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In this paper, we study regular and Fredholm solvability of a Neumann type boundary value problem for a second order elliptic type equation with operator coefficients for a separable Hilbert space on a finite domain. The conditions of regular and Fredholm solvability for the given problem in terms of only the coefficients of the equation are found. The estimates for intermediate derivatives operators are obtained. These estimates determine the solvability conditions for our problem. Note that the considered operator equations have variable coefficients. [ABSTRACT FROM AUTHOR]

Many researchers are interested in topics related to solutions of differential equations with variable coefficients. Here for the first time, we study the nonhomogeneous second-order Euler–Cauchy operator-differential equation. In this paper, we apply the Laplace transform method to find the general solution of the Euler–Cauchy equation with Neumann boundary conditions. We show that this method is powerful in finding the general solution of the Euler–Cauchy equation in terms of generalized exponential functions.

In this paper, we study two classes of operator-differential equations of the third order with a multiple characteristic, considered on the whole axis. We introduce the concept of a smooth regular solution of order 1 and obtain sufficient conditions for the "smoothly" regular solvability of these equations. [ABSTRACT FROM AUTHOR]

The paper considers a variable third-order operator-differential equation in a separable Hilbert space. Under certain assumptions, it is proved that this ODE has a unique solution. The proof is based on a classical Galerkin discretization of the separable Hilbert space in term of certain eigenfunctions. The approximation quality of the Galerkin approximations can be controlled in terms of the eigenvalues. We deduce estimates for the convergence rate of the approximate solutions to the exact one. An example provided as application to the investigated method. [ABSTRACT FROM AUTHOR]

In the work, we consider issues of well-posed and unique solvability of all possible boundary value problems for one class of operatordifferential equations of the third order with a multiple characteristic in Sobolev type spaces of the third order on the semi-axis. In addition, we find exact values of the norms of intermediate derivatives operators in Sobolev-type spaces and establish their connection with solvability conditions. Note that the solvability conditions obtained are expressed only by the operator coefficients of the equation, which makes it easy to check them in the application. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and uniqueness of the generalized solution of a second order partial operator-differential equation in Hilbert space. To this end, at first we prove lemmas on possibility of continuation of some functional generated by the minor terms of the equation on all the Hilbert space. Then we prove a theorem on the existence of a unique generalized solution of a binomial non-homogeneous equation. Using this theorem we prove the existence and uniqueness of the generalized solution of the given operator-differential equation. [ABSTRACT FROM AUTHOR]

In this article, through the coefficients of the second order parabolic operatordifferential equation, the regular solvability problem with initial-boundary conditions is proved in a weight space. The association between the lower bound of the spectrum of the self-adjoint operator in the main part of the differential equation and the weight exponent is clearly provided. A mixed problem of a partial differential equation is introduced as an applied result of this article. [ABSTRACT FROM AUTHOR]

Abstract In this study, we establish existence-uniqueness of a vector function in appropriate Sobolev-type space for a boundary value problem of a fifth-order operator differential equation. Proper conditions are obtained for the given problem to be well-posed. Much effort is devoted to develop the association between these conditions and the operator coefficients of the investigated equation. In this paper, accurate estimates of the norms of the intermediate derivatives operators are presented and used to determine the solvability conditions.

We consider the solvability of boundary value problems and some inverse problems for operator-differential equations of odd order which can be referred to as mixed type equations since the coefficient of the higher order time derivative can change sign. Studying the general classes of boundary value problems with nonlocal boundary conditions, we establish the existence and uniqueness theorems of regular solutions under some conditions on data. [ABSTRACT FROM AUTHOR]

This review paper is devoted to the description of the results of N.D. Kopachevsky and S. Krein (form 1964 till nowadays) related to spectral properties of strong dissipative hydrodynamical systems and corresponding operator-differential equations. Such spectral problems usually lead to eigenvalue problem for the so-called operator pencil of S. Krein with some possible modifications. We provide some abstract results for differential equations in Hilbert space, consider a number of problems on normal motions of a heavy viscous liquid under the different additional physical conditions, enumerate properties of the classical pencil of S. Krein and its modifications. [ABSTRACT FROM AUTHOR]

