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Abstract The theory of time scales was introduced by Hilger [30], in order to unify continuous and discrete analysis. Preliminary studies on some basic applications of calculus on time scales were presented by Agarwal and Bohner [3] and Hilger [29]. The study of dynamic equations on time scales has been established in order to unify the study of dierential and dierence equations. Stability theory is a very important problem in the theory and applications of dierential equations. The basic concept of stability emerged from the study of an equilibrium state of mechanical system, dated back to as early as 1644, when E. Torricelli studied the equilibrium of a rigid body under the natural force of gravity. The classical stability theorem of G. Lagrange formulated in 1788, is perhaps the best known about the stability of conservative mechanical systems, which states that if the potential energy of 1 a conservative mechanical systems, currently at the position of an isolated equilibrium and perhaps subject to some simple constraints has a minimum, then this equilibrium position of the system is stable [56]. The most fundamental concepts of stability were introduced by Lyapunov in the late 19th century [55]. The history of asymptotic stability of dynamic equations on time scale goes back to Aulbach and Hilger [8]. For a real scalar dynamic equation, stability and instability results were obtained by Gard and Hoacker [24]. Ptzche [63] provides sucient conditions for the uniform exponential stability in Banach spaces, as well as spectral stability conditions for time- varying systems on time scales. Doan, Kalauch, and Siegmund [20] established a necessary and sucient condition for the existence of uniform exponential stability and characterized the uniform exponential stability of a system by the spectrum of its matrix. Properties of exponential stability of a time varying dynamic equation on a time scale have been also investigated recently by Bohner and Martynyuk [9], Dacunha [17], Hoacker and Tisdell [32], and Peterson and Raoul [60]. 2 In 1992 Kaymakcalan [35] developed Lyapunov;s second method in the framework of general comparison principle so that one can cover and include several stability results for both types of equations at same time. Choi et al. [15] studied the hstability for linear dynamic systems by using the unied time scale quadratic Lyapunov functions. In 2012 Choi et al. [27] introduced a necessary and sucient condition for characterizing hstability for linear dynamic systems on time scales by using Lyapunov function. In [28] Alaa E. Hamza, K. Oraby studied many types of stability of the rst order linear dynamic equations on time scales. Finally, we investigate sucient conditions for stability of both of the abstract rst order linear dynamic equations on time scales of the form x(t) + A(t)x(t) = f(t); t 2 T; and the second order linear equations of the form x(t) + A(t)x(t) + R(t)x(t) = f(t); t 2 T; where A;R : T ! L(X) (the space of all bounded linear operators from In 1992 Kaymakcalan [35] developed Lyapunov;s second method in the framework of general comparison principle so that one can cover and include several stability results for both types of equations at same time. Choi et al. [15] studied the hstability for linear dynamic systems by using the unied time scale quadratic Lyapunov functions. In 2012 Choi et al. [27] introduced a necessary and sucient condition for characterizing hstability for linear dynamic systems on time scales by using Lyapunov function. In [28] Alaa E. Hamza, K. Oraby studied many types of stability of the rst order linear dynamic equations on time scales. Finally, we investigate sucient conditions for stability of both of the abstract rst order linear dynamic equations on time scales of the form x(t) + A(t)x(t) = f(t); t 2 T; and the second order linear equations of the form x(t) + A(t)x(t) + R(t)x(t) = f(t); t 2 T; where A;R : T ! L(X) (the space of all bounded linear operators froma Banach space X into itself), and f is rd-continuous from a time scale T to X. Some given illustrative examples show the applicability of the main results. Now, we describe the contents of the thesis as follows: Chapter One: In this Chapter, we rst present without proof several foundational de- nitions from the calculus on time scales in an excellent introductory texts by Bohner and Peterson [10, 11] which are necessary for the subsequent analysis. We exhibit the delta derivative and give a thorough development of the delta integral [26]. Chaptertwo: In this Chapter, we introduce the Hilger complex plane and cylinder transform in order to develop the generalized exponential function ep(t; s); t; s 2 T. we also exhibit the terminology and methods of solving linear dynamic systems on time scales of the form x(t) = A(t)x(t) + f(t); t 2 T; where A 2 CrdR(T;Mn(R)); n 2 N; where Mn(R) is the family of all n n real matrices, see [10, 11]. We also summarize some of the results concerning the generalized transition matrix eA(t; s); t; s 2 T for linear systems which plays a key role in the study of behavior of solutions of dynamic equations on time scales. Chapter three: This Chapter, contains the proofs of the main local and global existence theorems of non-linear dynamic equations on time scales in Banach spaces of the form x4 = F(t; x); x( ) 2 X; 2 T; (0.0.1) and F : T X ! X is rd-continuous in the rst argument. see [10]. Chapter four: In this Chapter, we introduce the Lyapunov Main Stability Theorem and the basic concepts of stability of dierential equations. Some important de- nitions about stability in real space R and the general time scale T are given. Chapter ve: In this Chapter, we introduce the Lyapunov Main Stability Theorem and the basic concepts of stability of dynamic equations on time scales. Some important denitions about stability on time scale T. And we investigate sucient conditions for stability of both of the abstract rst and second order linear dynamic equations on time scales. Some given illustrative examples show the applicability of the main results. The results of this chapter were prepared in an article which was accepted in ” Journal of Scientic Research for Science ”, 2017. |