الفهرس 
Preliminaries and IntroductionOnly 14 pages are availabe for public view
 |  | from | 145 |  |  |
| | | from | 145 | | |
Abstract
It is well known that obtaining analytic solutions of differential and difference equations is almost difficult especially in the case of nonlinear equations.
So, the need to the qualitative properties of solutions arised in the last four decades one of the important qualitative properties is the stability theory.
The idea of the time scales appeared by Hilger [42], to unify the results for both differential and difference equations. Since then many contribution offered (see [42, 24]).
Recently the stability theory of functional equations has been strongly developed. Very important contributions to this subject were brought by Ulam [112], Rassias [95], Hyers et al. [46], Jung [51], Guo et al. [36], Kolmanovski˘ı and Myshkis [62], and
Radu [94]. Our results are connected with some recent papers of Castro and Ramos [30] and Jung [51] (where integral and differential equations are considered), BotaBoriceanu and Petru¸sel [26], and Petru et al. [89] (where the Ulam–Hyers stability for operatorial equations and inclusions are discussed).
The fractional calculus extends the theory of differentiation and integration of integer order to real or complex order. Despite the fact that this calculus is as old as the classical one, scientists working on different areas have paid attention to it only in the last decades since good results were found out when the tools in this calculus were used to illuminate some models of real-world phenomena [43, 61, 72, 90, 102, 69].
Ulam,s type stability problem have been considered by a huge number of mathematicians . The study of this region has grown-up to be one of the vital subjects in mathematical analysis. For more details on Hyers-Ulam stability, we recommend [28, 44, 67, 68, 76, 108, 118, 119]. The authors in [64] employed Picard’s operator
technique, the abstract Gronwall lemma, and Pachpatte’s inequality. Their results improve and generalize those obtained by the authors in [30, 51, 83, 84, 99, 103, 111].
More recently, several interesting Ulam stability results of various type have been appeared in the literature (see for example [17, 33, 49, 50, 74, 76, 78, 91, 92, 99, 100, 121]). For the case of difference equations see [19, 18].
Furthermore, many studies recently appeared in the Hyers–Ulam stability of Dynamic equations on time scales [16, 66, 67, 105, 106]. Hamza and Yaseen [39] improved and extended the work of Douglas R. Anderson, Ben Gates and Dylan Heuer [16] for unbounded case on time scales.
This thesis is aim to discuss and prove some new different types of Ulam stability of first order nonlinear delay differential equations and nonlinear Voltera delay integro-differential equations with fractional, first-order nonlinear Volterra delay integro-differential equations with impulses and Nonlinear First order Volterra Delay Integro-dynamic Equation on Time Scales. The thesis consists of six chapters and is organized as follows:
Chapter 1: Preliminaries and Introduction.
In this chapter we present some basic concepts, definitions, basic inequalities and preliminary results of Fractional Calculus which are absolutely essential for completing the results and techniques used in subsequent chapters and basic definitions of Ulam-Hyers stability.
Chapter 2: Time Scales Calculus.
In this chapter we introduce basic concepts and terminologies of time scales calculus. The exponential function and operator exponential function on time scales with some essential properties are given. We introduce some of important inequalities on time scales like Bernoulli’s and Gronwall’s inequalities [9]. At the end of the chapter, the method of successive approximations within dynamic equations on time scales is applied to prove the existence and uniqueness of solutions of dynamic equations [109].
Chapter 3: Stability of Non-Linear Fractional Delay Differential Equations.
In this chapter we discuss different types of Ulam stability of nonlinear delay differential equations with fractional Riemann-Liouville Derivatives. Our analysis is based on a generalized Gronwall’s inequality and Picard operator theory. Applications are provided to illustrate the stability results obtained in the case of a finite interval.
The results of this chapter is submitted.
Chapter 4: Delay Differential Equations with a Generalized Caputo
Derivative.
In this chapter we discuss different types of Ulam stability of nonlinear delay differential equations with Caputo derivatives. To demonstrate our results two examples are presented.
Chapter 5: Impulsive Volterra Delay Integro-differential Equations.
In this chapter we state and prove some different types of Ulam stability of firstorder nonlinear Volterra delay integro-differential equations with impulses. Such types of equations allow the presence of two kinds of memory effects represented by the delay and the kernel of the used fractional integral operator. Our analysis is based on Pachpatte’s inequality, and the fixed-point approach represented by the Picard operators. Applications are provided to illustrate the stability results obtained in the case of a finite interval.
The results of this chapter have been published in Advances in Difference Equations, (2021) 2021:477.
Chapter 6: Nonlinear First order Volterra Delay Integro-dynamic Equation on Time Scales.
In this chapter, we improve and extend Hyers –Ulam stability and Hyers –UlamRassias stability for non–linear first order Volterra delay integro–dynamic equation on time scales.
The results of this chapter is submitted.