Iterative Approximation of the Minimal and Maximal Positive Solutions for Multipoint Fractional Boundary Value Problem on an Unbounded Domain

The fractional calculus has been recognized as an effective modeling methodology for describing hereditary properties of various materials and processes widely. For a lot of applications, we refer the reader to the books [1–5]. For some new development on the topic, see [6–17] and the references therein. Recently, there has been a significant development on boundary value problems for fractional differential equations on infinite intervals; see papers [18–26], in which authors are devoted to investigating the existence of solutions and positive solutions by employing some fixed point theorems, Leray-Schauder nonlinear alternative theorem, or fixed point index theory. By using Schauder’s fixed point theorem combined with the diagonalization method, Arara et al. [18] studied the existence of the bounded solution of the following problem on infinite intervals:

Recently, there has been a significant development on boundary value problems for fractional differential equations on infinite intervals; see papers [18][19][20][21][22][23][24][25][26], in which authors are devoted to investigating the existence of solutions and positive solutions by employing some fixed point theorems, Leray-Schauder nonlinear alternative theorem, or fixed point index theory.

0+
is the Caputo fractional derivative of order .
However, very interesting and important question is "If we know the existence of the solution, how can we find it?"This question motivates us to reconsider problem (3).In this paper, we not only establish the existence of two positive solutions for problem (3), but also develop two computable explicit monotone iterative sequences for approximating the minimal and maximal positive solutions of (3), which is indeed an important and useful contribution to the existing literature on the topic.In addition, to start our work, we employ the monotone iterative method, which is different from the ones used in [18][19][20][21][22][23][24][25][26].Let us state that this method was widely used for nonlinear problem; see, for instance, [27][28][29][30][31][32][33][34][35][36][37][38].

Preliminaries and Several Lemmas
In this section, we present some useful definitions and related theorems.
Definition 1 (see [2]).The Riemann-Liouville fractional derivative of order  for a continuous function  is defined by provided the right-hand side is pointwise defined on (0, ∞) and [] is the integer part of .
Definition 2 (see [2]).The Riemann-Liouville fractional integral of order  for a function  is defined as provided that such integral exists.

Main Results
In this section, we shall construct two explicit monotone iterative sequences which converge to the minimal and maximal positive solutions of (3).
Similarly, we can obtain Since Since (, 0) ̸ ≡ 0, for all  ∈ , then 0 is not a solution of problem (3).Thus, by (28), we know that  * and V * are the maximal and minimal positive solutions of (3) in (0,  −1 ], which can be obtained by the corresponding iterative sequences in (17).
This completes the proof.