By using the method of functional analysis,especially,the linear operator theory and C_0 semigroup theory on Banach space,the well-posedness of solution and the existence of positive solution are studied.

We give a complete introduction about C_0 semigroups in Banach space.

Irreducibility of the Positive Contraction C_0-semigroup Generated by M/G/1 Queueing Operator;

This paper discusses the existence of solutions of initial value problem for semilinear evolution equation with noncompact semigroup u (t)+Au(t)=f(t, u(t)), t≥0; u(0)=x_0 in a Banach space E, where -A is the infinitesimal generator of an equicontinuous C_0-semigroup, and f: [0, ∞)×E→E is continuous.

In this paper we have proved that a C--semigriop on Banach space X can be-come of C_0-semigroup by means of the method to restrict the C-semigroup onto a smallerBanach space F with a stronger norm; and they have the same analyticity.

Then,by the positive C_0-semigroup generated by system operator,it is shown that there exists a steady nonnegative solution of the system which is just the normalized eigen- vector corresponding to eigenvalue 0 of system operator.

Secondly, we can write the above system as an abstract system, then give the result that the corresponding C_0-semigroup is exponentially stable.

Besides, system operators can genetate a positive contraction C_0-semigroup in L~1space , so the solution of the system is nonnegative with probability character.