Application of some new contractions for existence and uniqueness of differential equations involving Caputo–Fabrizio derivative

*Correspondence: marasi@tabrizu.ac.ir 2Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran Full list of author information is available at the end of the article Abstract In this paper we study fractional initial value problems with Caputo–Fabrizio derivative which involves nonsingular kernel. First we apply α-contraction and α-type F-contraction mappings to study the existence and uniqueness of solutions for such problems. Finally, we use some contraction mappings in complete F-metric spaces for this purpose.


Introduction
Fractional calculus is a part of mathematical analysis that studies the performance of derivative and integral operations on non-integer orders. In the past years, early work in fractional calculus was limited to mathematics. But, in the last few years, extensive studies on the applications of fractional operators in the other disciplines have been conducted. Recently, this field had found many applications in various directions such as applied mathematics, electrochemistry, tracer in fluid flows, fractional-order multi-poles in electromagnetism, finance, signal processing, bio-engineering, viscoelasticity, fluid mechanics, and fluid dynamics [10,24]. These wide applications have led researchers to provide different definitions of fractional derivatives. The main difference between these definitions is related to possessing different kernels. Two famous fractional derivatives, namely the Riemann-Liouville and the Caputo derivatives, have received a lot of attention and so differential and integral equations containing these derivatives by several methods containing numerical and analytical methods ( [22,29,30], but these definitions included a singular kernel. Thus, recently Caputo and Fabrizio provided a definition with a nonsingular kernel which the properties of this new definition can be found in [25]. Various methods have been used by researchers to solve differential equations including Caputo-Fabrizio fractional derivative and multi-singular point-wise defined equations (see [16,18,32] and the references therein). One of the efficient methods in investigating the existence and uniqueness of the solutions of differential equations is using of fixed point theory and for this reason there is a long history of presenting various fixed point theorems (see [1-7, 9, 11, 13, 17, 19, 22, 26-28, 34]).
Wardowski et al. [35] proposed and investigated the F-contraction, then Abbas et al. [17,21] further generalized the concept of F-contraction and proved some fixed point results.
Wasfi Shatanawi and Erdal Karapınar in [34] introduced F S -contractions in the sense of Wardowski and Seghal and F J -contractions in the sense of Wardowski and Jachymski. Then they ensured some existence and uniqueness fixed point results. Throughout the article J denote [0, 1].
In this work, we consider the following differential equation with Caputo-Fabrizio derivatives via fixed point theorems: where D ς is the Caputo-Fabrizio derivative of order ς and f is continuous with In what we will have below it is supposed that (M, d) be a complete b-metric space and p 1 is its constant, also the elements of are increasing and continuous functions : [0, ∞) → [0, ∞) satisfying (qx) ≤ q (x) ≤ qx, q > 1; moreover, denotes the family of nondecreasing functions such that, for p 1 ≥ 1, we have : [0, ∞) → [0, 1 p 1 2 ).
Then ψ is called an α-admissible mapping. Now, we have the following fixed point theorem.
Then there exists a fixed point for the mapping ψ.
Definition 1.5 Let 0 < ς < 1. The Caputo-Fabrizio integral for a function j of order ς is defined by where p 1 = 2 is the constant of (M, d) and M = C(J, R).
In this paper we consider where D ς is the Caputo-Fabrizio derivative of order ς , also it is supposed that f : J × M → M satisfies in f (0, (0)) = 0 and is continuous. It is easy to prove the following lemma.

Main results
In this section, for existence and uniqueness of a solution for the problem be defined in (6) firs we apply an α--contraction, then we continue by using an α-type F-contraction and another contraction in complete F-metric space to examine the existence and uniqueness of solutions of the mentioned problem.
Then there exist at least one solution for the problem (6).
Now to define an α-type F-contraction mapping, let F be the family of strictly increasing functions F : R + → R such that there exists k ∈ (0, 1) for which lim α→0 + α k F(α) = 0 and also lim n→∞ F(α n ) = -∞ if and only if lim n→∞ α n = 0 for each sequence {α n } n∈N of positive numbers. We present the following theorem. Proof We show that has a fixed point. Thus,
Hence for , ℘ ∈ C(J), p ∈ J with j ( (p), ℘(p)) ≥ 0, we have -∞ else, Therefore we conclude that satisfies all conditions of definition of α-type F-contraction. By (H3), which shows that is α-admissible. From (H2) we have 0 ∈ C(J) such that α( 0 , 0 ) ≥ 1. According to (H4) and Theorem 2.3, we can obtain * ∈ C(J) where * = * which is a fixed point of and therefore a solution of the problem. Now to use in the next definition let us define F involved the functions ψ : (0, ∞) → R such that: Then d is called an F-metric on M, and the pair (M, d) is said to be a F-metric space.
Then ψ is said to be an α-orbital admissible.
Now we are ready to present the following theorem. Now, for a positive integer n we denote by h n the nth iterate of h, so that y = h 0 y and h n+1 y = h(h n y) for y ∈ X and n ∈ N. The triplet (X, d, h) represent a metric space (X, d) with a self-mapping h on it. We shall use (X * , d, h) to indicate the corresponding metric space is complete. Also on (X, d, h) an orbit of y 0 ∈ X is the set O(y 0 ) = h n y 0 : n = 0, 1, 2, . . . , and ρ(y 0 ) denote to the diameter of the set O(y 0 ). Note that for any subset B of X, ρ(B) = sup{d(u, y) : u, y ∈ B} is the diameter of B. We shall use the triplet (X 0 * , d, h) if for some y ∈ X, every Cauchy sequence from O(y) converges in X. In this case, the corresponding space is called orbitally complete.

Corollary 2.9
For (X, d * , h) with p : X → N, we suppose there exists τ > 0 such that for v, w ∈ X d h p(y) y, h p(y) w ≤ e -τ d(y, w).
Assume there exists y 0 ∈ X such that 0 < ρ < ∞. Moreover, (X, d) is h-orbitally complete. Then h has a unique fixed point. We demonstrate that (6) has a unique solution. We get Hence condition (10) holds with p : X → N such that p( ) = 2, ∈ X. Accordingly all axioms of Corollary 2.9 are verified and consequently possesses a unique fixed point. So (6) possesses a unique solution.