Unique and Non-Unique Fixed Points and their ApplicationsView this Special Issue
Terminal Value Problem for Implicit Katugampola Fractional Differential Equations in -Metric Spaces
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 . In , 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) . 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.
Theorem 1 (). Let , be such that . Then, for any , where , we have (1)Inverse property:(2)Linearity property: for all , we have
Lemma 1 (). Let . ; then the fractional differential equation , has a unique solution where with .
Lemma 2. Let . If and , then for some constant .
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
From Lemma 2, we get
for some . If we use the terminal condition in (21), we find
Henceforth, we deduce (18).
Contrariwise, if achieves (18), then ; for and .
We set the following: .
For some , we set .
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 regularThen 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 .
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
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,
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
Thus, hypothesis (H1) is achieved with
Define the functions with and with .
No data is used. No data is available in this work.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
U. N. Katugampola, “A new approach to generalized fractional derivatives,” Bulletin of Mathematical Analysis and Applications, vol. 6, p. 115, 2014.View at: Google Scholar
Y. Arioua, B. Basti, and N. Benhamidouche, “Initial value problem for nonlinear implicit fractional differential equations with Katugampola derivative,” Applied Mathematics E - Notes, vol. 19, pp. 397–412, 2019.View at: Google Scholar
S. Abbas, M. Benchohra, J. R. Graef, and J. Henderson, Implicit fractional differential and integral equations: existence and stability, De Gruyter, Berlin, 2018.View at: Publisher Site
S. Abbas, M. Benchohra, and G. M. N'Guérékata, Topics in fractional differential equations, Springer, New York, 2012.View at: Publisher Site
S. Abbas, M. Benchohra, and G. M. N'Guerekata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.
A. Ashyralyev, “A survey of results in the theory of fractional spaces generated by positive operators,” TWMS Journal of Pure and Applied Mathematics, vol. 6, no. 2, pp. 129–157, 2015.View at: Google Scholar
Z. Baitiche, C. Derbazi, and M. Benchora, “Caputo fractional differential equations with multipoint boundary conditions by topological degree theory,” Results in Nonlinear Analysis, vol. 3, no. 4, article 167178, 2020.View at: Google Scholar
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V, Amsterdam, 2006.
S. G. Samko, A. A. Kilbas, and O. I. Marichev, “Fractional Integrals and Derivatives,” in Theory and Applications, English translation from the Russian, Gordon and Breach, Amsterdam, 1987.View at: Google Scholar
Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.View at: Publisher Site
H. Shojaat, H. Afshari, and M. S. Asgari, “A new class of mixed monotone operators with concavity and applications to fractional differential equations,” TWMS Journal of Applied and Engineering Mathematics, vol. 11, no. 1, pp. 122–133, 2021.View at: Google Scholar
S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, vol. 46, no. 2, Atti del Seminario Matematico e Fisico dell' Universita di Modena, 1998.
S. Czerwik, “Contraction mappings in b-metric spaces,” Acta mathematica et informatica universitatis ostraviensis, vol. 1, pp. 5–11, 1993.View at: Google Scholar
H. Aydi and M. F. Bota, “A common fixed point for weak øcontractions on b-metric spaces,” Fixed Point Theory, vol. 13, no. 2, pp. 337–346, 2012.View at: Google Scholar
S. K. Panda, E. Karapinar, and A. Atangana, “A numerical schemes and comparisons for fixed point results with applications to the solutions of Volterra integral equations in dislocatedextendedb- metricspace,” Alexandria Engineering Journal, vol. 59, no. 2, pp. 815–827, 2020.View at: Publisher Site | Google Scholar