For linear differential systems, the Sacker–Sell spectrum (dichotomy spectrum) and the contractible set are the same. However, we claim that this is not true for the linear difference equations. A counterexample is given. For the convenience of research, we study the relations between the dichotomy spectrum and the contractible set under the framework on time scales. In fact, by a counterexample, we show that the contractible set could be different from dichotomy spectrum on time scales established by Siegmund [J. Comput. Appl. Math., 2002]. Furthermore, we find that there is no bijection between them. In particular, for the linear difference equations, the contractible set is not equal to the dichotomy spectrum. To counter this mismatch, we propose a new notion called generalized contractible set and we prove that the generalized contractible set is exactly the dichotomy spectrum. Our approach is based on roughness theory and Perron's transformation. In this paper, a new method for roughness theory on time scales is provided. Moreover, we provide a time-scaled version of the Perron's transformation. However, the standard argument is invalid for Perron's transformation. Thus, some novel techniques should be employed to deal with this problem. Finally, an example is given to verify the theoretical results.

Using known operator-valued Fourier multiplier results on vector-valued Hölder continuous function spaces ${{C}^{\alpha }}(\mathbb{R};\,X)$, we completely characterize the ${{C}^{\alpha }}$-well-posedness of the first order degenerate differential equations with finite delay $(Mu{)}'(t)\,=\,Au(t)\,+\,F{{u}_{t}}\,+\,f(t)$ for $t\,\in \,\mathbb{R}$ by the boundedness of the $(M,\,F)$-resolvent of A under suitable assumption on the delay operator $F$, where $A,M$ are closed linear operators on a Banach space $X$ satisfying $D(A)\,\cap \,D(M)\,\ne \,\{0\}$, the delay operator $F$ is a bounded linear operator from $C([-r,0];X)$ to $X$, and $r\,>\,0$ is fixed.

We introduce the principal matrix solution Z(t, s) of the linear Volterra-type vector integro-dynamic equation and prove that it is the unique matrix solution of We also show that the solution of is unique and given by the variation of parameters formula

In this paper, by using the Leggett–Williams fixed point theorem, the existence of three positive periodic solutions for differential equations with piecewise constant argument and impulse on time scales is investigated. Some easily verifiable sufficient criteria are established. Finally, an example is given to illustrate the results.