3,200 research outputs found
The Natural Logarithm on Time Scales
We define an appropriate logarithm function on time scales and present its
main properties. This gives answer to a question posed by M. Bohner in [J.
Difference Equ. Appl. {\bf 11} (2005), no. 15, 1305--1306].Comment: 6 page
Higher-Order Calculus of Variations on Time Scales
We prove a version of the Euler-Lagrange equations for certain problems of
the calculus of variations on time scales with higher-order delta derivatives.Comment: Corrected minor typo
A generalization of Ostrowski inequality on time scales for k points
In this paper we first generalize the Ostrowski inequality on time scales for
k points and then unify corresponding continuous and discrete versions. We also
point out some particular Ostrowski type inequalities on time scales as special
cases.Comment: 10 page
Halanay type inequalities on time scales with applications
This paper aims to introduce Halanay type inequalities on time scales. By
means of these inequalities we derive new global stability conditions for
nonlinear dynamic equations on time scales. Giving several examples we show
that beside generalization and extension to q-difference case, our results also
provide improvements for the existing theory regarding differential and
difference inequalites, which are the most important particular cases of
dynamic inequalities on time scales
Noether's Theorem on Time Scales
We show that for any variational symmetry of the problem of the calculus of
variations on time scales there exists a conserved quantity along the
respective Euler-Lagrange extremals.Comment: Partially presented at the 6th International ISAAC Congress, August
13 to August 18, 2007, Middle East Technical University, Ankara, Turke
Optimality conditions for the calculus of variations with higher-order delta derivatives
We prove the Euler-Lagrange delta-differential equations for problems of the
calculus of variations on arbitrary time scales with delta-integral functionals
depending on higher-order delta derivatives.Comment: Submitted 26/Jul/2009; Revised 04/Aug/2010; Accepted 09/Aug/2010; for
publication in "Applied Mathematics Letters
Symmetric Differentiation on Time Scales
We define a symmetric derivative on an arbitrary nonempty closed subset of
the real numbers and derive some of its properties. It is shown that
real-valued functions defined on time scales that are neither delta nor nabla
differentiable can be symmetric differentiable.Comment: This is a preprint of a paper whose final and definite form will be
published in Applied Mathematics Letters. Submitted 30-Jul-2012; revised
07-Sept-2012; accepted 10-Sept-201
Leitmann's direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales
The fundamental problem of the calculus of variations on time scales concerns
the minimization of a delta-integral over all trajectories satisfying given
boundary conditions. This includes the discrete-time, the quantum, and the
continuous/classical calculus of variations as particular cases. In this note
we follow Leitmann's direct method to give explicit solutions for some concrete
optimal control problems on an arbitrary time scale.Comment: Accepted for publication (9/January/2010) in Applied Mathematics and
Computatio
Noether's symmetry theorem for nabla problems of the calculus of variations
We prove a Noether-type symmetry theorem and a DuBois-Reymond necessary
optimality condition for nabla problems of the calculus of variations on time
scales.Comment: Submitted 20/Oct/2009; Revised 27/Jan/2010; Accepted 28/July/2010;
for publication in Applied Mathematics Letter
- …