The initial boundary value problems for linear non-autonomous first-order evolution equations are examined. Our assumptions provide a unified treatment which is applicable to many situations, where the domains of the operators may change with t. We study existence, uniqueness and maximal regularity of solutions in Sobolev spaces. In contrast to the previous results, we use only the continuity assumption on the operators in the main part of the equation. [ABSTRACT FROM AUTHOR]

In this work, conditions for the regular solvability of one boundary-value problem for fourth-order operator-differential equations with a variable coefficient on the semi-axis are found. The obtained conditions are expressed by the properties of the coefficients of the considered operator-differential equation. Moreover, the norms of intermediate derivatives operators are estimated in terms of the norm of the right-hand side of the equation and these estimates are related to the solvability conditions of the boundary value problem. [ABSTRACT FROM AUTHOR]

In the paper we obtain conditions of Fredholm property for third order operator-differential equations in a finite domain. All condi- tions were expressed by the coefficients of the given equation. [ABSTRACT FROM AUTHOR]

In this paper, we obtain suffcient conditions for the well-posed and unique solvability in a weighted Sobolev space for a class of inverse parabolic operator-differential equations of third order. The main part of the equation under consideration has a multiple characteristic. We establish a connection between the solvability conditions and the values of the norms of intermediate derivatives operators. These norms are estimated with respect to the norm of the operator generated by the main part of considered equation. The obtained results show the role of the lower boundary of the spectrum of an abstract operator appearing in the main part of the equation. [ABSTRACT FROM AUTHOR]

We study solvability of the inverse problems of recovering an unknown function on the nonlinear right-hand side of a first order operator-differential equation in some Banach space. The equation is furnished with the Cauchy data, and the overdetermination condition is the value of some operator at a solution. The existence and uniqueness theorems local in time are established. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study a class of second order coefficient operators differential equation with general (possibly non local) boundary conditions. We obtain new results extending those given in a previous paper . Existence, uniqueness and optimal regularity of the strict solution are proved in UMD spaces, using the well‐known Dore–Venni theorem. [ABSTRACT FROM AUTHOR]

The well-known Lagrange method for linear inhomogeneous differential equations is generalized to the case of second-order equations with constant operator coefficients in locally convex spaces. The solutions are expressed in terms of uniformly convergent functional vector-valued series generated by a pair of elements of a locally convex space. Sufficient conditions for the continuous dependence of solutions on the generating pair are obtained. The solution of the Cauchy problem for the equations under consideration is also obtained and conditions for its existence and uniqueness are given. In addition, under certain conditions, the so-called general solution of the equations (a function of most general form from which any particular solution can be derived) is obtained. The study is carried out using the characteristics (order and type) of an operator and of a sequence of operators. Also, the convergence of operator series with respect to equicontinuous bornology is used. [ABSTRACT FROM AUTHOR]

We study a third-order operator-differential equation on the semi-axis with a discontinuous coefficient and boundary conditions which include an abstract linear operator. Sufficient conditions for the well-posed and unique solvability are found by means of properties of the operator coefficients in a Sobolev-type space.

In this work, we give new results concerning existence, uniqueness and maximal regularity of the strict solution of a class of elliptic equations with non-local boundary conditions containing an unbounded linear operator. This study is performed in the framework of UMD Banach spaces. [ABSTRACT FROM AUTHOR]

We study solvability of boundary value problems for the so-called kinetic operator-differential equations of the form B( t) u− L( t) u = f, where L( t) and B( t) are families of linear operators defined in a complex Hilbert space E. We do not assume that the operator B is invertible and that the spectrum of the pencil L − λ B is included into one of the half-planes Re λ < a or Re λ > a $${(a\in {\mathbb{R}})}$$ . Under certain conditions on the above operators, we prove several existence and uniqueness theorems and study smoothness questions in weighted Sobolev spaces for solutions. [ABSTRACT FROM AUTHOR]

