Opial’s inequality and oscillation of 2nd order equations

doi:10.1090/S0002-9939-97-03907-5

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Opial’s inequality and oscillation of 2nd order equations

Journal Article•DOI•

Opial’s inequality and oscillation of 2nd order equations

R. Brown¹, D. Hinton¹•Institutions (1)

University of Alabama¹

01 Jan 1997-Vol. 125, Iss: 4, pp 1123-1129

TL;DR: For a second-order differential equation, this article derived Opial's inequality lower bounds for the spacing between two zeros of a solution or between a zero of the solution and its derivative, in terms of antiderivatives of the potential.

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Abstract: For a second-order differential equation, we obtain from Opial’s inequality lower bounds for the spacing between two zeros of a solution or between a zero of a solution and a zero of its derivative. These bounds are expressed in terms of antiderivatives of the potential, and in particular we derive some new Liapunov type inequalities from them.

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Journal Article•DOI•

Lyapunov-type integral inequalities for certain higher order differential equations

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Devrim Çakmak¹•Institutions (1)

Gazi University¹

01 Mar 2010-Applied Mathematics and Computation

TL;DR: First a short survey of the most basic results on Lyapunov inequality is given, and next this-type integral inequalities for certain higher order differential equations are obtained.

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105 citations

Cites methods from "Opial’s inequality and oscillation ..."

...For authors who contributed the Lyapunov-type inequalities, we also refer to Brown and Hinton [1], Çakmak and Tiryaki [3], Došlý and Řehák [4], Kwong [6], Lee et al....
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...For authors who contributed the Lyapunov-type inequalities, we also refer to Brown and Hinton [1], Çakmak and Tiryaki [3], Došlý and Řehák [4], Kwong [6], Lee et al. [7], Pachpatte [9–11], Panigrahi [12], Parhi and Panigrahi [14], Tiryaki, Ünal and Çakmak [15], Ünal, Çakmak and Tiryaki [16], Ünal and Çakmak [17], and Yang [19]....
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Journal Article•DOI•

A Lyapunov-type stability criterion using ^{} norms

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Meirong Zhang¹, Weigu Li²•Institutions (2)

Tsinghua University¹, Peking University²

25 Mar 2002

TL;DR: In this article, a relation between the (anti-)periodic and Dirichlet eigenvalues is established to establish lower bounds for the first anti-periodic eigenvalue.

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Abstract: Let q(t) be a T-periodic potential such that ∫ T 0 q(t) dt < 0. The classical Lyapunov criterion to stability of Hill's equation -x + q(t)x = 0 is ∥q-∥ 1 = ∫ T 0 |q- (t)|dt < 4/T, where q- is the negative part of q. In this paper, we will use a relation between the (anti-)periodic and the Dirichlet eigenvalues to establish some lower bounds for the first anti-periodic eigenvalue. As a result, we will find the best Lyapunov-type stability criterion using L α norms of q-, 1 < α < ∞. The numerical simulation to Mathieu's equation shows that the new criterion approximates the first stability region very well.

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83 citations

Cites methods from "Opial’s inequality and oscillation ..."

...This is based essentially on the L 2 norm of the primitive W (t) = ∫ t w(t)dt and is proved using Opial’s inequality in [1, 2]....
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Book Chapter•DOI•

Lyapunov Inequalities and their Applications

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R. C. Brown¹, Don B. Hinton²•Institutions (2)

University of Alabama¹, University of Tennessee²

01 Jan 2000

TL;DR: For nearly 50 years Lyapunov inequalities have been an important tool in the study of differential equations as discussed by the authors, and they have been used extensively in the analysis of Hamiltonian systems.

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Abstract: For nearly 50 years Lyapunov inequalities have been an important tool in the study of differential equations. In this survey, building on an excellent 1991 historical survey by Cheng, we sketch some new developments in the theory of Lyapunov inequalities and present some recent disconjugacy results relating to second and higher order differential equations as well as Hamiltonian systems.

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67 citations

Recent Developments of Lyapunov-type Inequalities

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Aydin Tiryaki

01 Jan 2010

TL;DR: A survey of Lyapunov-type inequalities can be found in this article, where some recent developments related to this type of inequalities are sketched and discussed. But this survey is limited to a single inequality.

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Abstract: In this work, first we give a survey of the most basic results on Lyapunov-type inequality, and next we sketch some recent developments related to this type of inequalities.

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63 citations

Cites result from "Opial’s inequality and oscillation ..."

...However, stronger results were obtained in Brown and Hinton [3] and Kwong [17]....
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Journal Article•DOI•

On inequalities of Lyapunov type

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Xiaojing Yang¹•Institutions (1)

Tsinghua University¹

01 Jan 2003-Applied Mathematics and Computation

TL;DR: This work generalizes the classical Lyapunov inequality for second-order linear differential equations to nonlinear differential equations of second order and then to higher order Linear differential equations.

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60 citations

Cites background or methods from "Opial’s inequality and oscillation ..."

