Higher Monotonicity Properties of Certain Sturm-Liouville Functions. III

1970 ◽  
Vol 22 (6) ◽  
pp. 1238-1265 ◽  
Author(s):  
Lee Lorch ◽  
M. E. Muldoon ◽  
Peter Szego
Keyword(s):  
Differential Equation ◽  
Trivial Solution ◽  
Infinite Sequence ◽  
Liouville Function ◽  
The Real ◽  
Linearly Independent ◽  
Monotonicity Properties ◽  
Real Domain ◽  
Sturm Liouville ◽  
Image Position

A Sturm-Liouville function is simply a non-trivial solution of the Sturm-Liouville differential equation(1.1)considered, together with everything else in this study, in the real domain. The associated quantities whose higher monotonicity properties are determined here are defined, for fixed λ > –1, to be(1.2)where y(x) is an arbitrary (non-trivial) solution of (1.1) and x1, x2, … is any finite or infinite sequence of consecutive zeros of any non-trivial solution z(x) of (1.1) which may or may not be linearly independent of y(x). The condition λ > –1 is required to assure convergence of the integral defining Mk, and the function W(x) is taken subject to the same restriction.

Higher Monotonicity Properties of Certain Sturm-Liouville Functions. IV

1972 ◽  
Vol 24 (2) ◽  
pp. 349-368 ◽  
Author(s):  
Lee Lorch ◽  
Martin E. Muldoon ◽  
Peter Szego
Keyword(s):  
Differential Equation ◽  
Linearly Independent ◽  
Monotonicity Properties ◽  
Sturm Liouville ◽  
Definition Of ◽  
Image Position

The Sturm-Liouville functions considered in this instalment are real (as are all other quantities discussed here) non-trivial solutions of the differential equation1.1Higher monotonicity properties, as defined in § 2, are investigated for a number of sequences (finite or infinite) associated with these functions. One such sequence, discussed in detail later, has the kth term1.2where the constant X > — 1 (to assure convergence of each integral), W(x) possesses higher monotonicity properties and, moreover, is such that, again, each integral converges, and X1, X2, … is a sequence (finite or infinite) of consecutive zeros of a solution of (1.1), which may or may not be linearly independent of y(x), in the interval of definition of the functions under consideration.

Bounds for the point spectrum for a Sturm-Liouville equation

1978 ◽  
Vol 80 (1-2) ◽  
pp. 57-66 ◽  
Author(s):  
F. V. Atkinson ◽  
W. N. Everitt
Keyword(s):  
Differential Equation ◽  
Liouville Equation ◽  
Upper Bound ◽  
Trivial Solution ◽  
Point Spectrum ◽  
Real Parameter ◽  
The Real ◽  
Sturm Liouville ◽  
Image Position ◽  
General Conditions

SynopsisThis paper obtains, under certain general conditions on the coefficient q, a best-possible upper bound on the real parameter λ for the differential equationto have a non-trivial solution in the integrable-square space L2 (a, ∞).

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

26.—The Strong Limit-2 Case of Fourth-order Differential Equations

1974 ◽  
Vol 71 (4) ◽  
pp. 297-304
Author(s):  
V. Krishna Kumar
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Order Equation ◽  
Complex Numbers ◽  
Strong Limit ◽  
Fourth Order Equation ◽  
Linearly Independent ◽  
Image Position ◽  
Adjoint Operators ◽  
Bounded Below

SynopsisThe fourth-order equation considered isConditions are given on the coefficients r, p and q which ensure that this differential equation (*) is in the strong limit-2 case at ∞, i.e. is limit-2 at ∞. This implies that (*) has exactly two linearly independent solutions which are in the integrable-square space ℒ2(0, ∞) for all complex numbers λ with im [λ] ≠ 0. Additionally the conditions imply that self-adjoint operators generated by M[·] in ℒ2(0, ∞) are semi-bounded below. The results obtained are applied to the case when the coefficients r, p and q are powers of x ∈ [0, ∞).

The Schwarzian Derivative and Disconjugacy of nth order Linear Differential Equations

1969 ◽  
Vol 21 ◽  
pp. 235-249 ◽  
Author(s):  
Meira Lavie
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Schwarzian Derivative ◽  
Order Linear ◽  
Second Order Differential Equations ◽  
Number Of Zeros ◽  
Linearly Independent ◽  
Basic Relationship ◽  
Linear Differential ◽  
Image Position

In this paper we deal with the number of zeros of a solution of the nth order linear differential equation1.1where the functions pj(z) (j = 0, 1, …, n – 2) are assumed to be regular in a given domain D of the complex plane. The differential equation (1.1) is called disconjugate in D, if no (non-trivial) solution of (1.1) has more than (n – 1) zeros in D. (The zeros are counted by their multiplicity.)The ideas of this paper are related to those of Nehari (7; 9) on second order differential equations. In (7), he pointed out the following basic relationship. The function1.2where y1(z) and y2(z) are two linearly independent solutions of1.3is univalent in D, if and only if no solution of equation(1.3) has more than one zero in D, i.e., if and only if(1.3) is disconjugate in D.

