Unique positive solutions for boundary value problem of p -Laplacian fractional differential equation with a sign-changed nonlinearity

. This paper investigates the existence of a unique positive solution for a class of boundary value problems of p -Laplacian fractional differential equations, where its nonlinearity is sign-changed and involves a fractional derivative term, and its boundary involves a nonlinear fractional integral term. By constructing an appropriate auxiliary boundary value problem and applying a generalized ﬁxed point theorem of sum operator and properties of Mittag-Lefﬂer function, some sufﬁcient conditions for the existence of a unique positive solution are presented, and a monotone iterative sequence uniformly converging to the unique solution is constructed. In addition, an example is given to illustrate the main result.


Introduction and preliminaries
Because of the extensive application in many fields such as physics, biology and engineering, etc., fractional differential equation has attracted considerable attention and has become an important area of investigation in differential equation theories. For a small sample of such work, we refer the reader to [1][2][3]11,13,16,20] and the references therein. At the same time, the differential equations with p-Laplacian operator are recognized as important mathematical models in various fields of non-Newtonian mechanics, population biology, elasticity theory, and so forth. More and more emphases have been put on the research of positive solutions for fractional boundary value problems with p-Laplacian operator, and excellent results from research into it emerge continuously. For some recent works on the subject, readers can see [4,7,8,10,12,14,15,17,18,24] and the references therein. In these literature, there are a few papers on the existence of a unique positive solution [7,17,24]. Xu and Dong [24] investigated the existence and uniqueness for the following Riemann-Liouville fractional boundary value problem with p-Laplacian operator x(t) , t ∈ (0, 1), where α ∈ (1, 2], β ∈ (3,4], η ∈ (0, 1), b ∈ (0, η (1−α)/(p−1) ), f ∈ C([0, 1] × R + , R + ).
Their analysis based on the Schauder fixed point theorem, the upper and lower solutions method and the idea of concave and increasing operator theory. But to our knowledge, there are few papers reported on the existence of a unique positive solution for p-Laplacian fractional boundary value problems involving a fractional derivative term in the nonlinearity and a nonlinear integral term in the boundary conditions. As is well known, the existence of a unique positive solution for nonlinear boundary value problems plays a very important role in theory and application, and the fixed point theory of operators with monotonicity and concavity (convexity) is an effective tool to deal with such problems. Many researchers have studied the existence and uniqueness of positive solutions by using different fixed point theorems of operators with monotonicity and concavity (convexity), for example, fixed point theorems of concave (such as ϕ-concave, δ-concave, u 0 -concave, ψ − (h, r)-concave) and increasing operators, see [5,7,17,25]; fixed point theorems of generalized δ-concave and increasing (generalized −δ-convex and decreasing) operators, see [22]; the fixed point theorem of sum operators (i.e., Lemma 7), see [26] and [27]; fixed point theorems of sum operators with concavityconvexity and mixed monotonicity, see [9,28,29]. As usual, while using this tool to study unique positive solutions of a boundary value problem, it is essential to require its nonlinearity to be nonnegative and satisfy monotonicity conditions and concavity (convexity) conditions. But, when the nonlinearity of the boundary value problem is a sign-changed function without monotonicity and concavity (convexity), we want to know whether the boundary value problem has a unique positive solution. More specifically, under what conditions and how to use this tool to prove the existence of the unique positive solution? To the best of authors' knowledge, there are no answers to these questions.
The purpose of this paper is to establish some sufficient conditions for the existence of a unique positive solution of BVP (1) where the nonlinearity f (t, x, y) may be signchanged and has neither monotonicity nor concavity (convexity), and construct a monotone iterative sequence uniformly converging to the unique positive solution. Our analysis relies on the cone theory, properties of Mittag-Leffler function, and a generalized fixed point theorem of a sum operator defined on an equivalence class in cone.
Secondly, since f involves D β 0 + x, the partially order used in this paper is related to D β 0 + x. In addition, note that BVP (1) involves φ p and I ω 0 + g(ξ, x(ξ)), so it is difficult to construct a valid equivalence class in cone, but it is essential for our work. Finally, to our knowledge, fixed point theorems in the existing literature can not be directly applied to our analysis.
The paper is organized as follows. In Section 2, we recall some useful preliminaries and lemmas. In particular, we generalize a fixed point theorem of sum operators on cone. In Section 3, based on the generalized fixed point theorem, some results on the existence of a unique positive solution for BVP (1) are presented and proved. In Section 4, an example is given to illustrate our main result.

