The Exact Iterative Solution of Fractional Differential Equation with Nonlocal Boundary Value Conditions

We deal with a singular nonlocal fractional differential equation with Riemann-Stieltjes integral conditions. The exact iterative solution is established under the iterative technique. The iterative sequences have been proved to converge uniformly to the exact solution, and estimation of the approximation error and the convergence rate have been derived. An example is also given to demonstrate the results.


Introduction
Fractional differential equations arise in many engineering and scientific disciplines; see [1][2][3][4][5].Much attention has been paid to study fractional differential equations both with initial and boundary conditions; see, for example, [6,7].In [8,9], they focused on sign-changing solution for some fractional differential equations.In [10], they get the existence of solutions for impulsive fractional differential equations.In [11][12][13], they get the existence and multiplicity of nontrivial solutions for a class of fractional differential equations.The mainly techniques authors need are fixed point theory, variational method, and global bifurcation techniques.
In [23], authors obtained results on the uniqueness of positive solution for problem D   () +  ()  (,  ()) +  () = 0,  ∈ (0, 1) , where 2 <  ≤ 3 is a real number.Under the assumption that where  ∈ [0,1), and  is the first eigenvalue of the corresponding linear operator.Motivated by the above works, we study the following nonlocal boundary value problems: where    denotes the left-handed Riemann-Liouville derivative of order q and 2 <  ≤ 3 is a real number.[] = ∫ 1 0 ()Λ() denotes a Stieltjes integral with a suitable function Λ of bounded variation.Different from [23] and other works, we only use the iterative methods to obtain the existence and uniqueness of positive solution.Moreover, the estimation of the approximation error and the convergence rate have also been derived.
For clarity in presentation, we also list below some assumptions to be used later in the paper.

Preliminaries
For the convenience of the reader, we present here some necessary definitions from fractional calculus theory.These definitions and properties can be found in the recent monograph [23].
Definition 1.The Riemann-Liouville fractional integral of order  > 0 of a function  : (0, ∞) →  is given by provided that the right-hand side is pointwise defined on (0, ∞).
Proof.We have the estimation where 13) holds.

The Main Results
Throughout this paper, we will work in the space  = [0, 1], which is a Banach space if it is endowed with the norm ‖  ‖= max ∈[0,1] |()| for any  ∈ .
Define the set  in  as follows: And define the operator  :  → .
Theorem 7. Assume that ( 1 )-( 3 ) hold.And Then BVP ( 3) has at least one positive solution (), and there exist constants 0 <   < 1 <   satisfying Proof.It is clear that  is a solution of (3) if and only if  is a fixed point of .
On the other hand, and since  0 ≤ V 0 and  is nondecreasing, by induction, (26) holds.
Let  0 = /, and then 0 <  0 < 1.It follows from (4) that And for any natural number , Thus, for any natural number  and  * , we have which implies that there exists  * ∈  such that (27) holds and Claim 2 holds.
Letting  → ∞ in   =  −1 and noting the fact that  is continuous, we obtain  * () =  * (), which is a positive solution of BVP (3).The proof of Theorem 7 is now complete.Theorem 8. Assume that ( 1 )-( 3 ) hold.Then (i) BVP ( 3) has unique positive solution  * (), and there exist constants ,  with 0 <  < 1 <  such that (ii) For any initial value  0 ∈ , there exists a sequence   () that uniformly converges to the unique positive solution  * (), and one has the error estimation where  is a constant with 0 <  < 1 and determined by  0 .
(i) It follows from Theorem 7 that BVP (3) has a positive solution  * () ∈ , which implies that there exist constants  and  with 0 < l <  < 1 such that  * () satisfies (18).Let V * () be another positive solution of BVP (3); then from Theorem 7 we have that there exist constants  1 and  2 with 0 <  1 < 1 <  2 such that Let  defined in (23) be small enough such that  <  1 and  defined in (23) be large enough such that  >  2 .Then Note that V * = V * and  is nondecreasing; we have Letting  → ∞ in (36), we obtain that V * =  * .Hence, the positive solution of BVP (3) is unique.(ii) From (i), we know that the positive solution  * to BVP (3) is unique.For any  0 ∈ , there exist constants  0 and  0 with 0 <  0 < 1 <  0 such that Similar to (i), we can let  and  defined by (23) satisfy  <  0 and  >  0 .Then Let   =  −1 ,  = 1, 2, . . . .Note that  is nondecreasing; we have Letting  → ∞ in (39), it follows that   uniformly converges to the unique positive solution  * for BVP (3), where At the same time, (33) follows from (31).Thus, the proof of the theorem is complete.