Operator Equations And Inequalities Stability

In this article, we study the solutions of several operator equations and the stability of operator equations and inequalities. We divide this article into two chapters.In Chapter 1, we fisrt prove that the quadratic mapping is even and obtain the explicit solutions of the quadratic operator equation under different conditions. Then we discuss the odd-even property of the cubic mapping and quartic mapping and give the explicit solutions of the cubic operator equation and quartic operator equation under different conditions. Furthermore, we solve the explicit solutions of the Cauchy-Jensen operator equation under some conditions. Finally, we introduce a new cubic operator equation and investigate its general solutions.In Chapter 2, we study the stability of several operator equations and inequalities.Firstly, we establish the Hyers-Ulam-Rassias stability problem for the quartic operator equation by using the fixed point alternative. Secondly, we prove that Butler-Rassias functional equation and Exponential functional equation have the Hyers-Ulam-Rassias stability. Finally, we prove that the operator inequality‖f(x) + f(y) + f(z)‖≤‖2f((x+y+z)/2)‖has the generalized Hyers-Ulam stability.