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This paper is devoted to the study of a nonlocal boundary value problem for a loaded equation of parabolic-hyperbolic type in a special domain. Using representations of the general regular solution, are proven the unique solvability of the problem posed. [ABSTRACT FROM AUTHOR]

This work is devoted to the study of the conditional correctness of the nonlocal boundary value problem for a nonhomogeneous parabolic type equation with two degenerate lines. In this article we investigated well-posed of the problem depending of coefficients in conditions for a problem. In the case of ill-posed, based on the idea of A.N. Tikhonov, the conditional correctness of the problem are proved, namely, the theorems of uniqueness and conditional stability on the set of correctness are proved. For all cases we get a priori estimate of the solution we used the logarithmic convexity method and the results of the spectral problem considered by S.G. Pyatkov. [ABSTRACT FROM AUTHOR]

Рассматривается линейное обыкновенное дифференциальное уравнение дробного порядка с композицией лево- и правосторонних операторов дробных производных в главной части. Уравнения, содержащие композицию операторов дифференцирования дробного порядка с различными началами, появляются при моделировании различных физических и геофизических явлений. К их появлению приводит использование понятия эффективной скорости изменения параметров моделируемых процессов. В частности, уравнения рассматриваемого в работе вида возникают при описании диссипативных колебательных систем. Дробное дифференцирование понимается в смысле РиманаЛиувилля и Герасимова-Капуто. Для исследуемого уравнения изучается нелокальная краевая задача. Нелокальное краевое условие задано в форме интегрального оператора от искомого решения. При определенном условии на ядро оператора, фигурирующего в нелокальном условии, рассматриваемая задача эквивалентно редуцируется к интегральному уравнению Фредгольма второго рода. Найдены достаточные условия разрешимости исследуемой задачи, включающее интегральное ограничение на переменный потенциал. В качестве следствия получено неравенство Ляпунова для решений рассматриваемой нелокальной задачи. Показано, что возникающее в решении задачи условие на ядро интегрального оператора из нелокального условия, является необходимым, в том смысле, что при нарушении этого условия единственность решения задачи теряется.

In this article, we prove existence of radial solutions for a nonlinear elliptic equation with nonlinear nonlocal boundary conditions. Our method is based on some fixed point theorem in a cone.

Как известно, в работе А.В. Бицадзе показано, что задача Дирихле для уравнения смешанного типа некорректна. Естественно возникает вопрос: нельзя ли заменить условия задачи Дирихле другими условиями, охватывающими всю границу, которые обеспечивают корректность задачи? Впервые такие краевые задачи (нелокальные краевые задачи) для уравнения смешанного типа были предложены и изучены в работах Ф.И. Франкля при решении газодинамической задачи об обтекании профилей потоком дозвуковой скорости со сверхзвуковой зоной, оканчивающейся прямым скачком уплотнения. Близкие по постановке задачи для уравнения смешанного типа второго рода второго порядка, имеются в работах А.Н. Терехова, С.Н. Глазатова, М.Г. Каратопраклиевой и С.З. Джамалова. В этих работах для уравнения смешанного типа второго рода второго порядка изучены нелокальные краевые задачи в ограниченных областях. Такие задачи для уравнения смешанного типа первого рода в трехмерном случае (в частости, для уравнения Трикоми) в неограниченных областях изучены в работах С.З. Джамалова и Х. Туракулова. Для уравнений смешанного типа второго рода в неограниченных областях нелокальные краевые задачи в многомерном случае практически не исследованы. С этой целью в данной работе в неограниченном параллелепипеде формулируется и изучается нелокальная краевая задача периодического типа для трехмерного уравнения смешанного типа второго рода второго порядка. Для доказательства единственности обобщённого решения используется метод интегралов энергии. Для доказательства существования обобщённого решения сначала используется преобразование Фурье и в результате получается новая задача на плоскости, а для разрешимости этой задачи используется методы «ε-регуляризации»и априорных оценок. Используя эти методы, и равенство Парсеваля, докажем единственность, существование и гладкость обобщённого решения одной нелокальной краевой задачи периодического типа для трехмерного уравнения смешанного типа второго рода второго порядка.

