scholarly journals The nonlocal problem with multi- point perturbations of the boundary conditions of the Sturm-type for an ordinary differential equation with involution of even order

2020 ◽  
Vol 54 (1) ◽  
pp. 64-78 ◽  
Author(s):  
Ya.O. Baranetskij ◽  
P.I. Kalenyuk ◽  
M. I. Kopach ◽  
A.V. Solomko
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Spectral Properties ◽  
Nonlocal Problem ◽  
Sufficient Conditions ◽  
Adjoint Problem ◽  
Multipoint Problem ◽  
Even Order

The spectral properties of the nonself-adjoint problem with multipoint perturbations of the Dirichlet conditions for differential operator of order $2n$ with involution are investigated. The system of eigenfunctions of a multipoint problem is constructed. Sufficient conditions have been established, under which this system is complete and, under some additional assumptions, forms the Riesz basis. The research is structured as follows. In section 2 we investigate the properties of the Sturm-type conditions and nonlocal problem with self-adjoint boundary conditions for the equation $$(-1)^ny^{(2n)}(x)+ a_{0}y^{(2n-1)}(x)+ a_{1}y^{(2n-1)}(1-x)=f(x),\,x\in (0,1).$$ In section 3 we study the spectral properties for nonlocal problem with nonself-adjoint boundary conditions for this equation. In sections 4 we construct a commutative group of transformation operators. Using spectral properties of multipoint problem and conditions for completeness the basis properties of the systems of eigenfunctions are established in section 5. In section 6 some analogous results are obtained for multipoint problems generated by differential equations with an involution and are proved the main theorems.

A nonlocal problem for a differential operator of even order with involution

2020 ◽  
Vol 26 (2) ◽  
pp. 297-307
Author(s):  
Petro I. Kalenyuk ◽  
Yaroslav O. Baranetskij ◽  
Lubov I. Kolyasa
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Riesz Basis ◽  
Existence And Uniqueness ◽  
Spectral Properties ◽  
Nonlocal Problem ◽  
Even Order

AbstractWe study a nonlocal problem for ordinary differential equations of {2n}-order with involution. Spectral properties of the operator of this problem are analyzed and conditions for the existence and uniqueness of its solution are established. It is also proved that the system of eigenfunctions of the analyzed problem forms a Riesz basis.

Construction of Green’s functional for a third order ordinary differential equation with general nonlocal conditions and variable principal coefficient

2020 ◽  
Vol 27 (4) ◽  
pp. 593-603 ◽  
Author(s):  
Kemal Özen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlocal Problem ◽  
Nonlocal Conditions ◽  
Adjoint Problem ◽  
Linear Ordinary Differential Equation ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
Order Linear ◽  
Theoretical Results

AbstractIn this work, the solvability of a generally nonlocal problem is investigated for a third order linear ordinary differential equation with variable principal coefficient. A novel adjoint problem and Green’s functional are constructed for a completely nonhomogeneous problem. Several illustrative applications for the theoretical results are provided.

On the existence and uniqueness of a positive solution to a boundary value problem for a nonlinear ordinary differential equation of even order

2021 ◽  
pp. 341-347
Author(s):  
Gusen E. Abduragimov ◽  
Patimat E. Abduragimova ◽  
Madina M. Kuramagomedova
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Ordinary Differential Equation ◽  
Even Order

In the article, we consider a boundary value problem for a nonlinear ordinary differential equation of even order which, obviously, has a trivial solution. Sufficient conditions for the existence and uniqueness of a positive solution to this problem are obtained. With the help of linear transformations of T. Y. Na [T. Y. Na, Computational Methods in Engineering Boundary Value Problems, Acad. Press, NY, 1979, ch. 7], the boundary value problem is reduced to the Cauchy problem, the initial conditions of which make it possible to uniquely determine the transformation parameter. It is shown that the transformations of T. Y. Na uniquely determine the solution of the original problem. In addition, based on the proof of the uniqueness of a positive solution to the boundary value problem, a sufficiently effective non–iterative numerical algorithm for constructing such a solution is obtained. A corresponding example is given.

On Existence of Nonoscillatory Solutions to Quasilinear Differential Equations

2007 ◽  
Vol 14 (2) ◽  
pp. 223-238
Author(s):  
Irina V. Astashova
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Positive Potential ◽  
Nonoscillatory Solutions ◽  
All Solutions ◽  
Even Order

Abstract Sufficient conditions are established for the existence of nonoscillatory solutions to a quasilinear ordinary differential equation of higher order. For the equation with a positive potential, a criterion is established for the existence of nonoscillatory solutions with nonzero limit at infinity. In the case of even order, a criterion is obtained for all solutions at infinity to be oscillatory.

