Frequent Convergence Of Solutions Of Difference Equations

In many natural sciences and edge sciences, such as information science, control engineering, medicine, biological mathematics, modem physics and social economics, many mathematical models of the problems are described by difference equations, therefore difference equations become essential mathematical tools for scientists now.The solutions of difference equations always appear in forms of sequences, so the properties of sequences are always reflected in the solutions of difference equation. The classic concept of limit is not enough to accurately describe the property of convergent sequence, however the definition of frequent convergence of sequence, defined by the concept of frequent measure, can get the better details of convergent sequence than the classic concept of convergence. Also the concept of frequent convergence can be better used in the complex power systems.In this thesis, we try to discuss the frequent convergence of solutions of the following difference equation, and get the corresponding theorem:Firstly, we discuss two difference equations: then deduce frequent convergent theorems of solutions for difference equations if the number of initial values inthe number of initial values in (the number ofinitial values in then the solutions of difference equation frequently belong to interval frequently belong to interval of degree also there are two frequent limits 0 and 1 with degreeSecondly, we discuss frequently convergent solutions of difference equations when m takes four different values 2,3,1/2,1/3, that is wediscuss the following four difference equations: through analyzing and concluding the above results, we discuss the frequent convergence of the difference equation of the general form xn+1= m-xn2. It is found that:only the equation has a frequent limit, while rest three equations have no frequent limit. Thus we have to discuss the influence of value the corresponding equation.Thirdly,during the analysis of effectively input points in the domain of equation ve have the conclusion:the fixed points of difference equation must be the fixed points of equation and but the fixed points of the equation need not to be the fixed points of equation xn+Finally, we discuss the influence of value m to the corresponding equation and find that:when and the initial value x then the difference equation xn+1=m-xn2 always have a solution-which is frequently convergent.