In this article, we prove existence of radial solutions for a nonlinear elliptic equation with nonlinear nonlocal boundary conditions. Our method is based on some fixed point theorem in a cone. [ABSTRACT FROM AUTHOR]

In this paper we consider a nonlinear nonlocal boundary value problem for one class of systems of hyperbolic equations with mixed derivatives. The problem under consideration is investigated by reducing it to a family of nonlinear boundary value problems for integro-differential equations. One algorithm modification's of D.S. Dzhumabaev parametrization method is proposed and its convergence is proved. Sufficient conditions for the existence of isolated in some set solutions of nonlinear nonlocal boundary value problem for systems of hyperbolic equations with mixed derivatives are revealed. [ABSTRACT FROM AUTHOR]

In this paper, we study the solvability of a new class of nonlocal boundary value problems for the generalized Helmholtz equation in the unit ball. These problems are a generalization of the classical Dirichlet boundary value problem to the Helmholtz equation. For the considered problems, existence and uniqueness theorems are proved. Integral representation of solutions is established. Corresponding spectral questions are also investigated, namely, eigenfunctions and eigenvalues of the problem are found. [ABSTRACT FROM AUTHOR]

In this paper we study Fourier problem for a fractional diffusion-wave equation with the Riemann–Liouville derivative. The correctness of the problem for the fractional diffusion–wave equation with necessary nonlocal conditions is established, and its solution in terms of the Green's function is constructed. [ABSTRACT FROM AUTHOR]

In this article, it is proved a theorem on the unique generalized solvability to a nonlocal boundary value problem for the three-dimensional Tricomi equation in a prismatic unbounded domain. The proof of the theorem is based on the Fourier transform, " -regularization" methods and a priori estimates. [ABSTRACT FROM AUTHOR]

The theory of boundary value problems for hyperbolic equations is one of the most important sections of the theory of partial differential equations, which is explained both by the theoretical significance of the results and by the presence of their practical applications in gas dynamics, in the theory of infinitesimal bendings of surfaces, in the momentless theory of shells, in magnetic hydrodynamics, mathematical biology and other areas. In the XX century in connection with applied technical and economic needs, it became necessary to solve control and optimization problems. The most famous works in this area are J.I. S. Pontryagin and his students V. G. Boltyanskiy, R. V. Gamkrelidze, E. F. Mishchenko, studied issues of control of processes described by systems of ordinary differential equations [18,19]. Further development of applied research leads to the need to control objects, the behavior of which is described using methods with partial differential equations. The corresponding control problems were considered in the works of A.G Butkovsky and co-authors [3,4], A.I Egorov [6-9], J.-L. Lyons [12], K. A. Lurie [13], T. K. Sirazetdinov [20], as well as V. A. Ilyin and E. I. Moiseeva [21,22], [14,15], A.T.Ramazanova [26], Kuliyev Hamlet [27]. Optimal control problems for the processes described by hyperbolic differential equations are encountered in various applications [1,2]. Such problems have been most fully investigated in cases where the boundary conditions for the equations of state are classical. However, there are numerous problems in physics, technology, biology, etc., in which the processes are described by hyperbolic equations, where the boundary conditions are nonclassical or nonlocal [23,24]. Among the nonlocal boundary value problems for the hyperbolic equations, a special place is occupied by boundary value problems with integral boundary conditions [25]. Optimal control problems for hyperbolic-type equations with nonlocal boundary conditions, including those with integral boundary conditions, have been little studied. [ABSTRACT FROM AUTHOR]

The problems of the unique classical solvability and the construction of the solution of a multidimensional boundary value problem for a homogeneous fifth order partial integro-differential equations with a degenerate kernel are studied. The multidi-mensional Fourier series method, based on the separation of many variables, is used. A system of countable systems of integral equations is derived. Iteration process of solving the problem is constructed. Suficient coeficient conditions for the unique classical solvability of the boundary value problem are established. [ABSTRACT FROM AUTHOR]

In this paper, we consider nonlocal boundary value problems for the Laplace operator in a ball, which are a multidimensional generalisation of the Samarskii–Ionkin problem. The well-posedness of the problems are investigated, and Fredholm property of the problems are studied. Moreover, we obtain integral representations of their solutions in explicit forms. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a nonlinear fractional differential equation involving two mixed fractional orders with nonlocal boundary conditions. By using some new techniques, we introduce a formula of solutions for above problem, which can be regarded as a novelty item. Moreover, under the weak assumptions and using Leray–Schauder degree theory, we obtain the existence result of solutions for above problem. Furthermore, we discuss the Ulam–Hyers stability of the above fractional differential equation. Three examples illustrate our results.

