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.

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 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.

scholarly journals Boundary Value Problems of Fractional Order Differential Equation with Integral Boundary Conditions and Not Instantaneous Impulses

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Peiluan Li ◽  
Changjin Xu
Keyword(s):  
Boundary Conditions ◽  
Fractional Order ◽  
Fixed Point Theorems ◽  
Order Differential Equation ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Integral Boundary Conditions ◽  
Fractional Order Differential Equation ◽  
Uniqueness Of Solutions ◽  
Integral Boundary

We investigate the existence of mild solutions for fractional order differential equations with integral boundary conditions and not instantaneous impulses. By some fixed-point theorems, we establish sufficient conditions for the existence and uniqueness of solutions. Finally, two interesting examples are given to illustrate our theory results.

scholarly journals Positive Solutions for the Eigenvalue Problem of Semipositone Fractional Order Differential Equation with Multipoint Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Ge Dong
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Positive Solution ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Fractional Order Differential Equation ◽  
Multipoint Boundary ◽  
Multipoint Boundary Conditions

We study the existence of positive solution for the eigenvalue problem of semipositone fractional order differential equation with multipoint boundary conditions by using known Krasnosel'skii's fixed point theorem. Some sufficient conditions that guarantee the existence of at least one positive solution for eigenvalues  λ>0sufficiently small andλ>0sufficiently large are established.

scholarly journals POSITIVE SOLUTIONS OF HIGHER-ORDER BOUNDARY-VALUE PROBLEMS

2005 ◽  
Vol 48 (2) ◽  
pp. 445-464 ◽  
Author(s):  
Lingju Kong ◽  
Qingkai Kong
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Boundary Conditions ◽  
Existence Of Positive Solutions ◽  
Eigenvalue Parameter ◽  
Even Order

AbstractWe consider a class of even-order boundary-value problems with nonlinear boundary conditions and an eigenvalue parameter $\lambda$ in the equations. Sufficient conditions are obtained for the existence and non-existence of positive solutions of the problems for different values of $\lambda$.

Pencils of differential operators containing the eigenvalue parameter in the boundary conditions

2003 ◽  
Vol 133 (4) ◽  
pp. 893-917 ◽  
Author(s):  
Marco Marletta ◽  
Andrei Shkalikov ◽  
Christiane Tretter
Keyword(s):  
Boundary Conditions ◽  
Spectral Properties ◽  
Differential Operators ◽  
Finite Interval ◽  
Linear Operators ◽  
Riesz Bases ◽  
Ordinary Differential Operators ◽  
Associated Functions ◽  
Eigenvalue Parameter

The paper deals with linear pencils N − λP of ordinary differential operators on a finite interval with λ-dependent boundary conditions. Three different problems of this form arising in elasticity and hydrodynamics are considered. So-called linearization pairs (W, T) are constructed for the problems in question. More precisely, functional spaces W densely embedded in L2 and linear operators T acting in W are constructed such that the eigenvalues and the eigen- and associated functions of T coincide with those of the original problems. The spectral properties of the linearized operators T are studied. In particular, it is proved that the eigen- and associated functions of all linearizations (and hence of the corresponding original problems) form Riesz bases in the spaces W and in other spaces which are obtained by interpolation between D(T) and W.

scholarly journals On the lower and upper solution method for higher order functional boundary value problems

2011 ◽  
Vol 5 (1) ◽  
pp. 133-146 ◽  
Author(s):  
John Graef ◽  
Lingju Kong ◽  
Feliz Minhós ◽  
João Fialho
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Solution Method ◽  
Order Differential Equation ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Boundary Functions ◽  
Lower And Upper Solution ◽  
Functional Boundary

The authors consider the nth-order differential equation ?(?(u(n?1)(x)))?= f(x, u(x), ..., u(n?1)(x)), for 2?(0, 1), where ?: R? R is an increasing homeomorphism such that ?(0) = 0, n?2, I:= [0,1], and f : I ?Rn ? R is a L1-Carath?odory function, together with the boundary conditions gi(u, u?, ..., u(n?2), u(i)(1)) = 0, i = 0, ..., n? 3, gn?2 (u, u?, ..., u(n?2), u(n?2)(0), u(n?1)(0)) = 0, gn?1 (u, u?, ..., u(n?2), u(n?2)(1), u(n?1)(1)) = 0, where gi : (C(I))n?1?R ? R, i = 0, ..., n?3, and gn?2, gn?1 : (C(I))n?1?R2 ? R are continuous functions satisfying certain monotonicity assumptions. The main result establishes sufficient conditions for the existence of solutions and some location sets for the solution and its derivatives up to order (n?1). Moreover, it is shown how the monotone properties of the nonlinearity and the boundary functions depend on n and upon the relation between lower and upper solutions and their derivatives.

scholarly journals The existence of multiple positive solutions of p-Laplacian boundary value problems

2007 ◽  
Vol 57 (3) ◽  
Author(s):  
Yuji Liu
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Multiple Positive Solutions ◽  
Nonlocal Boundary Value Problems

AbstractIn this paper, we establish sufficient conditions to guarantee the existence of at least three or 2n − 1 positive solutions of nonlocal boundary value problems consisting of the second-order differential equation with p-Laplacian (1) $$[\phi _p (x'(t))]' + f(t,x(t)) = 0, t \in (0,1),$$ and one of following boundary conditions (2) $$x(0) = \int\limits_0^1 {x(s) dh(s),} \phi _p (x'(1)) = \int\limits_0^1 {\phi _p (x'(s)) dg(s)} ,$$ and (3) $$\phi _p (x'(0)) = \int\limits_0^1 {\phi _p (x'(s)) dh(s),} x(1) = \int\limits_0^1 {x(s) dg(s)} .$$ Examples are presented to illustrate the main results.

scholarly journals Pontryagin’s Maximum Principle for the Optimal Control Problems with Multipoint Boundary Conditions

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
M. J. Mardanov ◽  
Y. A. Sharifov
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Boundary Conditions ◽  
Admissible Control ◽  
Sufficient Conditions ◽  
Pontryagin’S Maximum Principle ◽  
Pontryagin's Maximum Principle ◽  
Multipoint Boundary ◽  
Multipoint Boundary Conditions ◽  
Uniqueness Of The Solution

Optimal control problem with multipoint boundary conditions is considered. Sufficient conditions for the existence and uniqueness of the solution of boundary value problem for every fixed admissible control are obtained. First order increment formula for the functional is derived. Pontryagin’s maximum principle is proved by using the variations of admissible control.

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