...Recently Brown and Hinton [5] obtained the following result....
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...0þ, (1.10) yieldsZ b a jpðtÞjdt > 4=p: ð1:12Þ Inspired by the work of Brown and Hinton as well as Ha’s, we consider the following p-Laplacian type equation: ðpðtÞuaðx0ðtÞÞÞ 0 þ qðtÞuaðxðtÞÞ ¼ 0 ð1:13Þ and the following n-order linear differential equation: xðnÞðtÞ þ p1ðtÞxðn 1ÞðtÞ þ þ pn 1ðtÞx0ðtÞ þ pnðtÞ ¼ 0: ð1:14Þ If xðtÞ is a solution of (1.13) or (1.14) satisfying (1.7), we obtain inequalities similar to (1.3), (1.5) and (1.8)....
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...Our methods are different to the methods used in [3,5]....
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References

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Book•

Inequalities Involving Functions and Their Integrals and Derivatives

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Dragoslav S. Mitrinović, Josip Pečarić, A. M. Fink

31 Jul 1991

TL;DR: Gronwall inequalities in higher dimension as mentioned in this paper have been used to prove integral inequalities in other spaces: discrete, functional and abstract, and they have been shown to hold for functions with bounded derivatives.

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Abstract: I. Landau-Kolmogorov and related inequalities.- II. An inequality ascribed to Wirtinger and related results.- III. Opial's inequality.- IV. Hardy's, Carleman's and related inequalities.- V. Hilbert's and related inequalities.- VI. Inequalities of Lyapunov and of De la Vallee Poussin.- VII. Zmorovi?'s and related inequalities.- VIII. Carlson's and related inequalities.- IX. Inequalities involving kernels.- X. Convolution, rearrangement and related inequalities.- XI. Inequalities of Caplygin type.- XII. Inequalities of Gronwall type of a single variable.- XIII. Gronwall inequalities in higher dimension.- XIV. Gronwall inequalities on other spaces: discrete, functional and abstract.- XV. Integral inequalities involving functions with bounded derivatives.- XVI. Inequalities of Bernstein-Mordell type.- XVII. Methods of proofs for integral inequalities.- XVIII. Particular inequalities.- Name Index.

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905 citations

Journal Article•DOI•

Best constants in a class of integral inequalities

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David W. Boyd

01 Aug 1969-Pacific Journal of Mathematics

TL;DR: In this paper, it was shown that for a certain class of m and w, equality can be attained in the inequality for a given class of M and w by applying variational techniques to a nonlinear eigenvalue problem.

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Abstract: \\y\\*\\y\\*w(x)dχ£Ki\\ \\ y \\m{x)dx\\ where yia) = y(ά) = = y-\\a) = 0 and y~ is absolutely continuous. It is first shown that for a certain class of m and w, equality can be attained in the inequality. Applying variational techniques reduces the determination of the best constant to a nonlinear eigenvalue problem for an integral operator. If m and w are sufficiently smooth this reduces further to a boundary value problem for a differential equation. The method is illustrated by determining the best constants in case (a, b) is a finite interval, mix) = wix) = 1, and n = 1.

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63 citations

"Opial’s inequality and oscillation ..." refers methods in this paper

...License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use Results similar to Theorem 3.1 may be obtained by application of Boyd’s theorem: Theorem 3.2....
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...D. W. Boyd, Best constants in a class of integral inequalities, Pacific J. Math....
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...3) We also use another Opial inequality which is a special case of a more general result due to Boyd [2]....
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...(2.3) We also use another Opial inequality which is a special case of a more general result due to Boyd [2]....
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Journal Article•DOI•

Extensions of Opial’s inequality

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Paul Richard Beesack, Krishna Das

01 Aug 1968-Pacific Journal of Mathematics

47 citations

"Opial’s inequality and oscillation ..." refers background in this paper

...Two Opial type inequalities A special case of an inequality obtained by Beesack and Das [1] is the following (see also [4, p....
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...P. R. Beesack and K.M. Das, Extensions of Opial’s inequality, Pacific J. Math....
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...Two Opial type inequalities A special case of an inequality obtained by Beesack and Das [1] is the following (see also [4, p. 119])....
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Journal Article•DOI•

On the oscillation of differential equations with an oscillatory coefficient

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B. J. Harris, Qingkai Kong

01 May 1995-Transactions of the American Mathematical Society

TL;DR: In this paper, lower bounds for the distance between consecutive zeros of solutions of √ y + q(t)y = 0 when q takes both positive and negative values were derived.

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Abstract: We derive lower bounds for the distance between consecutive zeros of solutions of (*) y" + q(t)y = 0 when q takes both positive and negative values. We apply our results to the limit point/limit circle classifications of (*).

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29 citations

"Opial’s inequality and oscillation ..." refers methods in this paper

...By using the maximum of |Q| on [a, b] in (3.2) and (3.3), integrating, and then taking a square root we see that 1 (b− a) max a≤x≤b ∣∣∣∣∣ ∫ b x q(t) dt ∣∣∣∣∣(3.5) when y(a) = y′(b) = 0, and 1 (b− a) max a≤x≤b ∣∣∣∣∫ x a q(t) dt ∣∣∣∣(3.6) when y′(a) = y(b) = 0, which are the inequalities obtained by Harris and Kong....
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...Our motivation for this work comes from a recent paper of Harris and Kong [3]....
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...B. J. Harris and Q. Kong, On the oscillation of differential equations with an oscillatory coefficient, Trans....
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