In this paper the quasilinearization method is extended to finite systems of Riemann-Liouville fractional differential equations of order \(0\lt q\lt 1\). Existence and comparison results of the linear Riemann-Liouville fractional differential systems are recalled and modified where necessary. Using upper and lower solutions, sequences are constructed that are monotonic such that the weighted sequences converge uniformly and quadratically to the unique solution of the system. A numerical example illustrating the main result is given.

In this work we investigate integro-differential initial value problems with Riemann Liouville fractional derivatives where the forcing function is a sum of an increasing function and a decreasing function.We will apply the method of lower and upper solutions and develop two monotone iterative techniques by constructing two sequences that converge uniformly and monotonically to minimal and maximal solutions.In the first theorem we will construct two natural sequences and in the second theorem we will construct two intertwined sequences.Finally, we illustrate our results with an example.

The purpose of this work is to develop a Monotone Method for the anti-periodic boundary value problem with 0 < q < 1 on T = [0, T], (c)D(q)u(t) = f(t, u(t)) + g(t, u(t)), u(0) = -u(T), where f (t, u) is increasing in u and g(t, u) is decreasing in u. We will define coupled lower and upper solutions v(0)(t) and w(0)(t). Next we will construct two sequences {v(n)(t)}, {w(n)(t)} which converge uniformly and monotonically to coupled minimal and maximal solutions rho and r, respectively; i.e. rho and r satisfy the system D-c(q) rho(t) = f (t, rho(t)) + g(t, r(t)), rho(0) = -r(T), (c)D(q)r(t) = f (t, r(t)) + g(t, rho(t)), r(0) = -rho(T). Our iterates are solutions of initial value problems.

We develop a generalized monotone method using coupled lower and upper solutions for Caputo fractional differential equations with periodic boundary conditions of order , where . We develop results which provide natural monotone sequences or intertwined monotone sequences which converge uniformly and monotonically to coupled minimal and maximal periodic solutions. However, these monotone iterates are solutions of linear initial value problems which are easier to compute.

In this paper, by using upper and lower solutions, we develop monotone method for the nonlinear Caputo fractional boundary value problem of order a where 1 < alpha < 2. We construct two sequences which converge uniformly and monotonically to the extremal solutions of the nonlinear Caputo fractional boundary value problem.

In this work we develop monotone method for fractional boundary value problem of order α where 1 < α < 2. For that purpose we develop some basic fractional calculus results which are used as tools to develop comparison results relative to upper and lower solutions of Caputo fractional boundary value problems. Our results include the comparison theorem given in [2] for derivative of order α. We have also computed the Green’s function for the linear Caputo fractional boundary value problem. The integral representation of the linear fractional nonhomogeneous problem is useful in developing monotone method for the Caputo fractional boundary value problem.

In this paper we develop Monotone Method using upper and lower solutions for fractional differential equations with periodic boundary conditions. Initially we develop a comparison result and prove that the solution of the linear fractional differential equation with periodic boundary condition exists and is unique. Using this we develop iterates which converge uniformly monotonically to minimal and maximal solutions of the nonlinear fractional differential equations with periodic boundary conditions in the weighted norm.