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Journal ArticleDOI

New gaps between zeros of fourth-order differential equations via Opial inequalities

30 Aug 2012-Journal of Inequalities and Applications (Springer International Publishing)-Vol. 2012, Iss: 1, pp 182
TL;DR: In this paper, for a fourth-order differential equation, the authors established lower bounds for the distance between zeros of a nontrivial solution and their derivatives, and for the boundary value problems in the theory of bending of beams.
Abstract: In this paper, for a fourth-order differential equation, we will establish some lower bounds for the distance between zeros of a nontrivial solution and also lower bounds for the distance between zeros of a solution and/or its derivatives. We also give some new results related to some boundary value problems in the theory of bending of beams. The main results will be proved by making use of some generalizations of Opial and Wirtinger-type inequalities. Some examples are considered to illustrate the main results. MSC: 34K11; 34C10

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Citations
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TL;DR: In this article, the authors proved diamond-alpha dynamic inequalities of Opial type with one and two weight functions on time scales, which contain as special cases improvements of results given in the literature, and these improvements are new even in the important discrete case.
Abstract: In this paper, we prove some new diamond-alpha dynamic inequalities of Opial type with one and with two weight functions on time scales. These results contain as special cases improvements of results given in the literature, and these improvements are new even in the important discrete case. Mathematics subject classification (2010): 39A10, 39A12, 26D15.

15 citations

Journal ArticleDOI
TL;DR: In this paper, the authors present a new general dynamic inequality of Opial type, which is new even in both the continuous and discrete cases and is proved by making use of a recently introduced new technique for Opial dynamic inequalities, the time scales integration by parts formula.
Abstract: We present a new general dynamic inequality of Opial type. This inequality is new even in both the continuous and discrete cases. The inequality is proved by making use of a recently introduced new technique for Opial dynamic inequalities, the time scales integration by parts formula, the time scales chain rule, an d classical as well as time scales versions of Holder's ineq uality.

4 citations

Journal ArticleDOI
TL;DR: In this article, lower bounds on the distance between zeros of a nontrivial solution and their derivatives were established for self-adjoint fourth-order differential equations, by making use of some generalizations of Hardy, Opial and Wirtinger type inequalities.
Abstract: In this paper, for self-adjoint fourth order differential equations, we establish some lower bounds on the distance between zeros of a nontrivial solution and also lower bounds on the distance between zeros of a solution and/or its derivatives. We also give new results related to boundary value problems which arise in the bending of rods. The main results will be proved by making use of some generalizations of Hardy, Opial and Wirtinger type inequalities.
References
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Journal ArticleDOI
TL;DR: In this article, an inequality of Lyapunov type was derived to find conditions to ensure that the oscillatory solutions of equation (1) tend to zero as t → ∞.
Abstract: The purpose of this paper is to consider the general nonlinear n th order differential-difference equation and derive an inequality of Lyapunov type. Later we use this inequality to find conditions to ensure that the oscillatory solutions of equation (1) tend to zero as t → ∞. The conditions that ensure that the oscillatory solutions of equation (1) tend to zero, also cause all solutions of equation to be non-oscillatory.

15 citations

Journal ArticleDOI
01 May 1992
TL;DR: In this paper, the authors prove inequalities of the type ∫ 0 h |f (i) (x)f (j) (ax)|dx≤C(n,i,j,p)h 2n−i−j+1−2/p when f(0)=f'(0) =...=f (n−1) (0)=0.
Abstract: We prove inequalities of the type ∫ 0 h |f (i) (x)f (j) (x)|dx≤C(n,i,j,p)h 2n−i−j+1−2/p (∫ h |f (n) (x)| p dx) 2p when f(0)=f'(0)=...=f (n−1) (0)=0. We assume that f (n−1) is absolutely continuous and f (n) ∈L p (0,h), with p≥1, n≥2, and 0≤i≤j≤n−1

13 citations

Journal Article
TL;DR: In this article, distribution-valued solutions to a parabolic problem arising from a model of the Black-Scholes equation in option pricing were studied, and a minor generalization of known existence and uniqueness results for solutions in bounded domains was given to give existence of solutions for certain classes of distributions f ∈ D′(Ω).
Abstract: We study distribution-valued solutions to a parabolic problem that arises from a model of the Black-Scholes equation in option pricing. We give a minor generalization of known existence and uniqueness results for solutions in bounded domains Ω ⊂ Rn+1 to give existence of solutions for certain classes of distributions f ∈ D′(Ω). We also study growth conditions for smooth solutions of certain parabolic equations on Rn × (0, T ) that have initial values in the space of distributions.

13 citations

Journal ArticleDOI
TL;DR: In this article, the authors established new large spaces between the zeros of the Riemann zeta-function by employing some Wirtinger-type inequalities and showed that consecutive nontrivial zeros often differ by at least 6.1392 times the average spacing.
Abstract: On the hypothesis that the th moments of the Hardy -function are correctly predicted by random matrix theory and the moments of the derivative of are correctly predicted by the derivative of the characteristic polynomials of unitary matrices, we establish new large spaces between the zeros of the Riemann zeta-function by employing some Wirtinger-type inequalities. In particular, it is obtained that which means that consecutive nontrivial zeros often differ by at least 6.1392 times the average spacing.

12 citations

Journal ArticleDOI
TL;DR: For a second order differential equation with a damping term, this article established some new inequalities of Lyapunov type and derived conditions for disfocality and disconjugacy.
Abstract: For a second order differential equation with a damping term, we establish some new inequalities of Lyapunov type. These inequalities give implicit lower bounds on the distance between zeros of a nontrivial solution and also lower bounds for the spacing between zeros of a solution and/or its derivative. We also obtain a lower bound for the first eigenvalue of a boundary value problem. The main results are proved by applying the Holder inequality and some generalizations of Opial and Wirtinger type inequalities. The results yield conditions for disfocality and disconjugacy. An example is considered to illustrate the main results.

12 citations