Quasilinearization method and monotone iterative technique on time scale

This thesis consists of five chapters. In the first chapter of the thesis, the importance and history of time scale and the historical developments of the studies on this subject until today are mentioned. In the second chapter, time scale analysis is introduced and fundamental definitions and theorems and proofs of derivative and integral on time scale are examined by giving the basic concepts on time scale. In the third chapter, the existence and uniqueness theorems, dynamic inequalities, the Lipschitz condition, the existence of extremal solutions, and definitions of coupled lower and upper solutions, comparison theorems frequently used in proofs and Arzela-Ascoli theorem are given. In fourth chapter of the study, we consider the following non-linear initial value problem Monotone iterations have been applied to the solutions using coupled lower and upper solutions on time scale and it has been observed that the sequences converged uniformly and monotonically to the extremal solutions of initial value problem. Lower and upper solution of type III of given above problem was discussed in the proof as follow, respectively: In the fifth chapter, starting from the question of whether it is possible to improve monotone sequences that converge to a solution of the original problem, we extended to a large class by generalized quasilinearization. We consider given below non-linear initial value problem. We have investigated the generalized quasilinearization method under some convenient conditions for nonlinear initial value problem of dynamic equation on time scale and constructed by monotone sequences of function by using comparison theorem. The elements of the sequences are the solutions of the following linear system. Using coupled lower and upper solutions, it has been shown that the obtained sequences converge uniformly and monotonically to the unique solution of the problem. Also it has been observed that the convergence rate also changes when the conditions of the functions are changed. Firstly, by taking the natural type coupled lower-upper solution and assigning different conditions to the functions in the problem, cases where the convergence is quadratic, semi-quadratic or weakly quadratic were obtained. The natural lower and upper solution of given problem was discussed in the proof. it was observed that the monotone sequences converge to the unique solution of the original problem uniformly and monotonically. Furthermore, we observed that this convergence is quadratic. We also observed that similar results were obtained in parallel with the results given by the classical derivative. The novelty of the applied quasilinearization method is the change in the convergence speed under different conditions. Keywords: Time scale, Dynamic equations, Quasilinearization method, Monotone iterative technique, Comparison theorems, Coupled upper and lower solutions, Extremal solutions.