DIRICHLET PROBLEM FOR DIFFERENTIAL EQUATIONS OF EVEN ORDER OPERATOR COEFFICIENTS THAT CONTAIN AN INVOLUTION

2018 ◽  
pp. 26-37
Author(s):  
Y. O. Baranetskij
Keyword(s):  
Differential Equation ◽  
Dirichlet Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Sufficient Conditions ◽  
Riesz Bases ◽  
Order Operator ◽  
Eigenvalues And Eigenfunctions ◽  
Even Order ◽  
Adjoint Operators

We study a problem with Dirichlet conditions for a differential equation of order 2n, whose coefficients are non-self-adjoint operators. It is established that the task operator has two subspaces generated by the involution operator, and two subsystems of the system of eigenfunctions, which are Riesz bases in each of the subspaces. Eigenvalues and eigenfunctions are defined. Sufficient conditions are obtained under which the system of eigenfunctions is the Rees base. The conditions for the existence of unity of the solution of the problem with homogeneous boundary conditions, constructed only as a series on the system of eigenfunctions, are established.

scholarly journals The nonlocal problem for the differential-operator equation of the even order with the involution

2018 ◽  
Vol 9 (2) ◽  
pp. 109-119 ◽  
Author(s):  
Ya.O. Baranetskij ◽  
P.I. Kalenyuk ◽  
L.I. Kolyasa ◽  
M.I. Kopach
Keyword(s):  
Differential Equation ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Operator Equation ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Adjoint Operator ◽  
Nonlocal Problem ◽  
Even Order ◽  
Differential Operator Equation

In this paper, the problem with boundary nonself-adjoint conditions for a differential-operator equations of the order $2n$ with involution is studied. Spectral properties of operator of the problem is investigated. By analogy of separation of variables the nonlocal problem for the differential-operator equation of the even order is reduced to a sequence $ \{L_{k}\}_{k=1}^{\infty}$ of operators of boundary value problems for ordinary differential equations of even order. It is established that each element $L_{k}$, of this sequence, is an isospectral perturbation of the self-adjoint operator $L_{0,k}$ of the boundary value problem for some linear differential equation of order 2n. We construct a commutative group of transformation operators whose elements reflect the system $V(L_{0,k})$ of the eigenfunctions of the operator $L_{0,k}$ in the system $V(L_{k})$ of the eigenfunctions of the operators $L_{k}$. The eigenfunctions of the operator $L$ of the boundary value problem for a differential equation with involution are obtained as the result of the action of some specially constructed operator on eigenfunctions of the sequence of operators $L_{0,k}.$ The conditions under which the system of eigenfunctions of operator $L$ the studied problem is a Riesz basis is established.

scholarly journals The nonlocal problem for the $2n$ differential equations with unbounded operator coefficients and the involution

2018 ◽  
Vol 10 (1) ◽  
pp. 14-30 ◽  
Author(s):  
Ya.O. Baranetskij ◽  
I.I. Demkiv ◽  
I.Ya. Ivasiuk ◽  
M.I. Kopach
Keyword(s):  
Boundary Conditions ◽  
Order Differential Equation ◽  
Nonlocal Problem ◽  
Sufficient Conditions ◽  
Riesz Bases ◽  
Associated Functions ◽  
Uniqueness Of The Solution ◽  
Even Order ◽  
Adjoint Operators ◽  
Differential Operator Equation

We study a problem with periodic boundary conditions for a $2n$-order differential equation whose coefficients are non-self-adjoint operators. It is established that the operator of the problem has two invariant subspaces generated by the involution operator and two subsystems of the system of eigenfunctions which are Riesz bases in each of the subspaces. For a differential-operator equation of even order, we study a problem with non-self-adjoint boundary conditions which are perturbations of periodic conditions. We study cases when the perturbed conditions are Birkhoff regular but not strongly Birkhoff regular or nonregular. We found the eigenvalues and elements of the system $V$ of root functions of the operator which is complete and contains an infinite number of associated functions. Some sufficient conditions for which this system $V$ is a Riesz basis are obtained. Some conditions for the existence and uniqueness of the solution of the problem with homogeneous boundary conditions are obtained.

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

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