Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces

Abstract:We deal with some impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces. We make use of α φ-Geraghty-type contraction. An illustrative example is the subject of the last section.

On the other hand, the fixed point theory has made serious progress in the last few decades. One of the most improvements is to show the validity of the fixed point theorem in the setting of a b-metric space that is a natural extension of standard metric space. Roughly speaking, by replacing the triangle inequality axiom of the metric notion, Czerwik [17,18] observed this new structure. Several authors reported interesting fixed point results in the framework of complete b-metric spaces, see e.g., [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].
In this manuscript, we shall investigate the Cauchy problem of Caputo-Fabrizio impulsive fractional differential equations where ( ) M r is a normalization constant depending on r. Analogously, for a function ∈ ( ) φ C I 1 , the Caputo-Fabrizio fractional derivative of order < < r 0 1 is s u p , f o r a l l , .
ϑ I

2
It is clear that d is a b-metric with = c 2.
, f o r a l l , . 2 2 Clearly, d is not a metric, but is a b-metric space with ≥ r 2.
We use Φ to indicate the set of all continuous and increasing function for some ≥ c 1.
is called a generalized α ϕ-Geraghty contraction-type mapping whenever there exist × → + α M M : 0 and some ≥ L 0 such that Remark 1.8. In the case when = L 0 in Definition 1.7, and the fact that , forms a generalized α ϕ-Geraghty contraction-type mapping with the following additional assumptions: , then F has a unique fixed point.

Main results
Consider the Banach space : , 0, , , and there exist and , Then ( ) PC d , , 2 is a b-metric space in the sense of Definition 1.4.
Definition 2.1. By a solution of problem (1) we mean a function ∈ PC ω that satisfies ( ) = ( ) + Proof. Assume u satisfies (7). If ∈ ϑ I 0 , then  If ∈ ϑ I k , we get (6). Conversely, assume that ω satisfies (6). If Assumptions: Here, we list the necessary assumptions to state our main theorem in a proper form.
Let × → + PC PC α : 0 be the function defined by: Our following result is based on Theorem 1.11.
Next, we verify that ϒ is α-admissible: This implies from ( )

An example
Let the impulsive Caputo-Fabrizio fractional differential equation 0,1 0,1 0 is given by:  . Also, ( ) Ax 3 holds from the definition of the function δ. Hence, there exists at least one solution of (11).
Conflict of interest: Authors state no conflict of interest.