Some Qualitative Analysis Of Set Dynamic Equations On Time Scales

The dynamic equations on time scales, which have been created in order to unify thestudy of differential and difference equations，are a new area of mathematics that hasrecently received a lot of attention. Moreover, many results on this subject have beenwell documented in the monographs [1-21].Recently, the study of set equations has received much attention and has beeninitiated as an independent subject. Several results of interest can be found in [27-31].Especially, the study in the stability criteria of the trivial solutions is recentlyundergoing rapid development[25 44]. The interesting feature of the set differentialequations is that the results obtained in this framework become the correspondingresults of ordinary differential equations as the Hukuhara derivative and the integralused in formulating the set differential equations reduce to the ordinary vectorderivative and integral when the set under consideration is a single valued mappings.Hence, set differential equations are the extension of multivalued differentialinclusions and ordinary differential equations. Since set equations can be distinguishedinto set differential and difference equations, the theory of set dynamic equations ontime scales allows one to get some insight into and better understand the subtledifference between discrete and continuous systems，Moreover，to unify some relatedstudies of such problems for differential and difference equations. The study of setdynamic equations is new and interesting. There are few monographs [22-25] at thepresent, and many interesting problems in both theory and application are worthfurther researching.In this thesis, we mainly study exponential stability of the trivial solutions of setdynamic equations on time scales and stability criteria of set control dynamicequations on time scales. Our corresponding results are obtained via a class of newgeneralized Dini derivatives of the Lyapunov-like function on time scales.This paper is organized as follows.First chapter is an introduction. We mainly introduce some researchers’resultsand their methods. Moreover, we expatiate on the significance to study dynamicequations on timescales. At last, we enumerate the problems of which will be solved inthis thesis.In the second chapter, we recall some theoretical knowledge about time scales, which is the foundation of our subsequence works. In the first section of this chapter, we recall some basic concepts and theoretical results due to the investigation of Stefan Hilger and other mathematicians. In the second section, the basic concepts of set dynamic equations on time scales in this chapter are coming from the investigation of Professor Shihuang Hong,In the third chapter, we will consider exponential stability of set dynamic equations on time scales. By using a class of new generalized Dini derivatives of the Lyapunov-like function on time scales, we will study exponential stability of set dynamic equations on time scales.ΔHU= F(t,U), U(to)= Uo∈Kc(R), WhereΔH denotes the derivative of multivalued functions defining on time scales, Kc(R) denotes a complete metric space[22].In the fourth chapter, we will consider boundedness of solutions to set dynamic equations on time scales. By using a class of new generalized Dini derivatives of the Lyapunov-like function on time scales, we will study boundedness of solutions to set dynamic equations on time scales.ΔHU= F(t,U), U(to)= Uo∈eKc(R), WhereΔH denotes the derivative of multivalued functions defining on time scales, Kc(R) denotes a complete metric space[22].In the fifth chapter, we will consider stability criteria of set control dynamic equations on time scales. By using a class of new generalized Dini derivatives of the Lyapunov-like function on time scales, we will study stability criteria of set control dynamic equations on time scales.ΔHX=F(t,X,U), X(to)= Xo∈Kc(Rn), WhereΔH denotes the derivative of multivalued functions defining on time scales, Kc(R) denotes a complete metric space[22].We will use a class of new generalized Dini derivatives of the Lyapunov-like function on time scales to obtain the sufficient condition of exponential stability of set dynamic equations on time scales and stability criteria of set control dynamic equations on time scales.