الفهرس 
Basic concepts of time scales calculusOnly 14 pages are availabe for public view
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Abstract
In 1960, the Polish mathematician Z. Opial published a paper introducing an inequality known as Opial inequality, which is an inequality involving the integral of the product of a function and its derivative. Since then, there have been a wide interest in this type of inequalities and many extensions and generalizations have been developed in both continuous and discrete cases.In the recent years, particularly in 1988, Hilger introduced a new calculus, in his Ph.D. thesis, known as the time scale calculus, in which he defined a time scale T to be a non empty closed subset of the set of real numbers R. This calculus unifies the study of continuous and discrete analysis. Since then,there has been a wide interest in this calculus and many authors have studied dynamic equations on time scale.In the study of the qualitative properties of solutions of difference and differential equations, inequalities involving higher order derivatives of the integral type and the discrete type play an effective role.The importance of Opial inequalities appears in their applications in the theories of difference and differential equations such as the uniqueness of initial value problems,the existence and uniqueness of boundary value problems, and the upper bounds of solutions. This is the reason why we believe that the dynamic inequalities involving higher order derivatives on time scales will play the same effective role in the analysis of the qualitative properties of solutions of dynamic equations of higher order.Recently,there has been a wide interest in studying the uniqueness of the initial value problems, the uniqueness of the boundary value problem, the oscillatory behavior, and the upper bounds of solutions of first, second and third dynamic equations. On the contrary,the fourth order dynamic equations did not receive as much attention, and as far as we know, there is no results concerning the lower bounds of the distance between zeroes of some fourth-order dynamic equations.
In this thesis, we will prove some new Opial-type inequalities on time scales involving higher order derivatives and establish some of the applications of Opial’s inequalities on time scales on a fourth-order dynamic equation.The thesis is organized as follows:In Chapter 1, we recall the main concepts and definitions of time scales, the theorems, formulae and some basic inequalities on time scales required in the proof of the main theorems, such as Hölder’s and reverse Hölder’s inequalities,...etc.In Chapter 2, we give a brief introduction to Opial inequalities and their generalizations, extensions and some results related to our work where T= R, and T=N.In Chapter 3, we prove some new multiplicative dynamic inequalities of higher orders of Opial’s type on time scales by using Hölder’s inequality, a new chain rule and some new dynamic inequalities designed and proved for this purpose. The results as a special case when T=R, will be reduced to some improved inequalities in the differential forms similar to those proved by Cheung. The results in this chapter are published in Mathematical Inequalities and Applications.In Chapter 4, we use Hölder’s inequality, reverse Hölder’s inequality, chain rule, some elementary inequalities and a new estimation of the power rule of integration on time scales to prove some new Opial-type inequalities and improve some of the results proved previously. The results in this chapter are published in Vietnam Journal of Mathematics.In Chapter 5, we present some Hardy type inequalities, and some Opial type inequalities which we will use to introduce some applications on a fourth-order dynamic equation to establish some lower bounds of the distance between zeroes of a solution and it’s derivatives. The fourth-order dynamic equation is of the form〖(p(x) y^∆∆ (x))〗^∆∆+q(x) y^σ (x)=0,
with the boundary conditions y^∆∆ (a)=0,y^(∆_k ) (b)=0, for k=0,1,2, where p(x) is a nonnegative, non decreasing rd-continuous function on 〖[a,b]〗_T, q(x) is a nonnegative rd-continuous function on〖[a,b]〗_T with q(x)=Q^∆ (x).