Resolvents for weakly singular kernels and fractional differential equations

The solution R(t,s)R(t,s) of the resolvent equation equation(1)R(t,s)=B(t,s)+∫stB(t,u)R(u,s)du, where B(t,s)B(t,s) denotes a given weakly singular matrix, is obtained by means of fixed point mappings. The result is a series that begins with some singular terms after which the remainder of the terms defines a continuous function. In particular, the resolvent is calculated for the kernel B(t,s)=λ(t−s)q−1B(t,s)=λ(t−s)q−1 of the scalar Abel integral equation of the second kind equation(2)x(t)=a(t)+λ∫0t1(t−s)1−qx(s)ds where 0