The present work is devoted to the study of the solvability questions for a nonlocal problem with an integrodifferential conjugation condition for a fourth-order mixed-type equation with a generalized RiemannLiouville operator. Under certain conditions on the given parameters and functions, we prove the theorems of uniqueness and existence of the solution to the problem. In the paper, given example indicates that if these conditions are violated, the formulated problem will have a nontrivial solution. To prove uniqueness and existence theorems of a solution to the problem, the method of separation of variables is used. The solution to the problem is constructed as a sum of an absolutely and uniformly converging series in eigenfunctions of the corresponding one-dimensional spectral problem. The Cauchy problem for a fractional equation with a generalized integro-differentiation operator is studied. A simple method is illustrated for finding a solution to this problem by reducing it to an integral equation equivalent in the sense of solvability. The authors of the article also establish the stability of the solution to the considered problem with respect to the nonlocal condition.

This paper is devoted to the existence of solutions for a class of nonlinear boundary value problems with integral boundary conditions and generalized 𝑝-Laplacian on the positive half-line. We establish sufficient conditions to guarantee the existence of solutions in a special function space by using Leray–Schauder-type arguments. Examples are also given to illustrate the main results. [ABSTRACT FROM AUTHOR]

A boundary value problem for a third-order parabolic equation with a fractional Caputo derivative is considered. A priori estimation of the solution of a generalized nonlocal boundary value problem for an equation with multiple characteristics with a fractional Caputo derivative in time is obtained by the method of energy inequalities.

For the fractional diffusion equation, we study a nonlocal boundary value problem of the first kind. The problem nonlocality is manifested in the fact that a linear combination of the values of the desired function is specified in the boundary condition. The theorem on the existence and uniqueness of a solution to the problem is proved.

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 paper proposes a nonlocal formulation regarding the modeling of Antarctic Circumpolar Current by introducing flow functions to encode horizontal flow components without considering vertical motion. Using topological degree, zero exponent theory and fixed point technique, we show the existence of positive solutions to nonlocal boundary value problems with nonlinear vorticity. [ABSTRACT FROM AUTHOR]

Differential and difference interpretations of a nonlocal boundary value problem for Poisson's equation in open rectangular domain are studied. New solvability conditions are obtained in respect of existence, uniqueness and a priori estimate of the classical solution. Second order of accuracy difference scheme is presented. [ABSTRACT FROM AUTHOR]

In the present paper, differential and difference variants of nonlocal boundary value problem (NLBVP) for Poisson’s equation in open rectangular domain are studied. The existence, uniqueness and a priori estimate of classical solution are established. The second order of accuracy difference scheme is presented. The applications with weighted integral condition are provided in differential and difference variants.

Abstract The exact solution of fractional telegraph partial differential equation of nonlocal boundary value problem is obtained. The theorem of stability estimates is presented for this equation. Difference schemes are constructed for both the implicit finite difference scheme and the Dufort–Frankel finite difference scheme (DFFDS). The stability of these difference schemes for this problem are proved. With the help of these methods, numerical solutions of the fractional telegraph differential equation defined by Caputo fractional derivative for fractional orders α=0.1,0.5,0.9 $\alpha=0.1, 0.5, 0.9$ are calculated. Numerical results are compared with the exact solution and the accuracy and effectiveness of the proposed methods are investigated.

In the current paper, in the domain $D=\{(t,x): t\in(0,T), x\in(0,L)\}$ we investigate the boundary value problem for the equation of motion of a homogeneous elastic beam $$ u_{tt}(t,x)+a^{2}u_{xxxx}(t,x)+b u_{xx}(t,x)+c u(t,x)=0, $$ where $a,b,c \in \mathbb{R}$, $b^22$, then for almost all (with respect to Lebesgue measure in $\mathbb{R}$) numbers $a$ exists a unique solution $u\in\mathbf{C}^{\,2}([0,T];\mathbf{H}_{q})$ of the problem considered.

