SOLUTIONS OF FRACTIONAL HYBRID DIFFERENTIAL EQUATIONS VIA FIXED POINT THEOREMS AND PICARD APPROXIMATIONS

We investigate the following fractional hybrid differential equation: { D-t0(alpha) + [x(t) - integral 1(t, x(t))] = integral 2(t, x(t)) a.e t is an element of J. x(t(0)) = x(0) is an element of R, where D-t0(alpha) + is the Riemann-Liouville differential operator order of a > 0, J = [ t(0), t(0) + a], for some t(0) is an element of R, a > 0, f(1) is an element of C ( J x R, R) f(2) is an element of L(alpha)p ( J x R, R) p >= 1 and satisfies certain conditions. We investigate such equations in two cases: a is an element of ( 0, 1) and a >= 1. In the first case, we prove the existence and uniqueness of a solution of (1.0), which extends the main result of [1]. Moreover, we show that the Picard iteration associated to an operator T : C ( J x R) -> C( J x R) converges to the unique solution of (1.0) for any initial guess x is an element of C( J x R). In particular, the rate of convergence is n(-1) n In the second case, we investigate this equation in the space of k times differentiable functions. Naturally, the initial condition x(t(0)) x(0) is replaced by x((k)),(t0) = x0, 0 <= k <= n(alpha,p)(-1) and the existence and uniqueness of a solution of (1.0) is established. Moreover, the convergence of the Picard iterations to the unique solution of (1.0) is shown. In particular, the rate of convergence is n(-1) n Finally, we provide some examples to show the applicability of the abstract results. These examples cannot be solved by the methods demonstrated in [1].