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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.
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. 1. Introduction. Much recent attention has been given to dynamic equations… (More)
In this paper we consider problems that consist of symplectic difference systems depending on an eigen-value parameter, together with self-adjoint boundary conditions. Such symplectic difference systems contain as important cases linear Hamiltonian difference systems and also Sturm-Liouville difference equations of second and of higher order. The main… (More)
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 solutions of… (More)
We consider linear dynamic systems on time scales, which contain as special cases linear differential systems, difference systems, or other dynamic systems. We give an asymptotic representation for a fundamental solution matrix that reduces the study of systems in the sense of asymptotic behavior to the study of scalar dynamic equations. In order to… (More)
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)
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. Acknowledgements: The authors thank the referees for their careful reading of the manuscript and insightful comments.
" Contemporary Mathematics and Its Applications " is a book series of monographs, textbooks, and edited volumes in all areas of pure and applied mathematics. Authors and/or editors should send their proposals to the Series Editors directly. For general information about the series, T his book is devoted to a rapidly developing branch of the qualitative… (More)