Positive Solutions for BVP of Fractional Differential Equation with Integral Boundary Conditions

In this paper, we consider a class of boundary value problems of nonlinear fractional differential equation with integral boundary conditions. By applying the monotone iterative method and some inequalities associated with Green’s function, we obtain the existence of minimal and maximal positive solutions and establish two iterative sequences for approximating the solutions to the above problem. It is worth mentioning that these iterative sequences start off with zero function or linear function, which is useful and feasible for computational purpose. An example is also included to illustrate the main result of this paper.


Introduction
Fractional calculus has widespread applications in many fields of science and engineering, for example, viscoelasticity, continuum mechanics, bioengineering, rheology, electrical networks, control theory of dynamical systems, and optics and signal processing [1,2].
Recently, the monotone iterative method has been applied to study BVPs of nonlinear fractional differential equations. For example, in [20], Cui et al. discussed the BVP D q 0+ u (t) + f(t, u(t)) � 0, t ∈ (0, 1), where 2 < q ≤ 3 and D q 0+ denotes the standard Riemann-Liouville fractional derivative of order q. e authors obtained the existence of maximal and minimal solutions and the uniqueness result for BVP (1). In 2014, Sun and Zhao [21] investigated the following BVP with integral boundary conditions: where 2 < q ≤ 3 and D q 0+ is the standard Riemann-Liouville fractional derivative of order q. By means of the monotone iterative method, they proved the existence of a positive solution and established an iterative sequence for approximating the solution to BVP (2). For relevant results, one can refer to [22][23][24][25].

Preliminaries
First, we present the definitions of Riemann-Liouville fractional integral and fractional derivative and Caputo fractional derivative on a finite interval of the real line, which may be found in [1].
In the remainder of this paper, we always assume that the following condition is fulfilled: Now, we define Lemma 5. H(t, s) has the following property: Proof. In view of the definition of H(t, s) and Lemma 4, it is obvious.

Main Results
For convenience, we let 1] x(t).
(28) Theorem 2. Assume that f(t, 0) ≡ 0 for t ∈ [0, 1] and the following condition is satisfied: en, K is a normal cone in Banach space E. Note that this induces an order relation "

□
Step 1. We assert that T is monotone increasing on

H(t, s)f(s, u(s))ds
is shows that Tu ≾ Tv.
Step 2. We prove that υ 0 is a lower solution of T.
Step 3. We show that ω 0 is an upper solution of T.
Step 4. We claim that BVP (3) possesses a minimal positive solution υ and a maximal positive solution ω.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.