешимость полученного уравнения. Кроме того, используя методы операторов интегро — дифференцирования дробного порядка, теории специальных функций, априорных оценок, теория интегральных уравнений доказываются теоремы единственности, существования и вольтерровость сопряженной задачи в области с отходом от характеристики для уравнения смешанного типа дробного порядка. Полученные результаты новые и отличаются от результатов М.А. Садыбекова и А.С. Бердышева. In this article, it was proved the classical, strong solvability and Volterra property of the adjoint problem with departure from the characteristic for an equation of mixed parabolic-hyperbolic type with a fractional order operator in the sense of Gerasimov-Caputo. The aim of the research is to solve the conjugate problem for the equation of a mixed parabolic-hyperbolic type of fractional order. Taking into account the properties of fractional order operators, the adjoint operator is found and the statements of the adjoint problem are applied. To study the formulated problem in the parabolic part of the mixed domain, the first boundary value problem for a parabolic type equation of fractional order in the sense of Gerasimov-Caputo is solved. Using the properties of the Wright function, a functional relation is obtained on the transition line. In the same way, solving the Cauchy problem with the hyperbolic part of the mixed domain, we find a functional relation. Consequently, the problem posed reduces in an equivalent way to a Volterra integral equation of the second kind with a weak singularity. According to the theory of Volterra integral equations of the second kind, the unique solvability of the resulting equation is proved. In addition, using the methods of integro-differentiation operators of fractional order, the theory of special functions, a priori estimates, the theory of integral equations, uniqueness, existence and Volterra theorems for the adjoint problem in a domain with deviation out of the characteristic for a mixed-type equation of fractional order are proved. The results obtained are new and differ from the results of M. A. Sadybekov and A. S. Berdyshev.

Criterion for the Volterra Property of the Cauchy Problem for the Pantograph Equation

In this paper, spectral properties of the Cauchy problem of the pantograph equation are examined. We establish the boundaries of the parameter interval in which the problem remains a Volterra problem.

Some applications of Maia's fixed point theorem in the case of operators with Volterra property with respect to a subinterval

Some applications of Maia's fixed point theorem in the case of operators with Volterra property with respect to a subinterval

Some applications of Maia's fixed point theorem in the case of operators with Volterra property with respect to a subinterval

The aim of this paper is to improve some fixed point results given by I.A. Rus in (On a fixed point theorem of Maia, Studia Univ. Babes¸-Bolyai Math., 22(1977), 40-42), (Basic problem for Maia’s theorem, Sem. on Fixed Point Theory, Preprint 3(1981), Babes¸-Bolyai Univ. ClujNapoca, 112-115) and (Data dependence of the fixed points in a set with two metrics, Fixed Point Theory, 8(2007), no. 1, 115-123), by applying, instead of contraction principle, some variants of Maia’s fixed point theorem.

Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle

Following the idea of T.A. Burton, of progressive contractions, presented in some examples (T.A. Burton, \emph{A note on existence and uniqueness for integral equations with sum of two operators: progressive contractions}, Fixed Point Theory, 20 (2019), No. 1, 107-113) and the forward step method (I.A. Rus, \emph{Abstract models of step method which imply the convergence of successive approximations}, Fixed Point Theory, 9 (2008), No. 1, 293-307), in this paper we give some variants of contraction principle in the case of operators with Volterra property. The basic ingredient in the theory of step by step contraction is $G$-contraction (I.A. Rus, \emph{Cyclic representations and fixed points}, Ann. T. Popoviciu Seminar of Functional Eq. Approxim. Convexity, 3 (2005), 171-178). The relevance of step by step contraction principle is illustrated by applications in the theory of differential and integral equations.

Volterra operators in the edge-calculus

We study the Volterra property of a class of anisotropic pseudo-differential operators on $$\mathbb {R}\times B$$ for a manifold B with edge Y and time-variable t. This exposition belongs to a program for studying parabolicity in such a situation. In the present consideration we establish non-smoothing elements in a subalgebra with anisotropic operator-valued symbols of Mellin type with holomorphic symbols in the complex Mellin covariable from the cone theory, where the covariable $$\tau $$ of t extends to symbols with respect to $$\tau $$ to the lower complex v half-plane. The resulting space of Volterra operators enlarges an approach of Buchholz (Parabolische Pseudodifferentialoperatoren mit operatorwertigen Symbolen. Ph.D. thesis, Universitat Potsdam, 1996) by necessary elements to a new operator algebra containing Volterra parametrices under an appropriate condition of anisotropic ellipticity. Our approach avoids some difficulty in choosing Volterra quantizations in the edge case by generalizing specific achievements from the isotropic edge-calculus, obtained by Seiler (Pseudodifferential calculus on manifolds with non-compact edges, Ph.D. thesis, University of Potsdam, 1997), see also Gil et al. (in: Demuth et al (eds) Mathematical research, vol 100. Akademic Verlag, Berlin, pp 113–137, 1997; Osaka J Math 37:221–260, 2000).

