Monotone Iterative Technique for Systems of Nonlinear Caputo Fractional Differential Equations

In this work, we deal with the existence of extremal quasisolutions for the following finite system of nonlinear fractional differential equation (C)D(q)u (t) + f (t, u (t)) = 0 in (0, 1), u(0) - alpha u' (0) = lambda, u(1) + beta u'(1) = mu,where 1 < q < 2,alpha,beta is an element of (R+)(n) , lambda, mu epsilon R-n and f is an element of C([0, 1] x R-n ,R-n) and D-C(q) is the Caputo fractional derivative of order q. We shall prove constructive existence results for a class of nonlinear equations by the use of iterative method technique combined with upper and lower quasisolutions. We construct a pair of sequences of coupled lower and upper quasisolutions which converge uniformly to extremal quasisolutions. Then, a uniqueness result is given under additional conditions on the nonlinearity f.