Some Problems About Meromorphic Functions Normality And Difference

With the development of science and technology,agricultural electrification and automation in our country are getting higher and higher.Mathematics can be regarded as the basis of agricultural electrification and automation.Especially chaos and fractal theory are widely used in agricultural engineering.For example,the combination of chaos and neural network can be used in the planning of associative memory robot;Chaos optimization aims at finding the best points by using the randomness,ergodicity and regularity of chaotic motion,so,it can be used in many aspects such as system identification,optimal parameter design and so on.Chaos and fractal theory are widely used in agricultural production,crop cultivation,agricultural product classification,and diseases and insect pests in agriculture and forestry.Chaotic phenomena are inevitable problems in the study of chaos and fractal theory.And chaotic phenomena in deterministic dynamic systems can be regarded as the characteristics of the iterative process,which has a very close relationship with the complex dynamic system.The basic conceptions such as Julia set and Fatou set in complex dynamic system are all derived from the normal family.And the normal family theory is based on the value distribution theory of the meromorphic function which established by Nevanlinna.This doctoral dissertation is mainly about the mathematical basic theory of complex dynamic system,some important problems in normal families and difference of meromorphic functions.Firstly,we briefly introduce the role of chaos and fractal theory in agricultural engineering,and gives a brief overview of meromorphic function value distribution theory and normal family theory which are closely related to chaotic fractal theory.Secondly,we studied the periodic points and normal families concerning multiplicity by Maple,we provedLet F be a family of meromorphic functions in domain D,and let k?2,l are two positive integer satisfying l? 4 for k=2,l?3 for k?3.If for each f?F,all zeros of f(z)-z have multiplicity at least l,and fk has at most one fix point in D,then F is normal in D.The result improved the results due to Chang and Fang(2005).Thirdly,we discuss the normal criteria of meromorphic functions concerning sequence of functions by the method of mathematical induction,and we provedLet {fn} be a family of meromorphic functions in domain D,let {hn} be a sequence of holomorphic functions in D satisfying hn(z)(?)h(z),where h(z)(?0,?)is a holomorphic function in D,and let k be a positive integer.If for each n?N+,fn(z)?0,and fm(k)(z)-hn(z)have at most k distinct zeros in D,then {fm} is normal in D.To some extent,we extended the results due to Chen et al.(2013).Fourthly,we use the method of construct a new sequence of functions to separate the zero,and studied the normality concerning Hayman question and the normal criteria about differential polynomial sharing functions.And we obtained some interesting results.Our results extended and improved the results due to Lei and Fang(2010),Meng and Hu(2015).Then,by using the characteristics of the difference operator,and the method of classical value distribution,we studied the uniqueness problem concerning meromorphic function of infinite order share sets with it's difference operator or shifts,and the uniqueness problem of meromorphic function sharing a polynomial with it's high-order difference operator,we provided some difference analogue of the results due to Chang and Fang,and extended the results due to Chen et al.(2012,2014).Finally,we summarized this paper and putted forward several questions that can be further studied.