In this paper we study the nonlocal Neumann boundary value problem of the following form $$ u'' =f(t,u,u'),\quad u'(0)=0, \quad u'(1)=\int_{0 }^{1}u'(s)dg(s), $$ where $f:[0,1]\times\mathbb R^n\times\mathbb R^n\to\mathbb R^n$ and $g=\mbox{diag}(g_1,\ldots,g_n)$ with $g_i:[0,1]\to\mathbb R$, $i=1,\ldots,n$. The case when the function $f$ does not depend on $u'$ is also considered. The existence of solutions is obtained by means of the generalized Miranda theorem. The main results can be applied to many problems of this type depending on which conditions will be imposed upon the function $f$.

We provide sufficient conditions for the existence of positive solutions of a nonlocal boundary value problem at resonance concerning a second order functional differential equation. The method is developed by inserting an exponential factor which depends on a suitable positive parameter $\lambda$. By this way a Green's kernel can be established and the problem is transformed into an operator equation $u=T_{\lambda}u$. As it can be shown the well known Krasnoselskii's fixed point theorem on cones in the Banach space C[0,1] cannot be applied. More exactly, there is no (positive) value of the parameter $\lambda$ for which the condensing property $\|T_{\lambda}u\|\leq\|u\|$, with $\|u\|=\rho(>0)$ is satisfied. To overcome this fact we enlarge the space $C[0,1]$ and work in ${\mathcal{L}^1[0,1]}$ where, now, Krasnoselskii's fixed point theorem is applicable. Compactness criteria in this space are, certainly, needed.

We describe the statements of boundary value problems for multidimensional higher order equations of mixed type and propose a method for constructing coercive boundary value problems. Also, we prove the generalized and regular solvability of these boundary value problems in Sobolev spaces. [ABSTRACT FROM AUTHOR]

Nonlocal boundary value problems for arbitrary order hyperbolic systems with one spatial variable are considered. A priori estimates for general nonlocal mixed problems for systems with smooth and piecewise smooth coefficients are obtained. The correct solvability of such problems is proved. Examples of additional conditions necessity are provided. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear fractional differential equation involving two mixed fractional orders with nonlocal boundary conditions. By using some new techniques, we introduce a formula of solutions for above problem, which can be regarded as a novelty item. Moreover, under the weak assumptions and using Leray–Schauder degree theory, we obtain the existence result of solutions for above problem. Furthermore, we discuss the Ulam–Hyers stability of the above fractional differential equation. Three examples illustrate our results. [ABSTRACT FROM AUTHOR]

This paper investigates the solvability of a class of higher order nonlocal boundary value problems of the form u (n) (t) = g(t, u(t), u' (t). . . u (n-1)(t)), a.e. t ∈ (0, ∞) subject to the boundary conditions u (n-1)(0) = (n - 1)!/ ξn-1 u(ξ), u(i) (0) = 0, i = 1, 2, . . . , n - 2, u (n-1)(∞) = ∫ξ0 u(n-1)(s)dA(s) where ξ > 0, g : [0, ∞) × Rn → R is a Caratheodory's function, A : [0, ξ] → [0, 1) is a non-decreasing function with A(0) = 0, A(ξ) = 1. The differential operator is a Fredholm map of index zero and non-invertible. We shall employ coicidence degree arguments and construct suitable operators to establish existence of solutions for the above higher order nonlocal boundary value problems at resonance. [ABSTRACT FROM AUTHOR]

In the three-dimensional domain a Boussinesq type nonlinear partial integro-differential equation of the fourth order with a degenerate kernel, integral form conditions, spectral parameters and reflecting argument is considered. The solution of this partial integro-differential equation is studied in the class of generality functions. The method of separation of variables and the method of a degenerate kernels are used. Using these methods, the nonlocal boundary value problem is integrated as a countable system of ordinary differential equations. When we define the arbitrary integration constants there are possible five cases with respect to the first spectral parameter. Calculated values of the spectral parameter for each case. Further, the problem is reduced to solving countable system of linear algebraic equations. Irregular values of the second spectral parameter are determined. At irregular values of the second spectral parameter the Fredholm determinant is degenerate. Other values of the second spectral parameter, for which the Fredholm determinant does not degenerate, are called regular values. Taking the values of the first spectral parameter into account for regular values of the second spectral parameter the corresponding solutions were constructed and we obtained the countable system of nonlinear integral equations for each of five cases. To establish the unique solvability of this countable system of nonlinear integral equations we use the method of successive approximations and the method of compressing mappings. Using the Cauchy–Schwarz inequality and the Bessel inequality, we proved the absolute and uniform convergence of the obtained Fourier series. The stability of the solution of the boundary value problem with respect to given functions in integral conditions is proved. The conditions under which the solution of the boundary value problem will be small are studied. For the irregular values of the second spectral parameter each of the five cases is checked separately. The orthogonality conditions are used. Cases are determined in which the problem has an infinite number of solutions and these solutions are constructed as Fourier series. For other cases, the absence of nontrivial solutions of the problem is proved. The corresponding theorems are formulated. [ABSTRACT FROM AUTHOR]