Preliminaries and fixed point theorems
where c 1 , c 2 , . . . , c n ∈ R, provided the above integral exists.
The following result can be easily derived by Lemma 1.
Arguing similarly to the proof of Lemma 1 in [19], we can show the following result.
If Γ(β + ω) = µξ β+ω−1 , then the fractional boundary value problem has a unique solution (7) is equivalent to the following problem: https://www.journals.vu.lt/nonlinear-analysis By Lemma 3 the unique solution of the initial value problem By Lemma 4 the unique solution of the boundary value problem Consequently, problem (8) has a unique solution which is the unique solution of BVP (7). The proof is complete.
Remark 5. According to the proof of Lemma 5, if x is a solution of BVP (7), then It is obvious that Lemma 6 follows from (6).
In the sequel, we present some concepts in ordered Banach spaces, which can be found in [6] and [26].
Let (E, · ) be a real Banach space which is partially ordered by a cone P ⊂ E, that is, x y iff y − x ∈ P . If x y and x = y, then we denote x ≺ y or y x. By θ we denote the zero element of E. A cone P is said to be normal if there exists a constant N > 0 such that θ x y implies x N y . In this case, the smallest constant satisfying this inequality is called the normality constant of P . For all x, y ∈ E, the notation x ∼ y means that there exist l 1 > 0, l 2 > 0 such that l 1 x y l 2 x. Clearly, ∼ is an equivalence relation. Given e θ (i.e., e ∈ P and e = θ), and the equivalence class of the element e is denoted by the set P e , that is, In [26], Zhai and Anderson obtained the following result.
Then the operator equation Ax + Bx = x has a unique solution x * in P e . Moreover, for any initial value x 0 ∈ P e , constructing successively the sequence x n = Ax n−1 + Bx n−1 (n = 1, 2, . . . ), we have lim n→+∞ x n − x * = 0.
However, in this paper, the operator B defined by (18) does not satisfy condition (G1) since Be / ∈ P e for any e θ. Therefore, we need to simply generalize Lemma 7. Set P e = x ∈ E ∃l(x) > 0 such that θ x l(x)e .
Clearly, P e ⊂ P e ⊂ P . So, the following condition (G1 ) is more extensive than (G1).
(G1 ) there is e θ such that Ae ∈ P e , Be ∈ P e .
In order to complete our analysis, we present the following result.
Theorem 1. Let P be a normal cone in E, A : P → P and B : P → P be increasing operators. Assume that (G1 ), (G2), and (G3) hold. Then the operator equation Ax + Bx = x has a unique solution x * in P e . Moreover, for any initial value x 0 ∈ P e , constructing successively the sequence x n = Ax n−1 + Bx n−1 (n = 1, 2, . . . ), we have lim n→+∞ x n − x * = 0.
Proof. Since Ae ∈ P e and Be ∈ P e , it is follows from (9) and (10) that there exist constants l 1 > 0, l 2 > 0 and l 3 > 0 such that l 1 e Ae l 2 e and 0 Be l 3 e, which implies that l 1 e Ae + Be (l 2 + l 3 )e.
So, Ae + Be ∈ P e . Define an operator T = A + B by T x = Ax + Bx, then T : P → P and T e ∈ P e . Next, to show that T (P e ) ⊂ P e . It is easy to see from (G2) that For any x ∈ P e , we can choose a sufficiently small number τ 0 ∈ (0, 1) such that τ 0 e x τ −1 0 e.
Noticing that T : P → P is increasing, we have Since l 2 τ −δ 0 + l 3 τ −1 0 > 0, l 1 τ δ 0 > 0, we get T x ∈ P e , that is, T (P e ) ⊂ P e . The rest of the proof is almost the same as that of Theorem 2.1 in [26]. The proof is complete.

Main results
In this section, by constructing an auxiliary boundary value problem and applying Theorem 1 we obtain some new results on unique positive solution for BVP (1).
Clearly, P is a cone, and X is endowed with a partial order given by the cone P , that is, Moreover, P is a normal cone and the normality constant is 1.
γ * Γ( α p−1 + β + 1) Γ( α p−1 + 1) Moreover, for any x 0 ∈ P , constructing successively the sequence x n (ξ) −µx n (ξ) +k , n = 0, 1, 2, . . . , (13) we have Proof. For any given x ∈ P , consider the auxiliary boundary value problem where L and µ are given in (H1) and (H2), respectively. By Lemma 5, BVP (15) has a unique solution given by Define two operators A and B by https://www.journals.vu.lt/nonlinear-analysis In view of Remark 5, we have Moreover, according to Lemmas 2, 6 and Remark 1, it is easy to show that A : P → P and B : P → P . In addition, from (15)- (18) we can assert that x * ∈ P is a fixed point of A + B if and only if x * is a solution of BVP (1) in P .
In order to get the conclusions, we verify that operators A and B satisfy all assumptions of Theorem 1 in the sequel.
Firstly, we show that operators A and B are increasing. For all x 1 , x 2 ∈ P with Furthermore, by (2) and (4) we obtain that is, Ax 1 Ax 2 and Bx 1 Bx 2 . Secondly, we prove that there exists e θ such that Ae ∈ P e and Be ∈ P e , that is, assumption (G1 ) holds. Set Clearly, which means that e θ.
https://www.journals.vu.lt/nonlinear-analysis Finally, applying Theorem 1, we obtain that A + B has a unique fixed point x * in P e , and for any x 0 ∈ P e , setting x n+1 = Ax n + Bx n , n = 0, 1, 2, . . . , we have lim n→∞ x n+1 − x * = 0. In addition, by using similar arguments as the proof of γ 0 e Ae, we can get γ 0 e Ax for x ∈ P . Moreover, (A + B)(P ) ⊂ P e . Therefore, BVP (1) has a unique positive solution x * satisfying (11) and (12), and for any x 0 ∈ P , we construct successively the sequence {x n+1 } as (13), then (14) is tenable. This ends the proof.
Remark 6. Corollary 1 is the special case of Theorem 2 where L = 0 in (H1), Corollary 2 is the special case of Theorem 2 where µ = 0 in (H2), and Corollary 3 is the special case of Theorem 2 where L = 0 in (H1), and µ = 0 in (H2). Although the above three corollaries are the special cases of Theorem 2, they are still new results.