The Research Of Several Functional Equations On Group

Functional equations are mathematical models for many problems of finance,physics,geometry,algebra,measure theory,etc.For example,the problem of rectangular domain in geometry,the problem of polygon interior Angle and the problem of simple interest in finance can be described by functional equations.The discussion of various functional equations was initially limited to the real number or complex number fields,and later breakthroughs were made in all directions,such as the discussion on groups,rings and fields.Results in Banach space or Hilbert space,hyers-ulam stability of functional equa-tions,problems related to measure theory,game theory,etc.In this paper,three types of functional equations with involutions are studied,and the solutions on the group are given and the expressions for the solutions are shown.The main contents are as follows:In the first chapter,we mainly introduces the research background of functional equation,including the research significance of functional equation,status at home and abroad,as well as the development of functional equation and the main content of this paper and the introduction of symbols.In the second chapter,we discussed the function equation of f(x)+f(y)=max{f(xy),f(xσy)} on the group of solution.Using the properties of the function f on the group,such as,symmetry,commutativity,etc.,to complete the proof of the main theorem.In the process of proof,we will through classification discussion to derive the contradiction,and then analyzed the properties of the function in detail to get the con-clusion.In the third chapter,we consider a kind of functional equation with two involutions∫(τx +σy+a)+g(x+y+α)=2∫(x)∫(y)on abelian group,through the analysis of the haracteristics of the equation itself,find the properties that satisfy the solution of this equation.Using the property of abelian group and the characteristic of pair mapping,the original equation is transformed into the related auxiliary equation of the known conclusion.Through the discussed of the auxiliary equation,and the conclusion have been applied to the original equation to complete the proof.In the fourth chapter,we discussed a class of functional equations f(x+σy+α)+g(x+τt+α)=2f(x)f(y)with two involutions on the abelian group.First,we derived the solutions of several functional equations related to the original equation on abelian group,and then we find the relation of several equations with group character and additive function.By using these results,we obtain the solutions of the equation and give the specific expressions of these solutions.