Systems-conjugate and focal points of fourth order nonselfadjoint differential equations

Cheng, Sui Sun

doi:10.1090/s0002-9947-1976-0473334-3

Cited by 7 publications

(5 citation statements)

References 9 publications

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“…Finally, in Section 5 we establish two criteria for the existence ofη 1 (a). Similar results were given in [3] and [6] for q(x) ≡ 0 and q(x) ≤ 0, respectively.…”

Section: Introductionsupporting

confidence: 85%

“…At the end of this work, he applied his results to equation (1.1) for q(x) ≤ 0 and the additional condition p − q ′′ /2 + q 2 /4r > 0. Note that the systems-focal point studied in [6] do not coincide with that defined above for (1.1) only for q ≡ const.…”

Section: Introductionmentioning

confidence: 82%

“…10.6] extended a part of Barrett's result to the case q(x) ≥ 0 (i.e., ifη 1 exists thenμ 1 (a) exists and a <μ 1 (a) <η 1 (a)). Cheng [6] also studied the existence and the relation betweenμ 1 (a) andη 1 (a) for a system of two second-order differential equations; in particular, he gave a physical interpretation of the numbersη 1 andμ 1 . At the end of this work, he applied his results to equation (1.1) for q(x) ≤ 0 and the additional condition p − q ′′ /2 + q 2 /4r > 0.…”

Section: Introductionmentioning

confidence: 99%

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Oscillation criteria for fourth‐order differential equations with middle term

Amara

2011

Mathematische Nachrichten

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Key words Boundary value problems, fourth-order linear differential equation, oscillation criteria MSC (2010) 34B05Oscillation criteria for self-adjoint fourth-order differential equationswere established for various conditions on the coefficients r(x) > 0, q(x) and p(x). However, most of these results deal with the case when limx→∞ x 1 q(s) ds < +∞. In this note we give a new oscillation criterion in the case when this condition is not fulfilled, in particular when q(x) +∞ (even with exponential growth).

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“…Finally, in Section 5 we establish two criteria for the existence ofη 1 (a). Similar results were given in [3] and [6] for q(x) ≡ 0 and q(x) ≤ 0, respectively.…”

Section: Introductionsupporting

confidence: 85%

Section: Introductionmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

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Oscillation criteria for fourth‐order differential equations with middle term

Amara

2011

Mathematische Nachrichten

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“…We remark that the particular choice of °U* in Corollary 4.2 was rather arbitrary -other combinations of zero and natural boundary conditions could also be used. However, this particular class leads to a natural generalization of the notion of systems zeros introduced by Barrett [3] and studied by Cheng [6] and Schmitt and Smith [22]. Also, for n = 2 they reduce to y(a) = y£(a) = 0 = y(b) = y*(b)> where y* = p 2 y"-As such, the prototype inequality of §1 is a very special case of Theorem 4.1.…”

Section: Natural Boundary Conditionsmentioning

confidence: 99%

Higher order Wirtinger inequalities

Kreith

Swanson

1980

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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SynopsisWirtinger-type inequalities of order n are inequalities between quadratic forms involving derivatives of order k ≦ n of admissible functions in an interval (a, b). Several methods for establishing these inequalities are investigated, leading to improvements of classical results as well as systematic generation of new ones. A Wirtinger inequality for Hamiltonian systems is obtained in which standard regularity hypotheses are weakened and singular intervals are permitted, and this is employed to generalize standard inequalities for linear differential operators of even order. In particular second order inequalities of Beesack's type are developed, in which the admissible functions satisfy only the null boundary conditions at the endpoints of [a, b] and b does not exceed the first systems conjugate point (a) of a. Another approach is presented involving the standard minimization theory of quadratic forms and the theory of “natural boundary conditions”. Finally, inequalities of order n + k are described in terms of (n, n)-disconjugacy of associated 2nth order differential operators.

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“…Therefore C(v o ) cannot go directly from II to I to II. Under the hypotheses b{i) > 0 and c(t) > 0, recent results in[2] can also be specialised to the system (2.1). Let C(0 O ) denote the trajectory of (2.1) satisfying and tan 0 O = ¥~ with -<9 0 <n. THEOREM 2.2(Cheng).…”

mentioning

confidence: 99%