There exists hysteresis effects in many automatic control systems. Models of mathematical describing harmful matter to reactdiffuse process are some kind of differential equations with hysteresis in air cleaning system. Due to the nonlinearity of hysteresis factors

such as the destruction of many excellent properties of functions

many models with hysteresis characteristics hardly can be analyzed. Therefore

in the modeling of general automatic control systems

hysteresis factors are omitted. First

a class of hysteresis operators are introduced according to the functional principle of air purification system

and some important properties of hysteresis operators are given through a series of detailed analysis. Then the existence and uniqueness of the global classical solution of differential equation with hysteresis are proved by using the fixed point theorem. Finally

a decision model for calculating the investment and benefit of purification equipment is proposed.