Oscillation of second order dynamic i̇nclusions on time scales

Time scale theory which was found in 1990 by Hilger to connect the continuous and discrete analysis, has received a lot of interest recently. Oscillation problems recently has become very attractive in the area of difference and differential equations. These areas, have been throughly studied by several authors with more generalised and strong theorems; since it was compiled as dynamic equations on time scales. Dynamic inclusions, represents an important generelation of dynamic equations. Solution of a dynamic inclusion instead of a single curve it is a reachable set. Dynamic inclusions are fundamental for some concepts of optimal control theory. Dynamic inclusions arise from several dynamical inequality situations; such as dynamic systems, dynamic Coulomb friction problems and fuzzy sets arithmetic. In the recent years there is a lot of research on oscillation or non-oscillation of several dynamic equations on time scales. However, there are few results dealing with the oscillation of second order dynamic inclusions on time scales and for second order dynamic inclusions with distributed deviating arguments. The purpose of this thesis is to investigate the some oscillation criteria of this type of second order dynamic inclusions. In the first chapter, some basic knowledge about time scale is given. Delta and nabla derivatives for the functions f : 𝕋 -> ℝ on time scales are explained and some basic definitions an examples are given regarding these derivatives. Also, concepts of delta and nabla integral are given and properties of delta and nabla integrals are investigated. In the second chapter, we look into the two studies of Ravi P. Agarwal, Said Grace and Donal O'Regan in 2003 and 2009, which are on the same second order differential inclusion. In these studies basic theorems on oscillation criteria of a second order differential inclusion, whose dynamic inclusion form will be used in chapter three, are given. In the third chapter we investigated the study of Elvan Akın-Bohner and Shurong Sun in 2012. In this work they investigated some oscillation criteria of second order strong superlinear and strong sublinear dynamic inclusions on time scales. In the last chapter we looked into a new study given by Said Grace, Elvan Akın and Rawi Agarwall in 2015. In this study oscillatory theorems for certain second order damped dynamic inclusions with distributed deviating arguments are given. Finally, some possible generalisations are investigated for the found results.