Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation

In this paper, the existence and uniqueness of solution for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative are studied. The estimation of error between the approximate solution and the solution for such equation is presented by employing the quasilinear iterative method, and an example is given to demonstrate the application of our main result.


(Communicated by Seak Weng Vong)
Abstract. In this paper, the existence and uniqueness of solution for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative are studied. The estimation of error between the approximate solution and the solution for such equation is presented by employing the quasilinear iterative method, and an example is given to demonstrate the application of our main result.
Recently some basic theory for the initial value problems of fractional differential equations involving Riemann-Liouville differential operator (0 < α < 1) has been discussed by Lakshmikantham et al [10,11,12]. In a series of papers (see [3,1]) the authors considered some classes of boundary value problems for differential equations involving Riemann-Liouville and Caputo fractional derivatives of order 0 < α < 1 and 2 < α < 3.
In this paper, we propose a quasilinear iterative method for the the estimations of error between the approximate solution and the solution of BVP (1), and an example is given to demonstrate the application of our main result.

2.
Preliminaries. In order to state our result, some notions and lemmas are given in this section. We denote by C(J, R) the Banach space of all continuous functions from J into R with the norm where Γ is the gamma function.

Lemma 2.4. [21]
Let α > 0. Then In particular, if a = 0, we have As a consequence of Lemma 2.4, we have the following result.
3. Main result. We consider the BVP (1). As a consequence of Lemma 2.5, it is easy to verify the following results. where G(t, s) is Green's function of BVP (1).
In fact, the approximate solution z(t) can be expressed as in which has only some limited discontinuous − points of the f irst kind}.
For x(t) ∈ R α , we define the norm It is easy to verify R α is a Banach space .
(ii) The sequence {x m (t)} converges to the unique solution z * (t) of the BVP (1).
(iii) A bound on the error is given by Proof. Firstly, we shall prove that {x m (t)} ⊆ S(z, N 1 ). From Lemma 3.1, defining an implicit operator T maps R α into itself, such that x m+1 = T x m , obviously, by norm of R α , ∀ x(t) ∈ D, we have (see [14]): whose form is patterned on the integral equation representation of (8), (9). Since x 0 (t) = z(t) ∈ S(z, N 1 ), we shall show that if x m (t) ∈ S(z, N 1 ), then (T x m )(t) ∈ S(z, N 1 ). Let x m (t) ∈ S(z, N 1 ), implies that x m (t) ∈ D, and hence from (8) and (13), we have Thus, it follows from Lemma 3.2 and (H1), (H2), (H3) that And hence Furthermore, we have N 1 ). Known by the mathematical induction, the part (i) follows. Next to prove part (ii). From (13) and x m+1 = T x m , we have Thus, from Lemma 3.2, (H1) and part (i), we get Furthermore, we obtain By induction, we have Because of ϑ β + βϑ < 1, the {x m (t)} is a Cauchy sequence, and converges to z * (t) ∈ S(z, N 1 ). It can be easily verified that z * (t) is the unique solution of (1).