Existence of an approximate solution for a class of fractional multi-point boundary value problems with the derivative term

where b > 0, Dα0+ , D β 0+ , D γ 0+ , D ν 0+ are the Riemann–Liouville fractional derivatives with n – 1 < α, β ≤ n, n – 2 < γ ≤ n – 1, n ≥ 2 (n ∈ N), α – γ – 1 > 0, 0 < ν ≤ γ , 0 < ξi, ηi, ζi < 1, i = 1, 2, 3, . . . , m – 2, m ≥ 2, ∑m–2 i=1 ξiη –1 i < 1, ∑m–2 i=1 ζiη β–1 i < 1. f , g : [0, 1] × (–∞, +∞) × (–∞, +∞) −→ (–∞, +∞) are continuous. In recent years, much attention has been paid to multi-point boundary value problems involving fractional order; see [1–29] and the references therein. We should mention related studies in [1–22], which motivated us to consider the problem (1.1). Lv [1] studied the existence of positive solutions of the following multi-point boundary value problem:

also investigated positive solutions for the nonlinear fractional differential equation with a derivative term. Goodrich [12] first obtained the Green function of the problem (1.4) when k(u(1)) = 0. In [13][14][15], they considered the fractional differential equations with integer order derivative, and they did not consider the boundary condition [D γ 0 + u(t)] t=1 . Zhang [16] considered the singular fractional differential equations with multiple derivative terms, and obtained the existence of positive solutions.
We should mention the work of Jong [3], which directly is related to our problem (1.1).
Jong investigated the following nonlinear fractional m-point boundary value problem with p-Laplacian operator: < 1, the p-Laplacian operator is defined as ϕ p (s) = |s| p-2 s, p > 1. Jong obtained that the problem (1.5) has a unique solution which is given by He first gave the Green function H(s, τ ). The main tool of [3] is the Banach contraction mapping principle. Furthermore, he also showed the uniqueness of the problem (1.5) in [4] by the classic fixed point theorem of mixed monotone operators. Li and Qi [17] focused on p-Laplacian boundary value problems of higher order nonlinear differential equations. Tan and Li [18] used Kuratowski's noncompactness measure and Sadovskii's fixed point theorem to study the problem (1.5) when the boundary condition ϕ p (D α Wang and Xiang [19] considered the problem (1.5) when all boundary conditions are replaced by Wang, Xiang and Liu [20] investigated the problem (1.5) when the boundary conditions are replaced by u(0) = 0, D α 0 + u(0) = 0 and u(1) = au(ξ ). We should point out that the main tools and methods adopted in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] are cone mapping theory. Therefore, nonlinearities in the problems (1.2)-(1.5) are usually required to be non-negative. But more and more authors are beginning to remove this restriction imposed on nonlinear terms. Very recently, Sang and Ren [21] dealt with the following fractional boundary value problem: are two continuous functions. In fact, Zhai and Wang [22] have introduced ϕ-(h, e) operators, and established the existence and uniqueness of a nontrivial solution for a class of nonlinear fractional equations by using partial order method. In this paper, the first goal is to obtain the fixed point of the solution of the operator equation M(x, x) + N(x, x) + e = x, where M and N are two mixed monotone operators. We will generalize the results of cone mapping to the non-cone case. Then we will provide some sufficient conditions under which the problem (1.1) has a unique solution and construct two iterative sequences of unique solution. Compared with [6,9], we do not demand the assumption that nonlinearities are non-negative, and the more general boundary conditions are adopted.
Our paper is organized as follows. In Sect. 2, we will introduce some definitions and give some lemmas to prove the main conclusions. In Sect. 3, the existence of fixed point for the operator equation associated with the problem (1.1) is established. Then, based on our abstract results, the existence and uniqueness of the solution of the problem (1.1) are proved.

Preliminaries and related lemmas
In this section, we give some definitions and lemmas that are useful for the proof of our main results.
In this paper, (E, · ) is a real Banach space. A partially ordered structure in E is induced by a cone P ⊂ E, i.e. x ≤ y if and only if yx ∈ P. θ is the zero element. P is called normal if there exists N > 0 such that θ ≤ x ≤ y ⇒ x ≤ N y . Given h > θ , we denote by P h where n = [α] + 1. The Riemann-Liouville fractional integral of order α > 0 of a function h is given by
be a continuous function. Then the problem (1.1) has the following unique solution: in which , consider the boundary value problem: Similarly, using Lemma 2.1, we deduce It follows from the condition v(0) = v (0) = · · · = v (n-2) (0) = 0 that c n = c n-1 = · · · = c 2 = 0. Thus The rest of our proof can be derived from Lemma 2.4 in [3].

