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with sin(2t) sin(3t) : 0 ≤ t ≤ π .
We introduce a version of the calculus of variations on time scales, which includes as special cases the classical calculus of variations and the discrete calculus of variations. Necessary conditions for weak local minima are established, among them the Euler condition, the Legendre condition, the strengthened Legendre condition, and the Jacobi condition.… (More)
Abstract. The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger (1988), is an area of mathematics which is currently receiving considerable attention. Although the basic aim of this is to unify the study of differential and difference equations, it also extends these classical cases to “in between”. In this paper we… (More)
Some people may be laughing when looking at you reading in your spare time. Some may be admired of you. And some may want be like you who have reading hobby. What about your own feel? Have you felt right? Reading is a need and a hobby at once. This condition is the on that will make you feel that you must read. If you know are looking for the book enPDFd… (More)
Abstract. We survey half-linear dynamic equations on time scales. These contain the well-known half-linear differential and half-linear difference equations as special cases, but also other kinds of half-linear equations. Special cases of half-linear equations are the well-studied linear equations of second order. We discuss existence and uniqueness of… (More)
For Sturm-Liouville eigenvalue problems on time scales with separated boundary conditions we give an oscillation theorem and establish Rayleigh's principle. Our results not only unifiy the corresponding theories for differential and difference equations, but are also new in the discrete case. © 1999 Published by Elsevier Science Inc. All rights reserved.… (More)
By means of Riccati transformation techniques, we establish some oscillation criteria for a second order nonlinear dynamic equation on time scales in terms of the coefficients. We give examples of dynamic equations to which previously known oscillation criteria are not applicable.
We obtain some oscillation criteria for solutions to the nonlinear dynamic equation x + q(t)x∆σ + p(t)(f ◦ x )= 0, on time scales. In particular, no explicit sign assumptions are made with respect to the coefficients p(t), q(t). We illustrate the results by several examples, including a superlinear Emden–Fowler dynamic equation. 2004 Elsevier Inc. All… (More)
We prove Ostrowski inequalities (regular and weighted cases) on time scales and thus unify and extend corresponding continuous and discrete versions from the literature. We also apply our results to the quantum calculus case.
We consider a linear Hamiltonian Difference System for the so-called singular case so that discrete Sturm]Liouville Equations of higher order are included in our theory. We introduce the concepts of focal points for matrix-valued and generalized zeros for vector-valued solutions of the system and define disconjugacy for linear Hamiltonian Difference… (More)