Instability theorems for certain fifth-order differential equations

1978 ◽  
Vol 84 (2) ◽  
pp. 343-350 ◽  
Author(s):  
J. O. C. Ezeilo
Keyword(s):  
Differential Equation ◽  
General Theory ◽  
Trivial Solution ◽  
Real Root ◽  
Order Differential Equation ◽  
Positive Real Part ◽  
Auxiliary Equation ◽  
Positive Real ◽  
Image Position ◽  
Fifth Order

1. Consider the constant-coefficient fifth-order differential equation:It is known from the general theory that the trivial solution of (1·1) is unstable if, and only if, the associated (auxiliary) equation:has at least one root with a positive real part. The existence of such a root naturally depends on (though not always all of) the coefficients a1, a2,…, a5. For example, ifit is clear from a consideration of the fact that the sum of the roots of (1·2) equals ( – a1) that at least one root of (1·2) has a positive real part for arbitrary values of a2,…, a5. A similar consideration, combined with the fact that the product of the roots of (1·2) equals ( – a5) will show that at least one root of (1·2) has a positive real part iffor arbitrary a2, a3 and a4. The condition a1 = 0 here in (1·4) is however superfluous whenfor then X(0) = a5 < 0 and X(R) > 0 if R > 0 is sufficiently large thus showing that there is a positive real root of (1·2) subject to (1·5) and for arbitrary a1, a2, a3 and a4.

On the global attractivity in a generalized delay-logistic differential equation

1986 ◽  
Vol 100 (1) ◽  
pp. 183-192 ◽  
Author(s):  
K. Gopalsamy
Keyword(s):  
Differential Equation ◽  
Trivial Solution ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Countable Number ◽  
Local Asymptotic Stability ◽  
Discrete Delays ◽  
Image Position ◽  
System 1

The purpose of this article is to derive a set of ‘easily verifiable’ sufficient conditions for the local asymptotic stability of the trivial solution ofand then examine the ‘size’ of the domain of attraction of the trivial solution of the nonlinear system (1·1) with a countable number of discrete delays.

On the Non-Existence of Conjugate Points

1970 ◽  
Vol 13 (1) ◽  
pp. 31-37
Author(s):  
G. J. Butler ◽  
L. H. Erbe ◽  
R. M. Mathsen
Keyword(s):  
Differential Equation ◽  
Real Line ◽  
Conjugate Points ◽  
Multiple Zeros ◽  
The Real ◽  
Image Position

In this paper we consider the types of pairs of multiple zeros which a solution to the differential equationcan possess on an interval I of the real line. The results obtained generalize those in [2] and (for n = 3) in [3].I. Let f satisfy the condition1.1for all t ∊ I, u0 ≠ 0, and all u1, … un-1.

X.—The Two Parameter Sturm-Liouville Problem for Ordinary Differential Equations

1971 ◽  
Vol 69 (2) ◽  
pp. 139-148
Author(s):  
B. D. Sleeman
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Liouville Problem ◽  
Sturm Liouville ◽  
Image Position ◽  
Two Parameter ◽  
General Conditions

SynopsisThis paper discusses the existence, under fairly general conditions, of solutions of the two-parameter eigenvalue problem denned by the differential equation,and three point Sturm-Liouville boundary conditions.

Higher monotonicity properties of certain Sturm-Liouville functions. V

1977 ◽  
Vol 77 (1-2) ◽  
pp. 23-37 ◽  
Author(s):  
M. E. Muldoon
Keyword(s):  
Bessel Function ◽  
Positive Zero ◽  
Monotonicity Properties ◽  
Sturm Liouville ◽  
Special Solutions ◽  
Image Position

SynopsisThe principal concern here is with conditions on f or on special solutions of the equationwhich ensure that the higher differences of the zeros and related quantities of solutions of (1) are regular in sign. In particular, by choosing f(x)= 2v−2x1/v−2, it is shown that if ⅓ ≦|v|<½, thenwhere cvk denotes the kth positive zero of a Bessel function of order v and Δµk = Δk+1 − µk. Lorch and Szego [15] conjectured that (2) should hold for the larger range | v | < ½ but the methods used here do not apply to the range | v <| ⅓.

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