Monotone Iterative Method for Fractional Differential Equations with Integral Boundary Conditions

<jats:p>In this paper, the existence of extremal solutions for fractional differential equations with integral boundary conditions is obtained by using the monotone iteration technique and the method of upper and lower solutions. The main equations studied are as follows:<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mfenced open="{" close="" separators="|"><mml:mrow><mml:mtable class="cases"><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mo>−</mml:mo><mml:msubsup><mml:mi>D</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>+</mml:mo></mml:mrow><mml:mi>α</mml:mi></mml:msubsup><mml:mi>u</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mfenced open="[" close="]" separators="|"><mml:mrow><mml:mn>0,1</mml:mn></mml:mrow></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mi>u</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mi>u</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mrow><mml:msubsup><mml:mo stretchy="false">∫</mml:mo><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:msubsup><mml:mrow><mml:mi>u</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mtext>d</mml:mtext><mml:mi>A</mml:mi></mml:mrow></mml:mrow></mml:mstyle><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:math>where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:msubsup><mml:mi>D</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>+</mml:mo></mml:mrow><mml:mi>α</mml:mi></mml:msubsup></mml:mrow></mml:math>is the standard Riemann–Liouville fractional derivative of order<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mn>1,2</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>A</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:math>is a positive measure function. Moreover, an example is given to illustrate the main results.</jats:p>

Based on the upper and lower solutions, this paper presents a method to prove the existence of solutions of Riemann-Liouville fractional differential equation (1). By using the monotone iteration technique coupled with the upper and lower solution method, a new comparison principle is established and the existence of the extremal solution of integral boundary value problems (1) is proved. is paper is mainly divided into the following two parts: Section 2 mainly introduces the preparation of this article, and then in Section 3, the monotone sequence of solutions is constructed, and the main result of integral boundary value problems (1) is given.

Preliminaries
In this section, we will briefly introduce some of the necessary definitions and results that will be used in the main results.
Definition 4. We say that v ∈ E is a lower solution of (1) if it satisfies It is easy to check that (see [57,58]) erefore, there exists a unique number b * > 0 such that ) is the Mittag-Leffler function (see [2,5]).
In this article, we list the following assumption for convenience: (H3): assume that w 0 , v 0 ∈ E are the upper and lower solutions of problem (1), respectively, Next, we will consider the following auxiliary linear boundary value problem:

Lemma 1. Suppose that (H1) and (H2) hold and y
en, fractional boundary value problem (9) has the following unique solution: where Proof. e main idea of Lemma 1 comes from [57]. By [2], we first find the solution of the fractional differential equations with two-point boundary condition which can be expressed by 2

Journal of Function Spaces
Hence, we conclude that which completes the proof.

Main Results
For

Theorem 1. Suppose (H1)-(H4) hold, and then there exist monotone iterative sequences
and v * , w * are a minimal and a maximal solution of (1) in Proof. For v n− 1 , w n− 1 ∈ E, n ≥ 1, we define two sequences v n , w n ⊂ E satisfying the following fractional differential equation: By consideration of Lemma 1, for any n ≥ 1, problems (28) and (29) have a unique solution v n+1 (t), w n+1 (t), respectively, which are well defined.
irdly, we prove that w 1 , v 1 are upper and lower solutions of problem (1), respectively. Note that And by assumption . is shows that v 1 is a lower solution of problem (1). Similarly, we can infer that w 1 is an upper solution of (1).
Using mathematical induction, it is easy to verify that Clearly, it is easy to conclude that v n and w n are uniformly bounded in C[0, 1]. Moreover, by Lemma 1, problems (28) and (29) are equivalent to the following integral equation: respectively. erefore, the continuity of the functions K(t, s) allows us to conclude that v n and w n are equicontinuous in C[0, 1]. Using (28) and (29) again, we know that D α 0+ v n and D α 0+ w n are uniformly bounded and equicontinuous in C[0, 1]. So, v n and w n are uniformly bounded and equicontinuous in E. Using the standard arguments, we have v n and w n converging, say, to v * and w * , uniformly on [0, 1], respectively. at is, Furthermore, v * (t) and w * (t) are the solutions of problem (1), Next, we need to prove that v * and w * are extremal solutions of (1) en, by assumption (H4), we obtain Applying mathematical induction, one has v n (t) ≤ u(t) ≤ w n (t) on [0, 1] for any n. Taking the limit, we e proof is complete.

Journal of Function Spaces
Data Availability e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.