Solutions of Dynamic Equations
Nesreen Abd El Hamed Abd El Hameed Yaseen;
Abstract
A time scale is an arbitrary non-empty closed subset of the set of real number. The object of the theory of dynamic equations on time scales is to unify continuous and discrete calculus which was introduced by Stefan Hilger in his Ph .D. in 1988 [29]. The theory presented a structure where, once a result is established for general time scale, special cases include a result for differential equations and a result for difference equations. A great deal of work has been done since 1988 unifying the theory of differential equations and the theory of difference equations by establishing the corresponding results in time scale setting. Recent two books of Bohner and Paterson [8,9] provide both an excellent introduction to the subject and up to-date coverage of much of the theory . Studying the theory of dynamic equations on time scales has attracted many authors in the last few years, because of the wide applications in engineering, industry, biology, economics and other field, see [5]. Also, the stability analysis of dynamic equations has become an important topic both theoretically and practically because dynamic equations occur in many areas such as mechanics, physics, and economics.
This work focuses on the Hyers-Ulam and Hyers-Ulam-Rassias stability of dynamic equations on time scales . In 1940 Ulam [63] posed his problem of stability which solved by Hyers [30], and the result of Hyers was generalized by Rassias [58]. Alsina and Ger [1] were the first authors who investigated the Hyers-Ulam stability of a differential equation. Also of interest, that many of articles were edited
by Rassias [57], dealing with Ulam, Hyers-Ulam and Hyers-Ulam-Rassias stability . Many papers introduced the Hyers-Ulam and Hyers-Ulam-Rassias stability for differential equations and integral equations[15,37]. On the other hand side the papers which were presented the Hyers- Ulam and Hyers-Ulam-Rassias stability of dynamic equations and integral dynamic equation on time scales are still very few, may be except the studies which were presented in the papers [4,7].
This thesis investigates the (Hyers-Ulam-Rassias) stability for first and second order abstract dynamic equations on time scales. Also, in this thesis we investigate the Hyers-Ulam-Rassias stability of integral dynamic equation on time scale.
The thesis is organized as follows:
Chapter I:
In this Chapter, we introduce the basic concepts and terminology of time scales calculus. We introduce basic theorems of delta derivative and delta integral which were presented in Bohner [10]. The exponential function and the operator exponential function on time scales with some properties of them are given. Also, we introduce the concept of a rd-continuous matrix, and a regressive matrix valued functions. Also this chapter presents definitions and some properties of operator exponential function in Banach spaces. Moreover, basic concepts of the theory of - Semigroup of bounded linear operators are introduced . Also some properties of a -Semigroup and its generator which were established in [2] are introduced in this chapter.
Chapter II
In this chapter we introduce the papers which presented the Hyers-Ulam and Hyers-Ulam-Rassias Stability of differential equation. Also, we introduce the papers which proved the Hyers-Ulam of integral equations. This chapter also, presents the articles dealing with the Hyers-Ulam stability of dynamic equations with constant coefficients on time scales[7]
Chapter III
In this chapter we investigate the Hyers-Ulam Stability of the abstract dynamic equation of the form
where (The space of all bounded linear operators from a Banach space into itself) and f is rd-continuous from a time scale to . This chapter deal with two cases, the first case if is regressive, and the second case if is not regressive. Some examples illustrate the applicability of the main results.
The results of this chapter were published in [4].
Chapter IV
This chapter is devoted to establishing Hyers-Ulam Stability of the abstract dynamic equation of the form
where , the space of all bounded linear operators from a Banach space into itself, and is rd-continuous from a time scale to . Some examples illustrate the applicability of the main result.
The results of this chapter have been prepared in an article which was submitted and accepted in Journal: International Journal of Math. Anal. [5].
Chapter V
In this chapter we establish the Hyers-Ulam-Rassias stability for Equations (1) and (2).
Also, some examples illustrate the applicability of the main result.
Chapter VI
In this chapter we investigate the Hyers-Ulam-Rassias stability for the Volterra integral equation of the form
Chapter VII
This chapter contains the conclusions and some points for future research.
