scholarly journals On boundedness of solutions of second order differential equations in the limit circle case

1975 ◽  
Vol 52 (1) ◽  
pp. 242-242 ◽  
Author(s):  
Man Kam Kwong
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
Boundedness Of Solutions ◽  
Limit Circle ◽  
Second Order Differential Equations ◽  
Limit Circle Case ◽  
Circle Case

scholarly journals Second-Order Differential Operators in the Limit Circle Case

2021 ◽  
Author(s):  
Dmitri R. Yafaev ◽  
◽  
◽  
Keyword(s):  
Differential Equation ◽  
Differential Operators ◽  
Second Order ◽  
Spectral Measures ◽  
Limit Circle ◽  
Jacobi Operators ◽  
Limit Circle Case ◽  
Circle Case ◽  
Stieltjes Transforms

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.

scholarly journals Boundedness of Solutions for Second Order Differential Equations

2020 ◽  
Vol 12 (4) ◽  
pp. 58
Author(s):  
Daniel C. Biles
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Boundedness Of Solutions ◽  
Second Order Differential Equations ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential

We present new theorems which specify sufficient conditions for the boundedness of all solutions for second order non-linear differential equations. Unboundedness of solutions is also considered.

20.—On the Limit-point and Limit-circle Theory of Second-order Differential Equations

1975 ◽  
Vol 72 (3) ◽  
pp. 245-256
Author(s):  
K. S. Ong
Keyword(s):  
Differential Equations ◽  
Weight Function ◽  
Limit Point ◽  
Circle Method ◽  
Second Order ◽  
Limit Circle ◽  
Second Order Differential Equations

SynopsisIn this paper the Weyl limit-point and limit-circle theory of second-order differential equations is extended to the case that the weight function is allowed to take on both positive and negative values—the polar case. This extension is achieved using Weyl's limit circle method.

scholarly journals Boundedness of Certain System of Second Order Differential Equations

2021 ◽  
Vol 45 (5) ◽  
pp. 787-796
Author(s):  
M. O. OMEIKE ◽  
◽  
A. A. ADEYANJU ◽  
D. O. ADAMS ◽  
A. L. OLUTIMO
Keyword(s):  
Lyapunov Function ◽  
Differential Equations ◽  
Second Order ◽  
System Of Equations ◽  
Boundedness Of Solutions ◽  
Ultimate Boundedness ◽  
Basic Technique ◽  
Second Order Differential Equations

This work is concerned with the ultimate boundedness of solutions of the system of vector differential equations X˙ = H (Y ), Y˙ = − F (X, Y )Y − G (X ) + P (t,X, Y ), where t ∈ ℝ+, X = X(t), Y = Y (t) ∈ ℝn, F : ℝn × ℝn → ℝn×n, G,H : ℝn → ℝn and P : ℝ+ × ℝn × ℝn → ℝn. By using a Lyapunov function as a basic technique, we prove that the solutions of the system of equations are ultimately bounded. In addition, result obtained includes and improves some related results in literature.

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