Research Article | Open Access

Salim Krim, Saïd Abbas, Mouffak Benchohra, Erdal Karapinar, "Terminal Value Problem for Implicit Katugampola Fractional Differential Equations in -Metric Spaces", *Journal of Function Spaces*, vol. 2021, Article ID 5535178, 7 pages, 2021. https://doi.org/10.1155/2021/5535178

# Terminal Value Problem for Implicit Katugampola Fractional Differential Equations in -Metric Spaces

**Academic Editor:**Santosh Kumar

#### 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.

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#### Copyright

Copyright © 2021 Salim Krim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.