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المستخلص Studing the dynamic equations on time scales was introduced by Stefan Hilger [28]. It is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is a notion that dynamic equations on time scales can build bridges between continuous and discrete mathematics. Further, studying time scales lead to several important applications, e.g., insect population models, neural networks, and heat transfer. A time scale T is a nonempty closed subset of the real numbers. When the time scale equals the set of real numbers, the obtained results yield results of ordinary dierential equations, while when the time scale equals the set of integers, the obtained results yield results of dierence equations. The new theory of the so - called ”dynamic equation” is not only unify the theories of dierential and dierence equations, but also extends the classical cases to the so - called q - dierence equations (when T = qN0 := fqt : t 2 N0; q > 1g or T = qZ = qZ [ f0g) which have important applications in quantum theory (see [31]). A neutral dierential equation with deviating arguments is a dierential equation in which the highest order derivative of the unknown function appears with and without deviating arguments. In recent years, there has been an increasing interest in studying oscillation and nonoscillation of solutions of neutral dynamic equations on time scales which seek to harmonize the oscillation of continuous and discrete mathematics, however the study was restricted to specic equations under certain conditions. The oscillation conditions of their equations are not applicable when these conditions change. For this reason we aim to generalize these equations. So we select the title of our thesis to be ”Oscillation of Second Order Dynamic Equations with Mixed Arguments on Time Scales” using the generalized Riccati transformation, the inequality technique and the generalized exponential function in establishing some new oscillation criteria for second order neutral dynamic equations with mixed arguments on time scales. iii SUMMARY This thesis is devoted to 1. Illustrate the new theory of Stefan Hilger by giving an introduction to the theory of dynamic equations on time scales, 2. Summarize some of the recent developments in oscillation of second order neutral dierential equations, oscillation of second order nonlinear neutral dynamic equations and oscillation of second order dynamic equations with damping on time scales. 3. Establish new sucient conditions to ensure that all solutions of second order nonlinear neutral dynamic equations with mixed arguments and second order nonlinear mixed neutral dynamic equations with nonpositive neutral term on time scales are oscillatory or almost oscillatory or tend to zero, so that the obtained results are more generalized than those obtained in previous studies. 4. Trying to nd some new oscillation criteria for second-order mixed nonlinear neutral dynamic equations with damping on time scales. 5. Give some examples to illustrate the importance of our results. This thesis consists of ve chapters :- Chapter 1, is an introductory chapter that contains the basic concepts of studying the oscillation of solutions for functional dierential equations as well as some previous results in studying the oscillation of second order neutral dierential equations. In Chapter 2, we give an introduction to the theory of dynamic equations on time scales, dierentiation and integration, and some examples of time scales. Moreover, we present various properties of generalized exponential function on time scales. Additionally, some previous studies for the oscillation theory of second order neutral dynamic equations and second order dynamic equations with damping on time scales are presented. In Chapter 3, we establish some new oscillation criteria for the second-order nonlinear neutral dynamic equation with mixed arguments on a time scale T r(t)[ x(t)+p1(t)x(1(t))+p2(t)x(2(t)) ] +f(t; x(1(t)))+g(t; x(2(t))) = 0; iv SUMMARY The results of this chapter generalize and extend the results of Tao Ji et al. [29], and published in: Journal of Basic and Applied Research International 17(1)(2016) 49-66. [5]. In Chapter 4, we present some new oscillation results for the second-order nonlinear mixed neutral dynamic equation with non positive neutral term on a time scale T (r(t)[(x(t)p1(t)x(1(t))+p2(t)x(2(t)))] )+f(t; x(1(t)))+g(t; x(2(t))) = 0; Our results not only generalize some existing results in [22], but also can be applied to some oscillation problems that do not covered before. Also, we give some examples to explain our results. The results of this chapter published in: Academic Journal of Applied Mathematical Sciences 3 (2)(2017)8-20 [6] Chapter 5 is concerned with the oscillatory behavior of all solutions of the second-order mixed nonlinear neutral dynamic equation with damping on a time scale T (r(t)(z(t))) + p(t)(z(t)) + f(t; x(1(t))) + g(t; x(2(t))) = 0; where (s) = jsj 1s and z(t) = x(t) + p1(t)x(1(t)) + p2(t)x(2(t)). Our results generalize the results of [22] and [30] which are considered as special cases of our results when taking = = ; p(t) = p2(t) = 0 and considering either g(t; x(2(t))) = 0 or f(t; x(1(t))) = 0. Also, we introduce an illustrated example to explain our results. |