Existence and uniqueness of nontrivial solution for nonlinear fractional multi-point boundary value problem with a parameter

where b > 0, Dα0+ and D β 0+ are the Riemann–Liouville fractional derivatives with n – 1 < α ≤ n, n – 2 < β ≤ n – 1, n ≥ 2 (n ∈ N), α – β – 1 > 0, 0 < ξi,ηi < 1, i = 1, 2, 3, . . . , m – 2, m ≥ 3, ∑m–2 i=1 ξiη α–β–1 i < 1. f , g : (0, 1) × (–∞, +∞) × (–∞, +∞) → (–∞, +∞) are continuous, and f , g may be singular at t = 0, 1, λ is a parameter. The problem (1.1) with λ = 0 has been investigated by many authors [1–8]. Li et al. [1] considered the following fractional three-point boundary value problem:

On the other hand, fractional boundary value problems with parameters have received considerable attention [18][19][20][21][22][23][24][25][26][27]. Tan, Tan and Zhou [18] considered the existence of positive solutions for fractional differential equations with a parameter as follows: where n -1 < α ≤ n, n ≥ 2, f 1 : , and λ is a parameter. In [19,20], the authors studied nonlinear boundary value problem with boundary condi- In addition, Graef and Kong [21] considered the boundary value problem with fractional q-derivatives, and studied the existence of positive solutions according to different ranges of parameter. Moreover, Li et al. [22] considered infinite point boundary value problem for fractional differential equations with perturbed parameter. In [24], Lee and Park considered non-local problems with the boundary value condition u(1) -1 0 g(s)u(s) ds = b. In [25], Wang and Guo studied fractional differential equations with boundary condition x(1) = 1 0 k(s)g(x(s)) ds + μ. Jia and Liu [26] discussed the effect of the mixed boundary condition m 2 u(1) + n 2 u (1) = 1 0 g(s)u(s) ds + a. In this paper, we first consider the Green function of the m-point boundary value problem (1.1) with a parameter. Then we define a new set, which is not a subset of a cone. So we extend the results of the cone mapping established in [17,18] to the non-cone cases. Finally, we will consider the singularity of f , g and provide some sufficient conditions to guarantee that the problem (1.1) has a unique solution and construct two iterative sequences of solutions.
The rest of this paper is structured as follows. In Sect. 2, we will give some definitions and related lemmas to prove the main result. In Sect. 3, the existence and uniqueness of the solution to the problem (1.1) is proved, and an example supporting conclusion is given.

Preliminaries and related lemmas
In this section, we will provide some necessary basic definitions and lemmas to prove our main theorem, which can be found in [28][29][30][31][32].
Throughout our article, we define its base space as a Banach space. Let E be a Banach space, and θ be the zero element of E. If there are (1) x ∈ P, λ ≥ 0 ⇒ λx ∈ P and (2) x ∈ P, -x ∈ P ⇒ x = θ , then we call that a nonempty closed convex set P ⊂ E is a cone. Define an ordered relation in E: x ≤ y if and only if yx ∈ P. If there exists a positive constant N such that, for all x, y ∈ E, θ ≤ x ≤ y ⇒ x ≤ N y , then P is called a normal cone. Given h > θ , we denote P h by where n = [α] + 1. The Riemann-Liouville fractional integral of order α > 0 is given by where c i ∈ R, i = 1, 2, . . . , n and n = [α] + 1.

From the boundary value condition
which yields Thus Therefore, the boundary value problem (2.1) has the unique solution The proof is complete.
Then the operator equation M(x, x) + N(x, x) + e = x has a unique solution x * in P h,e , for any initial values x 0 , y 0 ∈ P h,e , we can get the following iterative sequences: x n = M(x n-1 , y n-1 ) + N(x n-1 , y n-1 ) + e, y n = M(y n-1 , x n-1 ) + N(y n-1 , x n-1 ) + e, n = 1, 2, . . . , we have x n → x * and y n → x * in E as n → ∞.

Main result
In this section, we will consider the existence and uniqueness of the solution to the boundary value problem (1.1). For convenience in the proof, we work in a Banach space E = C[0, 1]. Let P ⊂ E be defined by P = {u ∈ E|u(t) ≥ 0, t ∈ [0, 1]}, it is clear that P is a normal cone. Let .
Proof By Lemma 2.2, we obtain .
. Hence, 0 < e(t) ≤ h(t) and P h,e = {u ∈ E|u + e ∈ P h }. By Lemma 2.3, the solution to problem (1.1) has the following expression:

Clearly, u(t) is the solution to problem (1.1), if and only if u(t) is the fixed point of the operator M(u, v)(t) + N(u, v)(t) + e(t).
Therefore, if it can be proved that the operators M, N satisfy all the conditions of the Lemma 2.4, then the conclusion of Theorem 3.1 holds.
In view of (C5), we get

G(t, s)f s, u(s), v(s) dse(t) < ∞,
Hence, M and N are two mixed monotone operators.

G(t, s)f s, h(s), h(s) ds
and
Remark 3.1 For problem (3.1), we can take some negative values in nonlinear term f + g -10. However, the authors of [18] required that the nonlinear term is non-negative. Therefore, Theorem 3.1 in [18] cannot be applied to dealing with the problem (3.1).

Conclusions
In this paper, we establish the existence and uniqueness theorem of the solution for fractional m-point boundary value problem. Our tool is mixed monotone fixed point theorem involving non-cone mapping. Furthermore, two iterative sequences to approximate the unique solution are also given.