Monotone iterative technique on time scale

This thesis consists of 5 chapters. In the first part, the importance and functionality of time scale, the studies made in the thesis are discussed in general. In the second part, firstly, time scale analysis is introduced. By giving the basic concepts on the time scale, it is aimed to give a better understanding of the isssue with a few additional examples. The basic concepts given here are limited to what will be used in the study. For the same reason, only the fundamental theorems and proofs of the derivative and integral on the time scale have been studied. In the third chapter, the concepts of Peano's type existence Perron's type uniqueness are given, and Arzela-Ascoli theorem, which is frequently used in the stuy, is mentioned. The content of dynamic inequalities, Lipschitz condition and extremal solutions are also the inormations given as a preparation fort the study. In the fourth chapter, definitions of lower and upper solutions are given. It has been shown that the lower and upper solutions form a closed interval for the solutions of dynamic equations on the time scale. Later, it is determined that the sequences to created with the monoton iteration technique converged uniformly and monoticially to extremal solutions of the problem. In the fifth chapter, the generalized monotone iterative technique is introduced. Necessary theorems and proofs of the technique are given, and then ewamples are shown. Later, the mixed monotony technique is introduce. Keywords: upper and lower solutions, extremal solutions, monotone iterative technique, time scale