Journal of Function Spaces
Volume 2021 (2021), Article ID 5535178, 7 pages
https://doi.org/10.1155/2021/5535178

Research Article

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

Salim Krim,¹ Saïd Abbas,¹ Mouffak Benchohra,² and Erdal Karapinar3,4,5

¹Laboratory of Mathematics, University of Saïda–Dr. Moulay Tahar, P.O. Box 138, EN-Nasr, 20000 Saïda, Algeria
²Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, P.O. Box 89, Sidi Bel-Abbes 22000, Algeria
³Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam
⁴Department of Mathematics, Cankaya University, 06790 Etimesgut, Ankara, Turkey
⁵Department of Medical Research, China Medical University Hospital, China Medical University, 40402 Taichung, Taiwan

Correspondence should be addressed to Erdal Karapinar; erdalkarapinar@tdmu.edu.vn

Received 27 February 2021; Revised 11 April 2021; Accepted 17 April 2021; Published 3 May 2021

Academic Editor: Santosh Kumar

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.

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]. Letbe 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)for all fixed points , either or , then the found fixed point is unique

This 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:

(H₁) There exist and so that for each , and

with

(H₂) There are (I) and , so that

with and

(H₃) For any , and , implies

with so that

(H₄) If with and , then

Theorem 3. We presume (H₁)–(H₄). 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 (H₁), 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 (H₃) that

which gives .

Ergo, is a -admissible.

Now, from (H₂), there exists such that

Finally, from (H₄), 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 (H₁) is achieved with

Define the functions with

and with .

Hypothesis (H₂) is satisfied with . Also, (H₃) 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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