On some Gamidov integral inequalities on time scales and applications (Q1750329)

The Gamidov inequality may be written as \[ u(t)\leq k+\int^t_0 g(s) u(s)\,ds+ \int^T_0 h(s)u(s)\,ds, \] where \(k\geq 0\) is a constant, an \(T\) is a positive real number. In [Tamkang J. Math. 33, No. 4, 353--358 (2002; Zbl 1029.26014)], \textit{B. G. Pachpatte} gave an extension of this inequality. Motivated by these results, the author extends Gamidov's inequality to time scales (first introduced by \textit{S. Hilger} [Result. Math. 18, No. 1--2, 18--56 (1990; Zbl 0722.39001)]). The obtained results can be used as tools in the study of certain properties of dynamical equations on time scales.