Multiple Positive Solutions for Fractional Three-Point Boundary Value Problem with p-Laplacian Operator

Nowadays, fractional calculus has been adapted to numerous fields, such as engineering, mechanics, physics, chemistry, and biology. Many essays and mongraphs studying various issues in fractional calculus have been researched (see [1–6]). In particular, fractional differential equations have been found to be a powerful tool in modeling various phenomena in many areas of science and engineering such as physics, fluid mechanics, and heat conduction. More details about research achievement on fractional differential equations and their applications are shown in [7–10]. Recently, fractional differential equations have gained considerable attention (see [11–15] and the references therein). Fractional differential equations and differential equations with p-Laplacian operators have attracted much attention from many mathematicians. As a result, meaningful research results have been drawn [16–20]. Fractional-order boundary value problems involving classical, multipoint, high-order, and integral boundary conditions have extensively been studied by many researchers and a variety of results can be found in recent literature on the topic [21–25]. In [15], Chai studied the boundary value problems of fractional differential equations with p-Laplacian operator as follows: D β 0+ φp D0+ u(t) ( 􏼁 􏼐 􏼑 + f(t, u(t)) � 0, 0< t< 1,


Introduction
Nowadays, fractional calculus has been adapted to numerous fields, such as engineering, mechanics, physics, chemistry, and biology. Many essays and mongraphs studying various issues in fractional calculus have been researched (see [1][2][3][4][5][6]). In particular, fractional differential equations have been found to be a powerful tool in modeling various phenomena in many areas of science and engineering such as physics, fluid mechanics, and heat conduction. More details about research achievement on fractional differential equations and their applications are shown in [7][8][9][10].
In [15], Chai studied the boundary value problems of fractional differential equations with p-Laplacian operator as follows: Some existence results of positive solutions are obtained by using the monotone iterative method.
In [16], by using Krasnosel'skii's fixed point theorem, Tian et al. obtained the existence of positive solutions for a boundary value problem of fractional differential equations with p-Laplacian operator as follows: In [17], by using the monotone iterative method, Tian et al. obtained the existence of positive solutions for a boundary value problem of fractional differential equations with p-Laplacian operator as follows: In [18], by means of the p-Laplacian operator, Han et al. obtained the existence of positive solutions for the boundary value problem of fractional differential equation as follows: where ϕ(s) � |s| p− 2 s, p > 1, α ∈ (2, 3], β ∈ (1, 2], f: (0, +∞) ⟶ (0, +∞) is continuous, and D α 0 + , D β 0 + are the Riemann-Liouville fractional derivatives.
Based on the above research, this paper analyzed the following fractional three-point boundary value problem with the p-Laplacian operator: e aim is to establish some existence and multiplicity results of positive solutions for BVP (5). is paper is organized as follows. In Section 2, some properties of Green's function will be given, which are needed later. In Section 3, the existence of multiplicity results of positive solutions of BVP will be discussed (5).

Preliminary Knowledge and Lemmas
Lemma 1 (see [14]). Assume that D α a + ∈ L 1 (a, b) with a fractional derivative of order α > 0. en, for some c i ∈ R, i � 1, 2, . . . , n, where n is the smallest integer greater than or equal to α .

Main Results
When E � [0, 1], any u ∈ E, ‖u‖ � max 0≤t≤1 |u(t)|, then E is a real Banach space. K ∈ E is a cone, which can be defined as K � u ∈ E: min t∈[0,1] u(t) ≥ 0, min t∈I u(t) ≥ (1/4) α− 1 ‖u‖ . Defining the operator T: E ⟶ E, for any u ∈ E, and for convenience, the following notation is introduced: (24) Theorem 1. If there are two positive numbers 0 < r 1 < r 2 such that the following conditions hold: then the fractional three-point boundary value problem (5) has at least one positive solution u and r 1 ≤ ‖u‖ ≤ r 2 .
Proof. From the continuity of G, H, f, it can be concluded that T: K ⟶ K is continuous. For (t, s) ∈ I × (0, 1), u ∈ K, by Lemma 4, we have min t∈I Tu(t) � min It means that T(K) ⊂ K. erefore, the Arzela-Ascoli theorem can prove that the operator T: K ⟶ K is completely continuous.
Let Ω 2 � u ∈ K: ‖u‖ ≤ λ 2 , for u ∈zΩ 2 . en, we also can conclude from Lemma 4 and (B 2 ) that (27) erefore, for u ∈zΩ 2 , ‖Tu‖ ≤ ‖u‖. In summary, by Lemma 5, the fractional three-point boundary value problem (5) has at least one positive solution u and r 1 ≤ ‖u‖ ≤ r 2 . □ Theorem 2. If there exist positive real numbers 0 < c < d < (1/4) α− 1 r such that the following conditions hold: then the fractional three-point boundary value problem (5) has at least three positive solutions u 1 , u 2 , and u 3 with Proof. Firstly, if u ∈ K r , then we may assert that T: K r ⟶ K r is a completely continuous operator. To see this, suppose u ∈ K r ; then, ‖u‖ ≤ r. It follows from Lemma 4 and (B 3 ) that (29) erefore, T: K r ⟶ K r . is together with Lemma 5 implies that T: K r ⟶ K r is a completely continuous operator. In the same way, if u ∈ K c , then assumption (B 5 ) yields ‖Tu‖ < c. Hence, condition (ii) of Lemma 6 is satisfied.
To sum up, all the conditions of Lemma 6 are satisfied; from Lemma 6, it follows that there exist three positive solutions u 1 , u 2 , and u 3 with

Conclusion
e existence of solutions to three-point boundary value problems of fractional differential equations with the p-Laplacian operators is discussed by using the fixed point exponential theorem and fixed point theorem of cone compression and cone tension. By extending the existence of solutions to boundary value problems, we have obtained the sufficient condition that the boundary value problem has multiple positive solutions or at least one positive solution.

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

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