In this paper we study the Galyorkin method with a special basis for a linear operator-differential equation of the third order in a separable Hilbert space. The projection method is based on the eigenvectors of the operator similar to the leading operator of equation. We obtain estimates for the convergence rate of approximate solutions in uniform topology. [ABSTRACT FROM AUTHOR]

In the current paper, we study a Petrov-Galerkin method for a Cauchy problem for an operator-differential equation with a leading self-adjoint operator A and a subordinate linear operator K( t) in a Hilbert space. Error estimates for the approximate solutions are obtained. We consider the full equation discretization based on a two-level difference scheme. New asymptotic estimates for the convergence rate of approximate solutions are obtained in uniform topology. The method is applied to the model parabolic problems. [ABSTRACT FROM AUTHOR]

We study boundary value problems for operator-differential equations of mixed type, prove existence of generalized solutions, and establish their smoothness in weighted Sobolev spaces. The results are applied to odd order forward-backward equations. [ABSTRACT FROM AUTHOR]

We study a third-order operator-differential equation on the semi-axis with a discontinuous coefficient and boundary conditions which include an abstract linear operator. Sufficient conditions for the well-posed and unique solvability are found by means of properties of the operator coefficients in a Sobolev-type space. [ABSTRACT FROM AUTHOR]

We obtain sufficient conditions for the regular solvability of initial boundary-value problems for a class of operator-differential equations of third order with variable coefficients on the semiaxis. These conditions are expressed only in terms of the operator coefficients of the equations under study. We obtain estimates of the norms of intermediate derivative operators via the discontinuous principal parts of the equations and also find relations between these estimates and the conditions for regular solvability. [ABSTRACT FROM AUTHOR]

We describe an analog of the Cauchy-Kovalevskaya sufficient conditions for the analytic solvability of the Cauchy problem for systems of operator-differential equations of arbitrary order in locally convex spaces; this analog is stated in terms of the order and type of the linear operator. [ABSTRACT FROM AUTHOR]

In the Sobolev-Besov spaces, we examine the question on solvability of nonlocal boundary value problems for operator-differential equations of the form where B is a linear operator, L is a positive operator, and γ is a real parameter. Under certain conditions on the parameter γ and the data, the existence and uniqueness theorems for solutions to this boundary value problem are proven. The results are applied to studying nonlocal boundary value problems for parabolic equations and systems. [ABSTRACT FROM AUTHOR]

We study a projection-difference method of solving the Cauchy problem for an operatordifferential equation with a selfadjoint leading operator A( t) and a nonlinear monotone subordinate operator K(·) in a Hilbert space. This method leads to a solution of a system of linear algebraic equations at each time level. Error estimates are derived for approximate solutions as well as for fractional powers of the operator A( t). The method is applied to a model parabolic problem. [ABSTRACT FROM AUTHOR]

We present conditions of solvability of a boundary value problem for a class of second order operator-differential equations on a finite segment, study the behavior of the resolvent of the corresponding operator pencil, prove the double completeness of a system of the derived chains of eigenvectors and associated vectors corresponding to a boundary value problem on a segment, and establish the completeness of elementary solutions to the homogeneous equation in the solution space. [ABSTRACT FROM AUTHOR]

In this paper we give definition of m-fold completeness and prove a theorem on completeness of elementary generalized solution of corresponding boundary value problems at which the equation describes the process of corrosive fracture of metals in aggressive media and the principal part of the equation has multiple characteristics. [ABSTRACT FROM AUTHOR]

In this note, we establish sufficient conditions for the correct and unique solvability of various boundary value problems for a class of even-order operator-differential equations on the half-axis. These conditions are unimprovable in terms of operator coefficients of the equation. We note that the principal part of the equation under study suffers a discontinuity. [ABSTRACT FROM AUTHOR]