Stability of abstract dynamic equations on time scales

Abstract

In this paper, we investigate many types of stability, like uniform stability, asymptotic stability, uniform asymptotic stability, global stability, global asymptotic stability, exponential stability, uniform exponential stability, of the homogeneous first-order linear dynamic equations of the form [Equation not available: see fulltext.] where A is the generator of a [InlineEquation not available: see fulltext.]-semigroup [InlineEquation not available: see fulltext.], the space of all bounded linear operators from a Banach space X into itself. Here, [InlineEquation not available: see fulltext.] is a time scale which is an additive semigroup with the property that [InlineEquation not available: see fulltext.] for any [InlineEquation not available: see fulltext.] such that [InlineEquation not available: see fulltext.]. Finally, we give an illustrative example for a nonregressive homogeneous first-order linear dynamic equation and we investigate its stability. © 2012 Hamza and Oraby; licensee Springer.