The Riesz representation theorem on time scales

Basic results for functional differential and dynamic equations involving impulses

“…Remark 4.3 Several authors have been interested in Stieltjes‐type integrals on time scales (see e.g. 10, 13). For example, the definition of the Riemann‐Stieltjes Δ‐integral \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\int _a^b f(t){\Delta }g(t)$\end{document} of a function \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$f:[a,b]_{\mathbb {T}}\rightarrow \mathbb {R}^n$\end{document} with respect to a function \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$g:[a,b]_{\mathbb {T}}\rightarrow \mathbb {R}$\end{document} can be obtained in a straightforward way by taking the definition of the Riemann Δ‐integral and replacing the usual integral sums by Alternatively, we can start with the definition of the Kurzweil‐Henstock Δ‐integral and modify the integral sums in the same way.…”

Pontryagin maximum principle for state constrained optimal sampled-data control problems on time scales

“…hold, and that a function x : [c, d] T → R n is of bounded variation if and only if x can be written as the difference x − x of two functions x , x : [c, d] T → R n with nondecreasing real components. We refer to Proposition 4 of [49] for details. One can easily deduce from the previous characterization that, if…”

Hopf bifurcation analysis and its preliminary control in a Hasting-Powell food chain model with two different delays

“…Regarding the center manifold C 0 , we have 10) where z andz are local coordinates for center manifold C 0 in the direction of q andq * . If u t is real, we note that W is real.…”