Existence of positive solutions for a class of fractional differential equations with the derivative term via a new fixed point theorem

In this paper, we firstly establish the existence and uniqueness of solutions of the operator equation A(x,x)+B(x,x)+C(x)+e=x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A(x,x)+ B(x,x)+C(x)+e = x$\end{document}, where A and B are two mixed monotone operators, C is a decreasing operator, and e∈P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$e\in P$\end{document} with θ≤e≤h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\theta \leq e \leq h$\end{document}. Then, using our abstract theorem, we prove a class of fractional boundary value problems with the derivative term to have a unique solution and construct the corresponding iterative sequences to approximate the unique solution.

Moreover, Yue and Zou [12] were concerned with a class of fractional Dirichlet boundary value problems with dependence on the first order derivative. Some sufficient conditions for the uniqueness of solutions for the above-mentioned problem were given. The main tool is also classical Banach's contraction mapping principle.
It is well known that problem (1.1) is the generalization of elastic beam equation [13]. In [14], Goodrich first studied the Green's function associated with problem (1.1) when k ≡ 0 and established the existence result on sublinear nonlinearity. Furthermore, Xu, Wei, and Dong [15] also considered sublinear problem (1.1) by using of the fixed point index theorem and spectral theory. Jleli and Samet [16] utilized a mixed monotone fixed point theorem to obtain a unique solution of problem (1.1) when b = 0. Moreover, Yang, Shen, and Xie [17] investigated the nonlinear term involving the first order derivative for problem (1.1).
Compared with problem (1.4), we add the derivative term D β 0 + u(t), the operator term (Hu)(t), and nonlinear boundary conditions k(u(1)) into problem (1.1). Furthermore, different from problems (1.2) and (1.3), we break through the restriction of positivity on nonlinearities f and g. The first goal of this paper is to establish the existence and uniqueness theorem of solution for the operator equation where A and B are both mixed monotone, C(x) is decreasing, and e ∈ P with P is a cone in Banach space E. Our abstract theorem generalizes the result on the cone mappings (see Theorem 3.1 in [19]) to non-cone case. Some sufficient conditions under which problem (1.1) has a unique solution are provided. Moreover, we also construct two iterative sequences for approximating a unique solution.
The structure of this paper includes the following sections. In Sect. 2, we introduce some definitions and give preliminary results to be used in the proof of our main theorems. In Sect. 3, we establish the existence and uniqueness of solutions for problem (1.1) based on a new fixed point theorem.

Preliminaries
In this section, we give some definitions and preliminary results that are used in this paper [23][24][25].
In this paper, (E, · ) is a real Banach space, which is partially ordered by a cone P ⊂ E, i.e., x ≤ y if and only if yx ∈ P. θ is the zero element in E. Recall that a nonempty closed convex set P ⊂ E is a cone if it satisfies: x ∈ P, λ ≥ 0 ⇒ λx ∈ P and x ∈ P, -x ∈ P ⇒ x = θ . P is called to be normal if there exists N > 0 such that θ ≤ x ≤ y ⇒ x ≤ N y . Given h > θ , we denote P h by Let e ∈ P with θ ≤ e ≤ h, we define P h,e = {x ∈ E|x + e ∈ P h }.

Lemma 2.1 ([21])
Let P be a normal cone of E and T : P h,e × P h,e − → E be a mixed monotone operator with T(h, h) ∈ P h,e , and the following condition is satisfied: for all u, v ∈ P h,e and λ ∈ (0, 1). Then (1) There exist u 0 , v 0 ∈ P h,e and s ∈ (0, 1) such that (2) T has a unique fixed point x * in P h,e ; (3) For any initial values x 0 , y 0 ∈ P h,e , by constructing successively the sequence as follows: we have x n → x * and y n → x * as n → ∞.

Lemma 2.2 ([19, 26]) Let h(t) ∈ C[0, 1], then the unique solution of the linear problem
is given by is the Green's function.

Lemma 2.3 ([19])
The Green's function G(t, s) in Lemma 2.2 has the following properties:

Theorem 2.1 Let P be a normal cone in E, and let A, B
: P h,e × P h,e − → E be two mixed monotone operators, C : P → P be a decreasing operator satisfying the following conditions: (A2) For all t ∈ (0, 1) and x, y ∈ P h,e , (A3) For all t ∈ (0, 1) and y ∈ P, we have For all x, y ∈ P h,e , there exists a constant δ > 0 such that Then the operator equation A(x, x) + B(x, x) + C(x) + e = x has a unique solution x * in P h,e , and for any initial values x 0 , y 0 ∈ P h,e , by setting two iterative sequences {x n } {y n } as follows: x n = A(x n-1 , y n-1 ) + B(x n-1 , y n-1 ) + C(y n-1 ) + e, n = 1, 2, . . . , we have x n → x * and y n → x * in E as n → ∞.
Proof We prove Theorem 2.1 in view of Lemma 2.1. Firstly, by condition (A4), and combining with Lemma 2.2 in [18], we have that there exist constants a i > 0 and b i > 0 (i = 1, 2, 3) such that

Main result
In the section, we use Theorem 2.1 to obtain the existence and uniqueness of a positive solution for problem (1.1).
Proof We will use Theorem 2.1 to prove Theorem 3.1.

G(t, s)g s, u(s), (Hu)(s) ds
For every t ∈ [0, 1] and u, v ∈ P h,e , we consider the following operators: and and (1) Firstly, for all u i , v i ∈ P h,e (i = 1, 2) with u 1 ≥ u 2 , v 1 ≤ v 2 , by (B5), we get that H(v 1 ) ≤ H(v 2 ). It follows from condition (B2), (3.2), and (3.5) that Thus, A is a mixed monotone operator. Similarly, we have from (3.3) and (3.6) that Hence, B is a mixed monotone operator. Since and Then C is a decreasing operator.