Spectral properties of boundary-value problem with a shift for wave equation

We consider a differential operator determined by wave equation with potential in characteristic triangle, and boundary-value conditions with shift on the characteristics, and with oblique derivative on non-characteristic part of a boundary. We obtain condition for validity of the Volterra property, and show completeness of the root functions in the rest cases. We study basis property for the system of root functions under assumption that the potential depends on a single variable.

On the Volterra property of a boundary problem with integral gluing condition for a mixed parabolic-hyperbolic equation

Abstract In the present work, we consider a boundary value problem with gluing conditions of an integral form for the parabolic-hyperbolic type equation. We prove that the considered problem has the Volterra property. The main tools used in the work are related to the method of the integral equations and functional analysis.

Linear Volterra operators in some metric spaces

We apply our definition of Volterra operator on abstract spaces to some problems arising in metric spaces. In contrast to those known before, our definition requires only the existence of a σ-algebra on a metric space. Note that, being applied to such spaces, the new definition substantially extends the classes of operators of an evolutionary nature. It also allows one to relate different properties of the Volterra-type operators. In particular, the problem of quasi-nilpotentness studied traditionally in the Banach spaces only (since it requires equality to zero of the spectral radius of an operator) allows interpretation in complete locally convex spaces. Apparently, the question on preserving the Volterra property by a conjugate operator is posed for the first time. It should be mentioned that, generally speaking, the dual space to a Frechet (i.e., complete locally convex) space is not a Frechet space.

Representation formula for solution of a functional equation with Volterra operator

The following functional equation is under consideration, (0.1) L x = f with a linear continuous operator L , defined on the Banach space X 0 ( Ω 0 , Σ 0 , μ 0 ; Y 0 ) of functions x 0 : Ω 0 → Y 0 and having values in the Banach space X 2 ( Ω 2 , Σ 2 , μ 2 ; Y 2 ) of functions x 2 : Ω 2 → Y 2 . The peculiarity of X 0 is that the convergence of a sequence x n 0 ∈ X 0 , n = 1 , 2 , … , to the function x 0 ∈ X 0 in the norm of X 0 implies the convergence x n 0 ( s ) → x 0 ( s ) , s ∈ Ω 0 , μ 0 -almost everywhere. The assumption on the space X 2 is that it is an ideal space. The suggested representation of solution to (0.1) is based on a notion of the Volterra property together with a special presentation of the equation using an isomorphism between X 0 and the direct product X 1 ( Ω 1 , Σ 1 , μ 1 ; Y 1 ) × Y 0 (here X 1 ( Ω 1 , Σ 1 , μ 1 ; Y 1 ) is the Banach space of measurable functions x 1 : Ω 1 → Y 1 ). The representation X 0 = X 1 × Y 0 leads to a decomposition of L : X 0 → X 2 for the pair of operators Q : X 1 → X 2 and A : Y 0 → X 2 . A series of basic properties of (0.1) is implied by the properties of operator Q.

ON δ* SETS AND THE GENERATED IDEAL

In this paper, the concept of δ* -sets is introduced and the ideal Ω generated by all δ* -sets in a topological space is considered. Some properties of δ* -sets, the ideal Ù and their relations with resolvability, irresolvability, the Volterra property and other ideals are explored. It is shown that Ω is compatible with the topology for any given space, which is an analogue to the Banach Category theorem.

ON <i>δ</i>* SETS AND THE GENERATED IDEAL

In this paper, the concept of a*-sets is introduced and the ideal Ω generated by all <5*-sets in a topological space is considered. Some properties of 5*-sets, the ideal Ω and their relations with resolvability, irresolvability, the Volterra property and other ideals are explored. It is shown that Ω is compatible with the topology for any given space, which is an analogue to the Banach Category theorem.

The ideal generated by σ-nowhere dense sets

In this paper, we consider the ideal I σ generated by all σ-nowhere dense sets in a topological space. Properties of this ideal and its relations with the Volterra property are explored. We show that I σ is compatible with the topology for any given topological space, an analogue to the Banach category theorem. Some applications of this result and the Banach category theorem are also given.

The Volterra Property of Some Problems with the Bitsadze-Samarskii-Type Conditions for a Mixed Parabolic-Hyperbolic Equation

We consider problems with the Bitsadze-Samarskii-type conditions for a mixed parabolic-hyperbolic equation with noncharacteristic type change curve. We prove theorems on the unique existence of regular and strong solutions and the Volterra property for the problems under consideration.

Reduction of orders in boundary value problems without transmission property

Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We employ specific smooth symbols of arbitrary real orders and with parameters, and we show that the associated operators induce isomorphisms between Sobolev spaces on a given manifold with boundary. Such operators for integer orders have the transmission property and belong to the calculus of Boutet de Monvel [1], cf. also [9]. In general, they fit to the algebra of boundary value problems without the transmission property in the sense of [17] and [24]. Order reducing elements of the present kind are useful for constructing parametrices of mixed elliptic problems. We show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We then investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary. We finally construct order reducing operators on a compact manifold with conical singularities and boundary.

A criterion for the Volterra property of boundary value problems for Sturm-Liouville equations

A criterion for the Volterra property of boundary value problems for Sturm-Liouville equations