GRONWALL AND BIHARI TYPE INEQUALITIES ON TIME SCALES S. A. ¨ OZG ¨ UN Department of Mathematics Education Hacettepe University, Beytepe 06532 Ankara, TURKEY and A. ZAFER and B. KAYMAKC ¸ ALAN Department of Mathematics Middle East Technical University 06531 Ankara, TURKEY Abstract In this article, in view of the recently developed theory of dynamic systems on time scales, Gronwall-Bihari type inequalities, which unify and extend some continuous and discrete versions at the same time, are developed. In the process of establishing these inequalities, a scalar com- parison result obtained for dynamic systems on time-scales is employed. 1 INTRODUCTION Recently initiated theory of dynamic systems on time-scales provides the framework to handle both continuous and dicrete dynamical sys- tems simutaneously so as to bring out better insight and understanding of the subtle differences of these two types of systems. Also from a mod- elling point of view, it is more realistic to consider dynamic systems on time-scales giving way to the incorporation of both the theory of dif- ference equations and differential equations. In [1,2,3] Aulbach and Hilger developed the necessary calculus of functions on time-scales and discussed certain basic results.