#### Abstract

This manuscript deals with a class of Katugampola implicit fractional differential equations in -metric spaces. The results are based on the -Geraghty type contraction and the fixed point theory. We express an illustrative example.

#### 1. Introduction and Preliminaries

An interesting extension and unification of fractional derivatives of the type Caputo and the type Caputo-Hadamard is called Katugampola fractional derivative that has been introduced by Katugampola [1, 2]. Some fundamental properties of this operator are presented in [3, 4]. Several results of implicit fractional differential equations have been recently provided (see [4–14] and the references therein). A new class of mixed monotone operators with concavity and applications to fractional differential equations has been considered in [15]. In [16], the authors presented some existence and uniqueness results for a class of terminal value problem for differential equations with Hilfer-Katugampola fractional derivative.

On the other side, a novel extension of -metric was suggested by Czerwik [17, 18]. Although the -metric standard looks very similar to the metric definition, it has a quite different structure and properties. For example, in the -metric topology framework, an open (closed) set is not open (closed). Additionally, the -metric function is not continuous. These weaknesses make this new structure more interesting (see [19–28]).

Throughout the paper, any mentioned set is nonempty. We consider the following type of terminal value problems of Katugampola implicit differential equations of noninteger orders: with and the function is continuous. Here, is the Katugampola fractional derivative of order .

Set . Then, forms a Banach space with the norm .

Set . Then, becomes a Banach space with the norm .

Set . Then, it forms a Banach space . Here, is called the weighted space of continuous functions.

*Definition 1 (Katugampola fractional integral) [1]. *The Katugampola fractional integrals of order and of a function are defined by

*Definition 2 (Katugampola fractional derivatives) [1, 2]. *The generalized fractional derivatives of order and corresponding to the Katugampola fractional integrals (2) defined for any by
where ; if the integrals exist.

*Remark 1 ([1, 2]). *As a basic example, we quote for and ,
Giving in particular,
In fact, for and , we have
If we put , we obtain from (6):
So, .

Theorem 1 ([2]). *Let , be such that . Then, for any , where , we have
*(1)*Inverse property:**(2)**Linearity property: for all , we have*

Lemma 1 ([2]). *Let . ; then the fractional differential equation , has a unique solution
where with .*

*Proof. *Let . from Remark 1, we have
Then, the fractional equation has a particular solution as follows:
Thus, the general solution of is a sum of particular solutions (12), i.e.

Lemma 2. *Let . If and , then
for some constant .*

*Proof. *Let be the fractional derivative (3) of order . If we apply the operator to and use the properties (8) and (9), we get
From the proof of Lemma 1, there exists , such that
which implies (14).

Lemma 3. *Let and and . A function forms a solution for
if and only if fulfills
*

*Proof. *Let . and . Suppose that satisfies (17). Employing the operator to the each side of the equation
we find
From Lemma 2, we get
for some . If we use the terminal condition in (21), we find
which shows
Henceforth, we deduce (18).

Contrariwise, if achieves (18), then ; for and .

Lemma 4. *Contemplate the problem (1), and set , and .**We presume achieves
**Then, forms a solution of (1).*

*Definition 3 [29, 30]. *A function is called *b*-metric if there is and fulfills
(i)(ii)(iii)for all . We say that the tripled () is -metric space (in short, b.m.s.).

*Example 1 [29, 30]. **Let**be described as*Ergo, is -metric space.

*Example 2 [29, 30]. *Set and be designated by
Henceforth, () with is -metric space.

We set the following: .

For some , we set .

*Definition 4 [29, 30]. *A self-operator , on a *b.m.s.* (), is called a generalized Geraghty contraction whenever there exists , and some such that for
we have
for all , where .

*Remark 2. *In the case when in Definition 4 and the fact that
the inequality (29) becomes

*Definition 5 [29, 30]. *Set . An operator , is admissible if
for all .

*Definition 6 [29, 30]. *Let () with be a *b.m.s* and .

We say that is regular if for any sequence in such that as and for each ; there exists subsequence of with for all .

Theorem 2 [29, 30]. *We presume that a self-operator over a complete b.m.s.*

*() with forms a generalized Geraghty contraction. Furthermore,*(i)

*is admissible with initial value for some*(ii)

*either is continuous or is regular*

*Then possesses a fixed point. Furthermore, if*(iii)

*f*or all fixed points , either or , then the found fixed point is uniqueThis manuscript launches the study of Katugampola implicit fractional differential equations on b.m.s.

#### 2. Main Results

Observe that is a complete b.m.s. with described as

A function is called a solution of (1) if it archives with .

In the sequel, we shall need the following hypotheses:

(*H*_{1}) There exist and so that for each , and with

(*H*_{2}) There are (I) and , so that
with and

(*H*_{3}) For any , and , implies
with so that

(*H*_{4}) If with and , then

Theorem 3. *We presume ( H_{1})–(H_{4}). Then, the problem (1) possesses at least a solution on .*

*Proof. *Take the operator into account that is described as
where , with .

On account of Lemma 4, we deduce that solutions of (1) are the fixed points of .

Let be the function defined by

First, we demonstrate that form a generalized -Geraghty operator. For any and each , we derive that where , , with

From (*H*_{1}), we have

Thus, where .

Next, we have

Thus,

Hence, where , with , and .

So, is generalized Geraghty operator.

Let such that

Accordingly, for any , we find

This implies from (*H*_{3}) that
which gives .

Ergo, is a -admissible.

Now, from (*H*_{2}), there exists such that

Finally, from (*H*_{4}), if with and , then,

Theorem 2 implies that fixed point of forms a solution for (1).

#### 3. An Example

The tripled is a complete b.m.s. with such that

We take the following fractional differential problem into consideration with

Let , and . If |, then

In the case when , we get

Hence,

Thus, hypothesis (*H*_{1}) is achieved with

Define the functions with and with .

Hypothesis (*H*_{2}) is satisfied with . Also, (*H*_{3}) holds the definition of the function . So, Theorem 3 yields that problem (57) admits a solution.

#### Data Availability

No data is used. No data is available in this work.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.