On a new fixed point theorem with an application on a coupled system of fractional differential equations

*Correspondence: tabdeljawad@psu.edu.sa 3Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, 11586, Riyadh, Saudi Arabia 4Department of Medical Research, China Medical University, 40402, Taichung, Taiwan Full list of author information is available at the end of the article Abstract In this work, new theorems and results related to fixed point theory are presented. The results obtained are used for the sake of proving the existence and uniqueness of a positive solution of a coupled system of equations that involves fractional derivatives in the Riemann–Liouville settings and is subject to boundary conditions in the form of integrals.


Introduction
For the last multiple of decades, the fractional calculus has drawn great interest of many sciences working on multifarious sciences on account of the good results obtained when this research applied the fractional tools to modeling of some real life problems [22,26,30,36,[46][47][48].
On the other side, even though the fixed point theory is a branch of pure mathematics, it is showed that this theory is one of the main tools to use in order to tackle the qualitative properties of differential and integral equations in general and the existence and uniqueness of solutions to these equations in particular. A significant number of mathematicians utilized the classical results in the fixed point theory to discuss solutions of initial and boundary value problems (see [2-21, 23, 24, 27-29, 31-34, 37, 39-43]). Meanwhile, others established new fixed point theorems and applied them to proving the existence and oneness of solutions to a variety of differential equations [13,25,38,44,45].
In this work, we will propose new theorems related to the fixed points of operators. We discuss the admissibility of two multi-valued mappings in the category of complete b-metric spaces to obtain the existence of a common fixed point. Using the triangular admissibility, we prove the uniqueness of the common fixed point. Then we utilize the findings of this problem to discuss one of the most considerable qualitative aspects for differential equation with fractional order; that is, the existence and uniqueness of the positive solution of Riemann-Liouville fractional coupled system governed by boundary integral conditions.
Let (X, d) be a b-metric space. Take CB(X) the set of bounded and closed sets in X. For x ∈ X and A, B ∈ CB(X), we define Define a mapping H : for every A, B ∈ CB(X). Then the mapping H forms a b-metric.
Throughout the article I will denote the interval [0, 1]. The definition for admissibility of a pair of single-valued mappings was first introduced in [1] and then generalized and used in [35] as well. Below we present the definition for the multi-valued case within the triangular admissibility. Definition 2.1 Let T 1 , T 2 : X → CB(X) be two multi-valued mappings and ρ : X × X → [0, +∞) be a function. Then the pair (T 1 , T 2 ) is said to be triangular ρ * -admissible if the following conditions hold: (ii) ρ(ς, u) ≥ 1 and ρ(u, η) ≥ 1 imply ρ(ς, η) ≥ 1.
In this paper, we discuss the local existence and uniqueness of positive solutions for the following coupled system of fractional boundary value problem subject to integral boundary conditions: where 1 < ρ, ≤ 2, φ, ϕ ∈ L 1 I are nonnegative and f , g ∈ C(I × [0, +∞), [0, +∞)) and D is the standard Riemann-Liouville fractional derivative. The functions φ(ς), ϕ(ς) satisfy the following conditions:

Main results
Now, we are ready to state and prove our main results.
The following key lemma is essential to proceed in proving the main results. It states that the admissibility of a pair of multi-valued functions will guarantee the existence of a sequence of points with diameter greater than 1.
We deduce that { n } is a Cauchy sequence in (X, d).

Corollary 3.3
Let (X, d) be an ρ-complete b-metric space, and ρ : X × X → [0, +∞) be a function. Suppose that T 1 , T 2 : X → X are mappings such that for ∈ Ω and ϕ ∈ χ . Moreover, suppose Then T 1 and T 2 have a common fixed point ∈ X. Moreover, if the following condition holds: We claim that, if Theorem 3.4 Suppose for κ ∈ I and η, z ∈ C(I) there exists ξ : R 2 → R and ϕ ∈ χ such that (i) . . .
So T 1 and T 2 obey all the conditions of Theorem 3.2. We find η * ∈ C(I) with η * = T 1 η * = T 2 η * . Now we claim that the solution of coupled system (1) is unique. By the condition H 2 we have the following.

Conclusion
Fixed point theory is one of the main tools of pure mathematics that are used to serve in the development in the qualitative theory of differential and integral equations. For the last few years, researchers have not only used the traditional fixed point theorems in proving the existence and uniqueness of solutions to various types of fractional differential equation, but have developed new fixed point theorems and applied them as well. In this article, we have developed a new fixed point theorem and utilized it to prove the local existence of positive solution to a coupled system of differential equations where the Riemann-Liouville fractional derivative is used. We believe that the new fixed point theorem considered here can be used to handle fractional differential equations in the setting of other derivatives, not only containing singular kernels, but nonsingular kernels as well.