solving fractional evolution problem in colombeau algebra by mean generalized fixed point

The present paper is devoted to the existence and uniqueness result of the fractional evolution equation dqcu(t)=g(t,u(t))=au(t)+f(t)‎ ‎for the real q∈(0,1) with the initial value u(0)=u0∈r~‎, ‎where r~ is the set of all generalized real numbers and a is an operator defined from g into itself‎. here the caputo fractional derivative dqc is used instead of the usual derivative. the introduction of locally convex spaces is to use their topology in order to define generalized semigroups and generalized fixed points, then to show our requested result.