Ulam stability for first order dynamic equations on time scale

In this research, Ulam stability in time scale for first order homogeneous and inhomogeneous dynamical equations was investigated. This stability theory, which was addressed by Hyres's answer to one of the questions Stanislav Ulam asked in a speech at the University of Wisconsin in 1940, is known today as the Hyres-Ulam stability. In this research, basic definitions and concepts such as first-order differential equations and then stability in these differential equations and Hyers-Ulam stability of functional equations are given. Then, the basic concepts of time scale, delta (Hilger) derivative, delta integral, complex Hilger plane, generalized exponential function and first-order linear dynamical equations are examined and their definitions are given. Then, taking into account the work of M. Onitsuka and D. R. Anderson, Hyers-Ulam stability (HUS) is studied in the time scale of constant coefficient first order linear dynamical equations including discrete and continuous dynamical equations in a few special cases. Here, the minimum Hyres-Ulam stability constants are searched for a few parameter values related to the granule function of the time scale. When we come to the last part, Ulam stability is discussed by applying the method of integrating the conjugate equation of linear first order dynamical equations, taking into account the work of Yonghong Shen. Then, the Ulam stability of the linear first-order dynamical equation is investigated through the properties of the generalized exponential function of this equation and the results of this equation.