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
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
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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Book•
31 Jan 1995
TL;DR: In this paper, generalizations of Opial's Inequality are discussed, including Opial Inequalities Involving Higher Order Derivatives and Discrete Opial inequalities in Several Independent Variables.
Abstract: Preface. 1: Opial's Inequality. 2: Generalizations of Opial's Inequality. 3: Opial Inequalities Involving Higher Order Derivatives. 4: Opial Inequalities in Several Independent Variables. 5: Discrete Opial Inequalities. 6: Applications. Name Index.
151 citations
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.
105 citations
TL;DR: In this article, a Liapunov-type inequality has been derived for a class of third-order differential equations of the form y + p t y = 0, wherepis a real-valued continuous function on [0, ∞] and the nature of the distance between consecutive two zeros or three zeros has been studied with the help of the inequality.
81 citations
TL;DR: The well-known Liapunov-type inequality for a class of third-order differential equations without damping force to 2n+1 order differential equations is generalized.
77 citations
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.
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.
67 citations