Boundary-value problems for loaded equations in the plane in the case of loaded parts consist of traces of an unknown solution or its first normal derivatives, are well studied. The multidimensional loaded differential equations are relatively less investigated. Moveover, when the loaded part consists of not only the traces of the solution or its first normal derivatives, but also second derivatives of the solutions, the classical methods are not effective. Therefore, in this paper, we propose a method which overcomes these difficulties. Under some conditions on coefficients of the loaded multidimensional Chaplygin's equation, we prove existence and uniqueness of a solution of a nonlocal boundary-value problem in the Sobolev space . [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the existence of positive solutions for a system of nonlinear fractional differential equations nonlocal boundary value problems with parameters and p-Laplacian operator. Under different combinations of superlinearity and sublinearity of the nonlinearities, various existence results for positive solutions are derived in terms of different values of parameters via the Guo-Krasnosel’skii fixed point theorem.

Abstract This paper focuses on the maximum and minimum solutions for a fractional order differential system, involving a p-Laplacian operator and nonlocal boundary conditions, which arises from many complex processes such as ecological economy phenomena and diffusive interaction. By introducing new type growth conditions and using the monotone iterative technique, some new results about the existence of maximal and minimal solutions for a fractional order differential system is established, and the estimation of the lower and upper bounds of the maximum and minimum solutions is also derived. In addition, the iterative schemes starting from some explicit initial values and converging to the exact maximum and minimum solutions are also constructed.

The problems of solvability and construction of solutions of a nonlocal boundary value problem for the second-order Fredholm integro-differential equation with degenerate kernel, integral conditions, spectral parameters and reflecting deviation are considered. Using the method of the degenerate kernel, the boundary value problem is integrated as an ordinary differential equation. When we define the arbitrary integration constants there are possible five cases with respect to the first spectral parameter. Calculated values of the spectral parameter for each case. Further, the problem is reduced to solving systems of linear algebraic equations. Irregular values of the second spectral parameter are determined. At irregular values of the second spectral parameter the Fredholm determinant is degenerate. Other values of the second spectral parameter, for which the Fredholm determinant does not degenerate, are called regular values. Taking the values of the first spectral parameter into account for regular values of the second spectral parameter the corresponding solutions were constructed for each of five cases. The stability of the solution of the boundary value problem for given values in integral conditions is proved. The conditions under which the solution of the boundary value problem will be small are studied. For the irregular values of the second spectral parameter each of the five cases is checked separately. The orthogonality conditions are used. Cases are determined in which the problem has an infinite number of solutions and these solutions are constructed. For other cases, the absence of nontrivial solutions of the problem is proved. [ABSTRACT FROM AUTHOR]

In this work, we study non-local value problem for second order mixed type differential equation. We show this problems are ill-posed and prove the theorems of uniqueness and conditional stability of solutions in the set of correctness. [ABSTRACT FROM AUTHOR]

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Abstract We consider the multipoint nonlocal boundary value problem for the two-dimensional Laplace equation in a rectangular domain. The solution of this problem is defined as a 9-point finite difference solution, with the fourth order gluing operator of the local Dirichlet boundary value problem, by constructing a special method to find a function as the boundary value on the side of the rectangle, where the nonlocal condition is given. Numerical experiments are illustrated to support the analysis made. [ABSTRACT FROM AUTHOR]

We consider nonlocal boundary value problems for a Sobolev-type equation with variable coefficients with fractional Gerasimov–Caputo derivative. The main result of the article consists in proving a priori estimates for solutions to nonlocal boundary value problems both in differential and difference form obtained under the assumption of the existence of a solution u(x, t) in a class of sufficiently smooth functions. These inequalities imply the uniqueness and stability of a solution with respect to the initial data and right-hand side and also the convergence of the solution to the difference problem to the solution to the differential problem. [ABSTRACT FROM AUTHOR]

