C .-solutions for the second type of generalized Feigenbaum.s functional equations

This work focuses on the second type of generalized Feigenbaum.s equation $\left\{ .(f(x))=f(f(.(x))),f(0)=1,0.f(x).1,x.[0,1], \right. $ where .(x) is C .-increasing function on [0, 1] and satisfies that .(0) = 0,0 < ..(x) < 1 (x . [0, 1]). Using constructive method, we discuss the existence of C .-single-valley solutions whose derivatives are not equal to 0 on origin of the above equation.