# 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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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

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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

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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.

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TL;DR: In this paper, the authors present an algebraic extension of LINEAR TRANSFORMATIONS and QUADRATIC FORMS, and apply it to EIGEN-VARIATIONS.

Abstract: Partial table of contents: THE ALGEBRA OF LINEAR TRANSFORMATIONS AND QUADRATIC FORMS. Transformation to Principal Axes of Quadratic and Hermitian Forms. Minimum-Maximum Property of Eigenvalues. SERIES EXPANSION OF ARBITRARY FUNCTIONS. Orthogonal Systems of Functions. Measure of Independence and Dimension Number. Fourier Series. Legendre Polynomials. LINEAR INTEGRAL EQUATIONS. The Expansion Theorem and Its Applications. Neumann Series and the Reciprocal Kernel. The Fredholm Formulas. THE CALCULUS OF VARIATIONS. Direct Solutions. The Euler Equations. VIBRATION AND EIGENVALUE PROBLEMS. Systems of a Finite Number of Degrees of Freedom. The Vibrating String. The Vibrating Membrane. Green's Function (Influence Function) and Reduction of Differential Equations to Integral Equations. APPLICATION OF THE CALCULUS OF VARIATIONS TO EIGENVALUE PROBLEMS. Completeness and Expansion Theorems. Nodes of Eigenfunctions. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS. Bessel Functions. Asymptotic Expansions. Additional Bibliography. Index.

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TL;DR: Zill's writing style has been highly praised by reviewers as easy-to-read, understandable and helpful to readers as discussed by the authors, and has been described as "easy to read, understandable, and helpful".

Abstract: Systems of Differential Equations Fourier Series and Boundary-Value Problems Numerical Analysis Complex Analysis. Zill's writing style has been highly praised by reviewers as easy-to-read, understandable and helpful to readers.

1,232 citations

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TL;DR: In this article, Sabatier implique l'accord avec les conditions générales d'utilisation (http://www.up-tlse.numdam.org/conditions).

Abstract: © Université Paul Sabatier, 1907, tous droits réservés. L’accès aux archives de la revue « Annales de la faculté des sciences de Toulouse » (http://picard.ups-tlse.fr/~annales/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

851 citations

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183 citations