Order Self-adjoint Differential Equations Research Articles

Oscillation and nonoscillation of forced second order dynamic equations

Oscillation and nonoscillation properties of second order Sturm?Liouville dynamic equations on time scales ? for example, second order self-adjoint differential equations and second order Sturm?Liouville difference equations ? have attracted much interest. Here we consider a given homogeneous equation and a corresponding equation with forcing term. We give new conditions implying that the latter equation inherits the oscillatory behavior of the homogeneous equation. We also give new conditions that introduce oscillation of the inhomogeneous equation while the homogeneous equation is nonoscillatory. Finally, we explain a gap in a result given in the literature for the continuous and the discrete case. A more useful result is presented, improving the theory even for the corresponding continuous and discrete cases. Examples illustrating the theoretical results are supplied.

Oscillation and Nonoscillation of Higher Order Self-Adjoint Differential Equations

Oscillation and nonoscillation criteria for the higher order self-adjoint differential equation (-1)n(talphay(n))(n)+q(t)y = 0 (*) are established. In these criteria, equation (*) is viewed as a perturbation of the conditionally oscillatory equation (-1)n(talphay(n))(n) - µ,α―t2n-αy = 0, where μ n, α is the critical constant in conditional oscillation. Some open problems in the theory of conditionally oscillatory, even order, self-adjoint equations are also discussed.

Positive functionals and oscillation criteria for second order differential systems

Let ℋ be a Hilbert space, let ℬ = (ℋ, ℋ) be the B*-algebra of bounded linear operators from ℋ to ℋ with the uniform operator topology, and let ℒ be the subset of ℬ consisting of the self-adjoint operators. This article is concerned with the second order self-adjoint differential equation

On the oscillatory and asymptotic behavior of solutions of fifth order selfadjoint differential equations

On the oscillatory and asymptotic behavior of solutions of fifth order selfadjoint differential equations

Related oscillation criteria for higher order self-adjoint differential equations

Related oscillation criteria for higher order self-adjoint differential equations

Quadratic Functionals of nth Order

It is well known that disconjugacy theorems for self-adjoint differential equations are very closely related to the study of the positivity of certain associated quadratic functionals. In this paper, this relationship is closely examined for quadratic functionals of nth order associated with selfadjoint differential equations of order 2n. The motivation of this work comes from a recent paper of W. Leighton [8] on quadratic functionals of second order. It is shown that the method used by W. Leighton extends in a straightforward manner to the study of quadratic functionals of nth order, provided one makes use of an identity due to G. Cimmino [1]. Several authors have considered related problems. In particular we wish to mention the recent papers of G. Ladas [6], W. Simons [13], D. B. Hinton [5]. The book by C. A. Swanson [14] also contains many other related results. As a consequence of our results we can obtain several known and some new criteria for oscillation of 2nth order self-adjoint differential equations.

Concerning solutions of third order self-adjoint differential equations

This work is concerned with the behavior of solutions of third order self-adjoint differential equations. It is well known that products of solutions of a second order self-adjoint equation are solutions of the third order equation. A converse of this fact is established. Existence of oscillatory solutions and asymptotic behavior of nonoscillatory solutions is established. Separation and comparison theorems are obtained for such equations.

Spectral theory for a singular linear system of Hamiltonian differential equations

In 1910 Weyl [9] inaugurated the modern theory of singular self-adjoint differential operators, considering, in particular, second-order differential operators with real coefficients. Since then, his results have been generalized by various authors to the case of arbitrary even order self-adjoint differential equations with real coefficients. Also, in 1954 Coddington [2] treated the case of an arbitrary n-th order self-adjoint differential equation with complex coefficients, He obtained the Parseval equality and spectral expansion associated with the singular case directly, through the consideration of such a problem as a limiting case of corresponding self-adjoint two-point boundary-value problems on compact subintervals of the reals. Using the same general concept, Coddington and Levinson [3] derived a Green’s function, which in turn enabled them to obtain the spectral matrix for a singular problem involving a linear homogeneous n-th order differential operator, and then proceeded to derive the Parseval equality and spectral expansions. Recently Brauer [I] treated similar problems which involve a definitely self-adjoint vector differential operator, of the sort that has been treated by Reid [6] and others. He obtained the Green’s matrix and spectral matrix through the use of the spectral theorem and the theory of direct integrals. The main purpose of the present paper is to employ the general method

Oscillation Properties of the 2-2 Disconjugate Fourth Order Selfadjoint Differential Equation

Oscillation Properties of the 2-2 Disconjugate Fourth Order Selfadjoint Differential Equation

Oscillation properties of the 2-2 disconjugate fourth order selfadjoint differential equation

This paper contains a proof that either all, or none, of the nontrivial solutions of the fourth order linear selfadjoint differential equation have an infinite number of zeros on a half line, provided that no nontrivial solution has more than one double zero on that half line.

A Comparison Theorem for Conjugate Points of General Selfadjoint Differential Equations

Comparison theorems for conjugate points of fourth order selfadjoint differential equations have been established by J. Barrett. In the present paper a more direct method of proof is used to generalize Barrett’s theorem to general selfadjoint equations of even order while replacing certain pointwise conditions on the coefficients by weaker integral conditions. Further generalizations are indicated to nonlinear equations, to differential inequalities, and to certain nonselfadjoint equations.

An eigenvalue comparison theorem for second order self-adjoint differential equations

This paper is concerned with a comparison of the eigenvalues for pairs of self-adjoint differential systems which arise from a single ordinary second-order differential equation. The sistems differ in the boundary conditions which are imposed, and it is these boundary conditions which will draw most of the attention. The principal results obtained deal with predicting alternation of the eigenvalues for two such systems from the boundary conditions alone, without special consideration of the differential equation.

Behavior of Solutions of Second Order Self-Adjoint Differential Equations

Behavior of Solutions of Second Order Self-Adjoint Differential Equations

Behavior of solutions of second order self-adjoint differential equations

Behavior of solutions of second order self-adjoint differential equations