Iterative Solutions to a Coupled System of Non-linear Fractional Differential Equation

In this article we investigate the existence of extremal(maximal and minimal) solutions for the following coupled system of integrodifferential equations of fractional order with the given boundary conditions of the form  Du(t) + f1(t, v(t), I v(t)) = 0, 0 < t < 1, Dv(t) + f2(t, u(t), I u(t)) = 0, 0 < t < 1, u(0) = 0, u′(1) = 0, v(0) = 0, v′(1) = 0. where 1 < α, β ≤ 2 and f1, f2 : [0, 1] × R × R → R are given functions satisfying Caratheodory conditions and D is standing for Riemann-liouvilledifferentiation of fractional order. We find some sufficient conditions for the existence and uniqueness of maximal and minimal solutions by using monotone iterative technique along with the method of upper and lower solutions. We also link our analysis for the problem to equivalent integral equations. We also give the error estimate and an example for the illustration of our results.