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العنوان
On Some Dynamic inequalities on time scales \
المؤلف
Ahmed, Fatma Mostafa Khamis.
هيئة الاعداد
باحث / فاطمة مصطفي خميس احمد
مشرف / حسن مصطفى العويض
مشرف / جمال علي فؤاد اسماعيل
مشرف / أحمد عبدالمنعم أحمد الديب
تاريخ النشر
2019.
عدد الصفحات
190 p. :
اللغة
الإنجليزية
الدرجة
ماجستير
التخصص
الجبر ونظرية الأعداد
تاريخ الإجازة
1/1/2019
مكان الإجازة
جامعة عين شمس - كلية البنات - لرياضيات
الفهرس
Only 14 pages are availabe for public view

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Abstract

Mathematical analysis has been the major and significant branch of mathe-matics for the last three centuries. During the twentieth century, discrete and integral inequalities played a fundamental role in mathematics and have a wide variety of applications in many areas of pure and applied mathematics.
In recent years the study of inequalities on time scales has received a lot of attention in the literature and has become a major field in pure and applied mathematics. These dynamic inequalities have a significant role in understand-ing the behavior of solutions of equations on time scales. The recent book dy-namic inequalities on time scales by Agarwal, O’Regan and Saker [5] contains most recent basic dynamic inequalities such as Young’s inequality, Jensen’s inequality, H¨older’s inequality, Minkowski’s inequality, Steffensen’s inequality, Hilbert’s inequality, Hardy’s inequality, Lyapunov’s inequality.
In 1988, Stefan Hilger [20] introduced time scales theory to unify continu-ous and discrete analysis. The general idea is to prove a result for a dynamic inequality where the domain of the unknown function a so-called time scale T, which is nonempty closed subset of the real numbers R, to avoid proving results twice, once in the continuous case which leads to a differential inequality and once again on a discrete case which leads to a difference inequality. The books of the subject of time scales by Bohner and Peterson [13],[14] summarize and organize much of time scales calculus.
Indeed, mathematical inequalities became an important branch of modern math¬ematics in twentieth century through the pioneering work entitled Inequalities by G. H. Hardy, J. E. Littlewood and G. P`olya [47], which was first published treatise in 1934. This unique publication represents a paradigm of precise logic, full of elegant inequalities with rigorous proofs and useful applications in math
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ematics.
In the year 1960, the Polish mathematician Zdzidlaw Opial published an in¬equality involving integrals of a function and its derivative, which now bears his name. While it has been shown that inequalities of this form can be de¬duced from those of Wirtinger and Hardy type, the importance of Opial’s result, however, is in the establishment of the best constant. Immediately after its publication, several simplifications of the original proof were offered, and gen¬eralizations and discrete analogues were considered. In the subsequent three decades, the study of Opial type inequalities has grown into a substantial field, with many important applications.
This thesis is devoted to prove some new inequalities of type Opial’s on time scales. This thesis consists of six chapters and is organized as follows:
Chapter 1: Preliminaries. We present some basic concepts, definitions, the basic inequalities and preliminary results of the calculus on a time scale T which are absolutely essential for completing the results and techniques used in subsequent chapters.
Chapter 2: Opial Type Inequalities. This chapter focus on the previous work and existing research effort in the field of Opial’s inequalities. Some former famous opial’s inequalities in some text books that have been established by different researchers and some known results in some recent periodicals are given as well. We collect some of the applications of Opial’s inequality and its many generalized versions.
Chapter 3: On Some Generalizations of Dynamic Opial-Type In-equalities. In this chapter, we will prove some new generalizations of dynamic Opial-type inequalities on time scales. from these inequalities, as special cases, we will formulate some integral and discrete inequalities proved in the literature and also extend some obtained dynamic inequalities on time scales. The main results will be proved by using some algebraic inequalities, H¨older inequality and a simple consequence of keller’s chain rule on time scales.
The contents of this chapter have been published in Journal of Advances in Difference Equations, 2019(323), 2019.
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Chapter 4: Weighted Dynamic Inequalities of Opial-Type on Time Scales. In this chapter, we will state and prove some weighted dynamic inequal¬ities of Opial-type involving integrals of powers of a function and of its derivative on time scales which not only extend some results in the literature but also im-prove some of them. As special cases of the obtained dynamic inequalities, we will get some continuous and discrete inequalities.
The contents of this chapter have been published in Journal of Advances in Difference Equations, 2019(393), 2019.
Chapter 5: Dynamic Opial Inequalities for Convex Functions. In this chapter, we present a convex version of Opial-type for time scales. Various generalizations of our results are offered as well. We prove some new inequalities of Opial-type for convex functions on an arbitrary time scale using the delta integrals. These inequalities extend and improve some known dynamic inequal¬ities in the literature. The main results will be proved by using some algebraic inequalities, H¨older inequality, Jensen inequality and a simple consequence of Keller’s chain rule on time scales. As special cases of the obtained dynamic inequalities, we will get some continuous and discrete inequalities.
The contents of this chapter is submitted in Journal of Symmetry.