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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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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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TL;DR: In this article, it is assumed that solutions of (3) and also of some generalizations of the form (3), both of which are real-valued and continuous on I.
Abstract: (3) (\\y'(t)\\~y'{t))' + q(t)\\y(t)ry(t) = o , P > 1 , where / € / = [0, and I contains the points a,b (a < b), the function q is real-valued and continuous on I . The problems of existence, uniqueness and other properties of solutions to equations of the form (3) are recently studied in [3]-[5]. In what follows it is assumed that solutions of (3), and also of some generalizations of (3), exist on I .

2 citations