A couple ($\sigma,\tau$) of lower and upper slopes for the resonant second order boundary value problem x" = f(t,x,x'), \quad x(0) = 0,\quad x'(1) = \int_0^1 x'(s) {\rm d}g(s), with $g$ increasing on $[0,1]$ such that $\int_0^1 dg = 1$, is a couple of functions $\sigma, \tau\in C^1([0,1])$ such that $\sigma(t) łeq\tau(t)$ for all $t \in[0,1]$, \begin{gather} \sigma'(t) \geq f(t,x,\sigma(t)), \quad\sigma(1) łeq\int_0^1 \sigma(s) {\rm d}g(s),\nonumber \tau'(t) łeq f(t,x,\tau(t)), \quad\tau(1) \geq\int_0^1 \tau(s) {\rm d}g(s),\nonumber\end{gather} in the stripe $\int_0^t\sigma(s) {\rm d}s łeq x łeq\int_0^t \tau(s) {\rm d}s$ and $t \in[0,1]$. It is proved that the existence of such a couple $(\sigma,\tau)$ implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.

In this paper, we use Bananch's contraction mapping principle and Schauder’s fixed point theorem to study the existence and uniqueness of solutions for nonlocal boundary value problems of fractional differential equations with the nonlinear term dependent in a fractional derivative of lower order on infinite intervals,  Dα0+u(t)=f(t,u(t),Dα-10+u(t)),t∈J,u(0)=0,Dα-10+u(∞)=∑ m-2 t=1 βiu(ηi)  and extend the existing relevant conclusions. We have given some examples to demonstrate the correctness of the main results.

In this article we show the existence of at least one solution to the system of nonlocal resonant boundary-value problem $$ x''=f(t,x), \quad x'(0)=0, \quad x'(1)=\int_{0 }^{1}x'(s)\,dg(s), $$ where $f:[0,1]\times\mathbb{R}^k\to\mathbb{R}^k$, $g:[0,1]\to\mathbb{R}^k$.

In the paper, under some conditions on coefficients of the loaded Chaplygin's equation in a rectangle, we prove the existence and uniqueness of a solution of a nonlocal boundary value problem in the Sobolev space W 3/2 (Q). [ABSTRACT FROM AUTHOR]

In this paper we prove unique solvability of a nonlocal boundaryvalue problem in the Sobolev space W 23 (Q), under certain conditions on the coefficients of a second kind second-order mixed type loaded equation in a rectangle. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions for the existence of positive solutions of a nonlocal boundary value problem at resonance concerning a second order functional differential equation. The method is developed by inserting an exponential factor which depends on a suitable positive parameter λ. By this way a Green's kernel can be established and the problem is transformed into an operator equation u = Tλu. As it can be shown the well known Krasnoselskii's fixed point theorem on cones in the Banach space C[0, 1] cannot be applied. More exactly, there is no (positive) value of the parameter λ for which the condensing property ǁTλuǁ ≤ ǁuǁ, with ǁuǁ = ρ(> 0) is satisfied. To overcome this fact we enlarge the space C[0, 1] and work in L1 [0, 1] where, now, Krasnoselskii's fixed point theorem is applicable. Compactness criteria in this space are, certainly, needed. [ABSTRACT FROM AUTHOR]

In the current paper, in the domain D = {(t, x) : t ∈ (0, T), x ∈ (0, L)} we investigate the boundary value problem for the equation of motion of a homogeneous elastic beam utt(t, x) + a2uxxxx(t, x) + buxx(t, x) + cu(t, x) = 0, where a, b, c ∈ R, b2 < 4a2 c, with nonlocal two-point conditions u(0, x) − u(T, x) = φ(x), ut(0, x) − ut(T, x) = ψ(x) and local boundary conditions u(t, 0) = u(t, L) = uxx(t, 0) = uxx(t, L) = 0. Solvability of this problem is connected with the problem of small denominators, whose estimation from below is based on the application of the metric approach. For almost all (with respect to Lebesgue measure) parameters of the problem, we establish conditions for the solvability of the problem in the Sobolev spaces. In particular, if φ ∈ Hq+ρ+2 and ψ ∈ Hq+ρ, where ρ > 2, then for almost all (with respect to Lebesgue measure in R) numbers a there exists a unique solution u ∈ C2 ([0, T]; Hq) of the problem. [ABSTRACT FROM AUTHOR]