الفهرس 
The Time Scales CalculusOnly 14 pages are availabe for public view
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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 di erential and di erence equations.
Stability theory is a very important problem in the theory and applications
of di erential 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 Ho acker [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], Ho acker and Tisdell
[32], and Peterson and Ra oul [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 uni ed 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 uni ed 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 di erential 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 de nitions 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 Scienti c Research for Science ”, 2017.