Oscillation criteria for fourth-order linear differential equations.

Howard, H. C.

doi:10.1090/s0002-9947-1960-0117379-x

Cited by 22 publications

(18 citation statements)

References 8 publications

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“…are satisfied nontrivially by a solution y{x) of (1). This is a generalization of the type of condition first used by Barrett [2,4] and Howard [8] as an intermediate condition to (10). …”

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confidence: 98%

“…Recently the special fourth-order case (n -2) has been investigated extensively by W. Leighton and Z. Nehari [10], by H. M. and R. L. Sternberg [13], by H. C. Howard [8], and by J. H. Barrett [2,3,4]. In the present paper some of the methods of Barrett [2,4] are extended to the general case; and, in so doing, some of the arguments used for n = 2 are simplified.…”

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confidence: 99%

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The behavior of solutions of ordinary, self-adjoint differential equations of arbitrary even order

Hunt¹

1962

Pacific J. Math.

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“…are satisfied nontrivially by a solution y{x) of (1). This is a generalization of the type of condition first used by Barrett [2,4] and Howard [8] as an intermediate condition to (10). …”

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confidence: 98%

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confidence: 99%

The behavior of solutions of ordinary, self-adjoint differential equations of arbitrary even order

Hunt¹

1962

Pacific J. Math.

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“…If it is desired to convert results concerning the presence or absence of focal points into statements about conjugate points-and thus, ultimately, about the zeros of solutions of the equations under consideration-it is necessary to have some information concerning the relations between conjugate and focal points. For n = 2,3,4, and the interval [0, °°), the situation is very simple: The equation is disconjugate if and only if it is disfocal [3], [5]. This statement may well be true for « > 4 but, in the general case, all we shall show here is that disfocality implies disconjugacy.…”

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confidence: 72%

Nonlinear Techniques for Linear Oscillation Problems

Nehari¹

1975

Transactions of the American Mathematical Society

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ABSTRACT. It is shown that for differential equations of the form .y'"' + py = 0 there exist associated sets of systems of nonlinear equations which play a role similar to that of the ordinary Riccati equation in the case n = 2. In particular, the existence of continuous solutions of the nonlinear system is equivalent to the absence of certain types of oscillatory solutions of the linear equation.If p is of constant sign, the coefficients of the "Riccati systems" are all nonnegative, and the resulting positivity and monotonicity properties make it possible to obtain explicit oscillation criteria for the original equation.1. One of the classical techniques employed in the discussion of oscillation problems for linear second-order differential equations consists in replacing the study of the original equation by that of the equivalent Riccati equation. While the linearity of the original problem is sacrificed in this process, the formal structure of the Riccati equation permits-indeed invites-analytic manipulations which were not possible, or at least not apparent, in the original formulation. In fact, as pointed out in E. Hille's fundamental paper on the subject [4], this seeming detour via the Riccati equation may in certain respects be regarded as the "natural" approach to the linear oscillation problem.In a recent paper [14] it was shown that a similar approach is available for differential equations of the form x' = ^4x, where x is an «-dimensional vector function and A a continuous n x n matrix. There exists a "Riccati system", i.e., a system of nonlinear differential equations which is equivalent to the original equation if and only if the latter is nonoscillatory in a certain sense, and this connection between the linear and nonlinear systems leads to effective ways of attacking various oscillation problems associated with the original equation. These Riccati systems, incidentally, are not related to the "matrix Riccati equations" [15], [2] which are used in the treatment of second-order vector-matrix equations and which represent a natural extension of the classical formalism.Since linear nth order differential equations are special'cases of the equation x' = Ax, they can be treated by the methods described in [14]. In fact, this treatment can be considerably simplified because of the particular character of the

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“…This work was stimulated by the recent investigations of fourthorder equations by Leighton and Nehari [11], Barrett [3,4,5] and Howard [8]. For example, if η^a) is defined to be the first point b > a for which there exist a solution of ( 2) {r{x)y tf T -p(x)y = 0 (r, p > 0, r e C 2 , p e C)…”

Section: Properties Of Solutions Of IV Th Order Linear Differential Ementioning

confidence: 99%

“…Subsequently Howard [8], using variational methods, discussed the relationships between focal points of (2) and eigenvalues of an eigenvalue problem, with focal point boundary conditions, associated with (2), making various assumptions on the coefficients. Barrett [3,5] extended these results to fourth order equations of more general form.…”

Section: // η^A) Exists Then μ^A) Exists and μ^A) < η^A)mentioning

confidence: 99%

Properties of solutions ofn-th order linear differential equations

Sherman¹

1965

Pacific J. Math.

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Oscillation criteria for fourth-order linear differential equations.

Cited by 22 publications

References 8 publications

The behavior of solutions of ordinary, self-adjoint differential equations of arbitrary even order

The behavior of solutions of ordinary, self-adjoint differential equations of arbitrary even order

Nonlinear Techniques for Linear Oscillation Problems

Properties of solutions ofn-th order linear differential equations

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