Notations
:= { 1,2,…..}
is the set of all integer numbers.
is the set of rational numbers.
is the set of all real numbers.
is the set of complex numbers.
is a time scale (i.e. a nonempty closed subset of ).
for (a closed interval of ).
is the forward jump operator on .
is the backward jump operator on T.
is the graininess function on .
is the family of regressive functions.
is the family of right dense continuous functions.
is the delta derivative ( derivative) of a function f.
is a Banach space.
This work focuses on the Hyers-Ulam and Hyers-Ulam-Rassias stability of dynamic equations on time scales . In 1940 Ulam [63] posed his problem of stability which solved by Hyers [30], and the result of Hyers was generalized by Rassias [58]. Alsina and Ger [1] were the first authors who investigated the Hyers-Ulam stability of a differential equation. Also of interest, that many of articles were edited
by Rassias [57], dealing with Ulam, Hyers-Ulam and Hyers-Ulam-Rassias stability . Many papers introduced the Hyers-Ulam and Hyers-Ulam-Rassias stability for differential equations and integral equations[15,37]. On the other hand side the papers which were presented the Hyers- Ulam and Hyers-Ulam-Rassias stability of dynamic equations and integral dynamic equation on time scales are still very few, may be except the studies which were presented in the papers [4,7].
This thesis investigates the (Hyers-Ulam-Rassias) stability for first and second order abstract dynamic equations on time scales. Also, in this thesis we investigate the Hyers-Ulam-Rassias stability of integral dynamic equation on time scale.
The thesis is organized as follows:
Chapter I:
In this Chapter, we introduce the basic concepts and terminology of time scales calculus. We introduce basic theorems of delta derivative and delta integral which were presented in Bohner [10]. The exponential function and the operator exponential function on time scales with some properties of them are given. Also, we introduce the concept of a rd-continuous matrix, and a regressive matrix valued functions. Also this chapter presents definitions and some properties of operator exponential function in Banach spaces. Moreover, basic concepts of the theory of - Semigroup of bounded linear operators are introduced . Also some properties of a -Semigroup and its generator which were established in [2] are introduced in this chapter.
Chapter II
In this chapter we introduce the papers which presented the Hyers-Ulam and Hyers-Ulam-Rassias Stability of differential equation. Also, we introduce the papers which proved the Hyers-Ulam of integral equations. This chapter also, presents the articles dealing with the Hyers-Ulam stability of dynamic equations with constant coefficients on time scales[7]
Chapter III
In this chapter we investigate the Hyers-Ulam Stability of the abstract dynamic equation of the form
where (The space of all bounded linear operators from a Banach space into itself) and f is rd-continuous from a time scale to . This chapter deal with two cases, the first case if is regressive, and the second case if is not regressive. Some examples illustrate the applicability of the main results.
The results of this chapter were published in [4].
Chapter IV
This chapter is devoted to establishing Hyers-Ulam Stability of the abstract dynamic equation of the form
where , the space of all bounded linear operators from a Banach space into itself, and is rd-continuous from a time scale to . Some examples illustrate the applicability of the main result.
The results of this chapter have been prepared in an article which was submitted and accepted in Journal: International Journal of Math. Anal. [5].
Chapter V
In this chapter we establish the Hyers-Ulam-Rassias stability for Equations (1) and (2).
Also, some examples illustrate the applicability of the main result.
Chapter VI
In this chapter we investigate the Hyers-Ulam-Rassias stability for the Volterra integral equation of the form
Chapter VII
This chapter contains the conclusions and some points for future research.
Notations
:= { 1,2,…..}
is the set of all integer numbers.
is the set of rational numbers.
is the set of all real numbers.
is the set of complex numbers.
is a time scale (i.e. a nonempty closed subset of ).
for (a closed interval of ).
is the forward jump operator on .
is the backward jump operator on T.
is the graininess function on .
is the family of regressive functions.
is the family of right dense continuous functions.
is the delta derivative ( derivative) of a function f.
is a Banach space.
Other data
| Title | Solutions of Dynamic Equations | Other Titles | حلول المعادلات الديناميكية | Authors | Nesreen Abd El Hamed Abd El Hameed Yaseen | Issue Date | 2014 |
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