Lemma 2.4 Let
Then the function G(t, s) defined in Lemma 2.2 satisfies the following conditions: At the same time, we have

Lemma 2.6 Let P be a normal cone and T : P h,e ×P h,e − → E be a mixed monotone operator.
Assume that the following conditions hold: (i) for every λ ∈ (0, 1) and u, v ∈ P h,e , there exists ϕ(λ, u, v) > λ such that Then: (3) for any x 0 , y 0 ∈ P h,e , taking the iterative sequences as follows: we have x n → x * and y n → x * as n → ∞.
Proof By (i), we have for every λ ∈ (0, 1), u, v ∈ P h,e . We can find a positive integer k with
Case 1: there is an integer N such that t N = t * . In this case, we have t n = t * for all n > N . Then We can get t * = t n+1 ≥ ϕ(t * ) > t * from the definition of t n+1 , which is a contradiction.
Case 2: for all n, t n < t * , we have By the definition of t n+1 , we have Let n → ∞, we have t * ≥ ϕ(t * , u 0 , v 0 ) > t * , which is a contradiction. Consequently t * = 1.
Since P is normal, we have where M is the normality constant. Let n → ∞, we get Therefore u n and v n are Cauchy sequences. Repeating the proof of Lemma 2.3 in Sang and Ren [21], we derive that our conclusions hold.

Lemma 2.7
Let P be a normal cone and T : P h,e ×P h,e − → E be a mixed monotone operator.
Assume that the condition (i) in Lemma 2.6 is satisfied. In addition, ϕ(t, u, v) is decreasing in u and increasing in v for every t ∈ (0, 1). Furthermore, there exists t 0 ∈ (0, 1) such that The rest of the proof is similar to that of Lemma 2.6, we omit it here.

Main results
In this section, we will establish the existence and uniqueness of nontrivial solution for the problem (1.1). The main tools are fixed point theorems of an operator equation.
Proof For every x i , y i ∈ P h,e (i = 1, 2) with x 1 ≥ x 2 , y 1 ≤ y 2 , the mixed monotone properties of M(x, y) and N(x, y) lead to Thus, T is a mixed monotone operator. Note that N(h, h) ∈ P h,e , there exist a 1 , a 2 ∈ P h,e such that a 1 h + (a 1 -1)e ≤ N(h, h) ≤ a 2 h + (a 2 -1)e.
Thus condition (i) in Lemma 2.6 is proved. By (L2), for every x 1 ≥ x 2 and y 1 ≤ y 2 with x i , y i ∈ P h,e , i = 1, 2, we have Therefore Thus we deduce condition (ii) in Lemma 2.6 to be met. According to Lemma 2.6, we get the conclusions of Theorem 3.1.
In terms of Lemma 2.7, we can establish the following theorem, which is parallel with Theorem 3.1.
Furthermore, it follows from Lemmas 2.4 and 2.5 that where L ≥ bDF β (α) (β) . Hence, 0 < e(t) ≤ h(t). By Lemma 2.3, we find that the problem (1.1) has the following expression: For every t ∈ [0, 1] and u, v ∈ P h,e , we consider the following operators: (3.4) and (1) Firstly, we show that M, N : P h.e × P h.e → E are two mixed monotone operators. By (H1) and (H2), for every u i , v i ∈ P h,e (i = 1, 2) with u 1 ≥ u 2 , v 1 ≤ v 2 , we have and Hence, M is a mixed monotone operator. Similarly, we deduce and Thus, N is also a mixed monotone operator.
(3) In view of (H6), we have and Thus,   Then l 2 h ≤ N(h, h) + e ≤ l 1 h, thus N(h, h) ∈ P h,e . Therefore, the condition (L4) of Theorem 3.1 is proved.
By the proof of Theorem 3.2, combining with Theorem 3.1 , we can obtain the following result. and y ∈ P h,e , ψ(λ, x, y) are decreasing in x ∈ P h,e and for fixed t ∈ [0, 1] and x ∈ P h,e , ψ(λ, x, y) are increasing in y ∈ P h,e . In addition, there exists t 0 ∈ (0, 1) such that Then the conclusions of Theorem 3.2 hold.
Lastly, let us give an example to illustrate our main results.
Example 3.1 Consider the following boundary value problem:  Then the problem (3.6) has a solution.

Conclusions
In this paper, we obtain two new mixed monotone fixed point theorems. By using our abstract results, we establish the existence and uniqueness theorems of the solution for a fractional m-point boundary value problem, which generalizes the well-known elastic beam equation. Furthermore, two iterative sequences to approximate the unique solution are also given.