EXISTENCE AND APPROXIMATION OF SOLUTIONS TO THREE-POINT BOUNDARY VALUE PROBLEMS FOR FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, we study existence and approximation of solutions to some three- point boundary value problems for fractional differential equations of the type c D q 0+u(t) + f(t,u(t)) = 0, t 2 (0,1),1 < q < 2 u 0 (0) = 0, u(1) = �u(�), where 0 < �, � 2 (0,1) and c D q is the fractional derivative in the sense of Caputo. For the existence of solution, we develop the method of upper and lower solutions and for the approximation of solutions, we develop the generalized quasilinearization technique (GQT). The GQT generates a monotone sequence of solutions of linear problems that converges monotonically and quadratically to solution of the original nonlinear problem.


Introduction
The study of fractional differential equations is of fundamental concern due to its important applications to real world problems. Many problems in applied sciences such as engineering and physics can be modeled by differential equations of fractional order [1,2,3]. It has been observed that the models with fractional differential equations provide more realistic and accurate results compared to the analogous models with integer order derivatives, see, [4,5]. Existence theory for solutions to boundary value problems for fractional differential equations have attracted the attention of many researcher quite recently, see for example [6,7,8,9,10,11,12] and the references therein. However, the method of upper and lower solutions for the existence of solution is less developed and hardly few results can be found in the literature dealing with the upper and lower solutions method to boundary value problems for fractional differential equations [13,14,15,16,17]. The method of quasilinearization is somewhat developed for initial value problems for fractional differential equations [18,19,20,21] but results dealing with quasilinearization to boundary value problems for fractional differential equations can hardly be seen in the literature. The paper seem to be an attempt to develop the generalized quasilinearization to three-point boundary value problems for fractional differential equations.

Preliminaries
We recall some basic definitions and lemmas from fractional calculus [4].
Definition 2.1. The fractional integral of order q > 0 of a function g : (0, ∞) → R is defined by provided the integral converges.
Definition 2.2. The Caputo fractional derivative of order q > 0 of a function g ∈ AC m [0, 1] is defined by provided that the right side is pointwise defined on (0, ∞).
has a unique solution given by Comparison results:

Main Results
Proof. Define the modification of f , Clearly, F is continuous, bounded on [0, 1] × R and is non-decreasing with respect to u for each fixed t ∈ [0, 1]. Hence, the modified BVP , is a solution of the BVP (2.1). We only need to show that where u is solution of the BVP (3.2). In view of the nondecreasing property of f , we obtain where u, y ∈ [ᾱ,β]. Using the non decreasing property of φ u with respect to u on [ᾱ,β] for each t ∈ [0, 1], we obtain where u, y ∈ [ᾱ,β] such that y ≤ c ≤ u. Substituting in (3.6), we have EJQTDE, 2011 No. 58, p. 4 We note that g(t, u, y) is continuous on [0, 1] × R × R and for u, y ∈ [ᾱ,β], using (3.8) and (3.9), we have Now, we develop the iterative scheme to approximate the solution. As an initial approximation, we choose w 0 = α and consider the linear problem The definition of lower and upper solutions and (3.10) imply that which imply that w 0 and β are lower and upper solutions of (3.11). Hence by Theorem 3.1, there exists a solution w 1 ∈ C[0, 1] of (3.11) such that w 0 ≤ w 1 ≤ β on [0, 1]. Again, from (3.10) and the fact that w 1 is a solution of (3.11), we obtain − c D q 0+ w 1 (t) = g(t, w 1 (t), w 0 (t)) ≤ f (t, w 1 (t)), t ∈ [0, 1], w ′ 1 (0) = 0, w 1 (1) = ξw 1 (η) (3.12) which implies that w 1 is a lower solution of (2.1).
The monotonicity and uniform boundedness of the sequence {w n } implies the existence of a pointwise limit w on [0, 1] such that w n → w uniformly. The dominated convergence theorem implies that for each t ∈ [0, 1], 1 0 G(t, s)g(s, w n (s), w n−1 (s))ds → 1 0 G(t, s)f (s, w(s))ds.