On the Continuous Dependence of the Solution of a Boundary Value Problem on Boundary Conditions: Elements of P-Regularity Theory

2019 ◽  
Vol 100 (2) ◽  
pp. 426-429
Author(s):  
B. Medak ◽  
A. A. Tret’yakov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Continuous Dependence ◽  
Boundary Value ◽  
Regularity Theory

scholarly journals On a boundary value problem of arbitrary orders differential inclusion with nonlocal, integral and infinite points boundary conditions

2021 ◽  
Vol 7 (3) ◽  
pp. 3896-3911
Author(s):  
A. M. A. El-Sayed ◽  
◽  
W. G. El-Sayed ◽  
Somyya S. Amrajaa ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Differential Inclusion ◽  
Continuous Dependence ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Existence Results ◽  
Sufficient Condition ◽  
Nonlocal Boundary Value Problem ◽  
Fractional Orders

<abstract><p>In this work, we are concerned with a boundary value problem of fractional orders differential inclusion with nonlocal, integral and infinite points boundary conditions. We prove some existence results for that nonlocal boundary value problem. Next, the existence of maximal and minimal solutions is proved. Finally, the sufficient condition for the uniqueness and continuous dependence of solution are studied.</p></abstract>

scholarly journals On the continuous dependence of the solution of a boundary value problem on boundary conditions. Elements of p-regularity theory

2019 ◽  
Vol 488 (2) ◽  
pp. 126-129
Author(s):  
B. Medak ◽  
A. A. Tret’yakov
Keyword(s):  
Boundary Value Problem ◽  
Small Parameter ◽  
Implicit Function Theorem ◽  
Continuous Dependence ◽  
Boundary Value ◽  
Regularity Theory ◽  
Regularity Property ◽  
Implicit Function ◽  
Left And Right ◽  
P Factor

The problem of the existence of a continuous dependence of the solution of a boundary value problem on a parameter is considered. In this paper, it is proved that, in the presence of the p-regularity property, there exists a solution that continuously depends on a small parameter. The main result of the paper is based on theorems representing different versions of the implicit function theorem. In the case of degenerate mappings, the theorems are used to analyze a boundary value problem with a small parameter. In the case of absolute degeneration, a -p-factor operator is found. The concept of the p-kernel of the operator is introduced, as well as the left and right inverse operators. Theorems are formulated that are special versions of the generalized Lyusternik theorem and the implicit function theorem in the degenerate case. An implicit function theorem is formulated and proved in the case of a nontrivial kernel.

Identification of the string boundary conditions using natural frequencies

2016 ◽  
Vol 11 (1) ◽  
pp. 38-52
Author(s):  
I.M. Utyashev ◽  
A.M. Akhtyamov
Keyword(s):  
Inverse Problems ◽  
Power Series ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Boundary Value ◽  
External Environment ◽  
Direct And Inverse Problems ◽  
Symmetric Potential ◽  
Modified Method

The paper discusses direct and inverse problems of oscillations of the string taking into account symmetrical characteristics of the external environment. In particular, we propose a modified method of finding natural frequencies using power series, and also the problem of identification of the boundary conditions type and parameters for the boundary value problem describing the vibrations of a string is solved. It is shown that to identify the form and parameters of the boundary conditions the two natural frequencies is enough in the case of a symmetric potential q(x). The estimation of the convergence of the proposed methods is done.

scholarly journals Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Maria Italia Gualtieri ◽  
Anna Napoli
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Interpolation Problem ◽  
Boundary Value ◽  
Computational Tools ◽  
Numerical Examples ◽  
Odd Order ◽  
Uniqueness Of The Solution ◽  
Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

scholarly journals The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients. II

2020 ◽  
Vol 12 (1) ◽  
pp. 173-188
Author(s):  
Ya.O. Baranetskij ◽  
P.I. Kalenyuk ◽  
M.I. Kopach ◽  
A.V. Solomko
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Conditions ◽  
Mixed Boundary Conditions ◽  
Multipoint Problem ◽  
Uniqueness Of The Solution

In this paper we continue to investigate the properties of the problem with nonlocal conditions, which are multipoint perturbations of mixed boundary conditions, started in the first part. In particular, we construct a generalized transform operator, which maps the solutions of the self-adjoint boundary-value problem with mixed boundary conditions to the solutions of the investigated multipoint problem. The system of root functions $V(L)$ of operator $L$ for multipoint problem is constructed. The conditions under which the system $V(L)$ is complete and minimal, and the conditions under which it is the Riesz basis are determined. In the case of an elliptic equation the conditions of existence and uniqueness of the solution for the problem are established.

scholarly journals Repeated alternating loading of a elastoplastic three-layer plate in a temperature field

2021 ◽  
Vol 21 (1) ◽  
pp. 60-75
Author(s):  
Eduard I. Starovoitov ◽  
◽  
Denis V. Leonenko ◽  
Keyword(s):  
Temperature Field ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Plastic Region ◽  
Local Load ◽  
Natural State ◽  
Elastic Plastic

Axisymmetric deformation of a three-layer circular plate under repeated alternating loading from the plastic region by a local load is considered. To describe kinematics of asymmetrical on the thickness of the plate pack is adopted the hypothesis of a broken line. In a thin elastic-plastic load-bearing layers are used the hypothesis of Kirchhoff. A non-linearly elastic relatively thick filler is incompressible in thickness. It is taken to be a hypothesis of Tymoshenko regarding the straightness and the incompressibility of the deformed normals with linear approximation of the displacements through the thickness layer. The work of the filler in the tangential direction is taken into account. The physical relations of stress-strain relations correspond to the theory of small elastic-plastic deformations. The effect of heat flow is taken into account. The temperature field in the plate was calculated by the formula obtained by averaging the thermophysical parameters over the thickness of the package. The system of differential equations of equilibrium under loading of the plate from the natural state is obtained by the Lagrange variational method. Boundary conditions on the plate contour are formulated. The solution of the corresponding boundary value problem is reduced to finding the three desired functions: deflection, shear and radial displacement of the shear surface of the filler. A non-uniform system of ordinary nonlinear differential equations is written for these functions. Its analytical iterative solution is obtained in Bessel functions by the method of elastic solutions of Ilyushin. In case of repeated alternating loading of the plate, the solution of the boundary value problem is constructed using the theory of variable loading of Moskvitin. In this case, the hypothesis of similarity of plasticity functions at each loading step is used. Their analytical form is taken independent of the point of unloading. However, the material constants included in the approximation formulas will be different. The cyclic hardening of the material of the bearing layers is taken into account. The parametric analysis of the obtained solutions under different boundary conditions in the case of a local load distributed in a circle is carried out. The influence of temperature and nonlinearity of layer materials on the displacements in the plate is numerically investigated.

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