The uniqueness of a positive solution to a higher-order nonlinear fractional differential equation with fractional multi-point boundary conditions

In this paper, we apply the iterative method to establish the existence of the positive solution for a type of nonlinear singular higher-order fractional differential equation with fractional multi-point boundary conditions. Explicit iterative sequences are given to approximate the solutions and the error estimations are also given. The result is illustrated with an example.

The first definition of fractional derivative was introduced at the end of the nineteenth century by Liouville and Riemann, but the concept of non-integer derivative and integral, as a generalization of the traditional integer order differential and integral calculus, was mentioned already in 1695 by Leibniz and L'Hospital.In fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes.The mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode's analysis of feedback amplifiers, capacitor theory, electrical circuits, electro-analytical chemistry, biology, control theory, fitting of experimental data, involves derivatives of fractional order.In consequence, the subject of fractional differential equations is gaining much importance and attention.For more details we refer the reader to [2,5,6,7,19,9,21,25,26,29,30,31,36] and the references cited therein.
Boundary value problems for nonlinear differential equations arise in a variety of areas of applied mathematics, physics and variational problems of control theory.A point of central importance in the study of nonlinear boundary value problems is to understand how the properties of nonlinearity in a problem influence the nature of the solutions to the boundary value problems.The multi-point boundary conditions are important in various physical problems of applied science when the controllers at the end points of the interval (under consideration) dissipate or add energy according to the sensors located, at intermediate points, see [8,10,18,32,35] and the references therein.
The techniques of nonlinear analysis, as the main method to deal with the problems of nonlinear fractional differential equations, plays an essential role in the research of this field, such as establishing the existence and the uniqueness or the multiplicity of solutions for nonlinear fractional differential equations, see [1,3,4,6,11,12,13,14,15,16,17,22,24,27,28,34,37,38,39] and the references therein.For instance, Zhang et al. [24] studied the existence of two positive solutions of following singular fractional boundary value problems: where D α 0+ , D β 0+ are the stantard Riemann-Liouville fractional derivative of order α ∈ In [23], the authors studied the boundary value problems of the fractional order differential equation: 0+ are the stantard Riemann-Liouville fractional derivative of order α.They obtained the multiple positive solutions by the Leray-Schauder nonlinear alternative and the fixed point theorem on cones.
Inspired and motivated by the works mentioned above, we focus on the uniqueness of positive solutions for the nonlocal boundary value problem (1.1) − (1.2) with the iterative method and properties of f (t, u), explicit iterative sequences are given to approximate the solutions and the error estimations are also given.The rest of this paper is organized as follows.After this section, we present some notations and lemmas that will be used to prove our main result in Section 2. We discuss the uniqueness in Section 3. Finally, we give an example to illustrate our result.

Preliminaries
In this section, we recall some definitions and facts which will be used in the later analysis.Definition 2.1.[31].The Riemann-Liouville fractional integral of order α > 0 of a function u : (0, ∞) → R is given by where Γ (•) is the Euler gamma function.
In the following Lemma, we present the Green function of fractional differential equation boundary value problem is given by where ) Proof.By using Lemma 2.4, the solution of the equation where c 1 , c 2 ..., c n are arbitrary real constants.By the boundary condition (2.1), one can Then, the unique solution of the problem (2.1) is given by The proof is complete.

Existence results
First, for the uniqueness results of problem (1.1) − (1.2), we need the following assumptions.
(A 2 ) For any r ∈ (0, 1), there exists a constant q ∈ (0, 1) such that f (t, ru) ≥ r q f (t, u) , (t, u) ∈ (0, 1) × [0, ∞) . (3.1) where In view of Lemma 2.5, we define an operator T as where G (t, s) is given by (2.3).By (A 1 ) it is easy to see that the operator T : 2) has a solution if and only if the operator T has a fixed point.Obviously, from (A 1 ) we obtain In what follows, we first prove T : D → D. In fact, for any u ∈ D, there exist a positive constants Then, from (A 1 ), f (t, u) non-decreasing respect to u and (A 2 ), we can imply that for s ∈ (0, 1) , q ∈ (0, 1) From (3.3) and Lemma 2.1, we obtain and for any initial function h 0 (t) ∈ D, then {h n (t)} must converge to u * (t) uniformly on [0, 1] and the rate of convergence is max where 0 < θ < 1, which depends on the initial function h 0 (t).
This together with (3.20) and uniqueness of limit imply that u * satisfy u * = T u * , that is u * ∈ D is a solution of BVP (1.1) − (1.2).