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 $2n$ 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.

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.

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.

scholarly journals UNAMBIGUOUS SOLVABILITY OF A PARTICULAR CASE OF SYSTEMS OF INTEGRODIFFERENTIAL EQUATIONS WITH A PULSED KAEV DISTANCE CONTAINING A PARAMETER

2020 ◽  
Vol 72 (4) ◽  
pp. 78-84
Author(s):  
Kh.I. Usmanov ◽  
◽  
A.S. Zhappar ◽  
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Integral Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
System Of Linear Equations ◽  
New Variables

We consider a special case of systems of integro-differential equations with a momentum boundary condition containing a parameter when the derivative of the desired function is contained in the right side of the equation. By integrating in parts, an integro-differential equation with a pulsed boundary condition is reduced to a loaded integrodifferential equation with a pulsed boundary condition. it is given in the system of integral-differential equations with impulse boundary conditions parametrically loaded. Then, by entering new parameters, as well as passing to new variables based on these parameters, the problem is reduced to an equivalent problem. Switching to new variables makes it possible to get the initial conditions for the equation. Based on this, the solution of the problem is reduced to solving a special Cauchy problem and a system of linear equations. Using the fundamental matrix of the main part of the differential equation, an integral equation of the Volterra type is obtained. The method of sequential approximation determines the unique solution of the integral equation. Based on this, we find a solution to the special Cauchy problem and put it in the boundary conditions. On the basis of the obtained system of linear equations, necessary and sufficient conditions for an unambiguous solution of the initial problem are established.

scholarly journals Oscillation and non-oscillation of some neutral differential equations of odd order

1992 ◽  
Vol 15 (3) ◽  
pp. 509-515 ◽  
Author(s):  
B. S. Lalli ◽  
B. G. Zhang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Nonoscillatory Solution ◽  
Neutral Differential Equation ◽  
Neutral Differential Equations ◽  
Neutral Term ◽  
Odd Order ◽  
Existence Criterion ◽  
Oscillation Of Solutions

An existence criterion for nonoscillatory solution for an odd order neutral differential equation is provided. Some sufficient conditions are also given for the oscillation of solutions of somenth order equations with nonlinearity in the neutral term.

A mechanical method for the solution of second-order linear differential equations

1931 ◽  
Vol 27 (4) ◽  
pp. 546-552 ◽  
Author(s):  
E. C. Bullard ◽  
P. B. Moon
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Order Differential Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Mechanical Method ◽  
Order Linear ◽  
Second Order Differential Equation ◽  
Linear Differential

A mechanical method of integrating a second-order differential equation, with any boundary conditions, is described and its applications are discussed.

scholarly journals Periodic solutions for periodic second-order differential equations with variable potentials

2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

Oscillatory Criteria for Nonlinear nth-Order Differential Equations with Quasiderivatives

1996 ◽  
Vol 3 (4) ◽  
pp. 301-314
Author(s):  
Miroslav Bartušek
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Sign Condition ◽  
Proper Solutions

Abstract Sufficient conditions are given for the existence of oscillatory proper solutions of a differential equation with quasiderivatives Lny = f(t, L 0 y, . . . , L n–1 y) under the validity of the sign condition f(t, x 1, . . . , xn )x 1 ≤ 0, f(t, 0, x 2, . . . , xn ) = 0 on .

A note on meromorphic functions with finite order and of bounded l-index

2021 ◽  
Vol 18 (1) ◽  
pp. 1-11
Author(s):  
Andriy Bandura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Finite Order ◽  
General Problem ◽  
Meromorphic Functions ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Riccati Differential Equation

We present a generalization of concept of bounded $l$-index for meromorphic functions of finite order. Using known results for entire functions of bounded $l$-index we obtain similar propositions for meromorphic functions. There are presented analogs of Hayman's theorem and logarithmic criterion for this class. The propositions are widely used to investigate $l$-index boundedness of entire solutions of differential equations. Taking this into account we raise a general problem of generalization of some results from theory of entire functions of bounded $l$-index by meromorphic functions of finite order and their applications to meromorphic solutions of differential equations. There are deduced sufficient conditions providing $l$-index boundedness of meromoprhic solutions of finite order for the Riccati differential equation. Also we proved that the Weierstrass $\wp$-function has bounded $l$-index with $l(z)=|z|.$

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