Global Solutions And Periodic Solutions Of Evolution Equations With Sector Operator

In this paper,by using the theory of operator semigroups and some fixed point theorems in abstract spaces,we deal with the existence,uniqueness and regularity of global solutions for the evolution equation with sector operator u'(l)+ Au(l)= ?(l,u(l)),l>0in Banach spaces.Besides,based on the discussion of global solutions for the initial problem,we consider the ?-periodic solutions of this equation further.Where,f is the nonlinear term and A is a sector operatorThe main results of this paper are as follows:1.Through using some fixed point theorems efficiently,while f satisfies different assumptions,we discuss the existence and uniqueness of saturated and global mild solutions for initial problems of the evolution equation with sector operator.2.With the aid of the properties of analytic semigroups,we consider the regular-ity of solutions for the evolution equation with sector operator and get the classical solutions.3.By the increasing operator fixed point theorem,we establish the existence and uniqueness of positive solutions for the equation without assuming the existence of upper and lower solution.4.While the analytic semigroup is exponentially stable,we consider the exis-tence and uniqueness of ?-periodic solutions for this equation further.