Page 88 - IJOCTA-15-1

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.1, pp.82-91 (2025)
https://doi.org/10.36922/ijocta.1653

RESEARCH ARTICLE

Existence of mild solution for fuzzy fractional differential equation
utilizing the Hilfer-Katugampola fractional derivative

Ramaraj Hariharan, Ramalingam Udhayakumar *

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore - 632 014,
Tamil Nadu, India
hariprf99@gmail.com, udhayaram.v@gmail.com

ARTICLE INFO ABSTRACT

Article History: This paper explores the existence of mild solutions for fuzzy fractional differ-
Received: 30 July 2024 ential equations involving the Hilfer-Katugampola fractional derivative. This
Accepted: 19 December 2024 derivative generalizes classical fractional derivatives, such as the Riemann-
Available Online: 24 January 2025 Liouville and Hadamard derivatives, offering a broader framework for frac-
tional calculus. The existence conditions for mild solutions are established
Keywords:
using fractional calculus, semigroup theory, and Schauder’s fixed point theo-
Fuzzy fractional differential equation
Semigroup rem. An example is provided to demonstrate the theoretical applications of
Hilfer-Katugampola fractional derivative the main results.
Mild solution
Fixed point theory
AMS Classification 2010:
20M05; 26A33; 47H10

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1. Introduction Lakshmikantham et al., and Zhou. These pub-
lications have significantly advanced both the
Fractional differential equations have become in- theoretical and practical aspects of fractional dif-
creasingly prominent in various scientific fields ferential equations, underscoring their importance
like physics, engineering, biology, finance, and in various scientific and engineering fields.
others. Their versatility and effectiveness in de-
scribing complex phenomena have made them in- In recent years, fuzzy fractional differential equa-
valuable for understanding real-world dynamics tions have gained significant attention for their
and creating advanced mathematical models. 1–5 ability to model complex systems with uncer-
tainty across various fields, including physics,
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Fractional Calculus began in 1695 when engineering, and finance. Arqub explores an
L’Hospital pondered the implications of a deriva- advanced computational method for addressing
tive with an order of 1/2. By 1697, Leibniz had fuzzy Fredholm–Volterra integrodifferential equa-
employed these half-order derivatives and made tions. These equations, which combine elements
some notes on them. In 1819, Lacroix discussed of integral and differential equations, play a sig-
derivatives of arbitrary order in his calculus book. nificant role in modeling various physical and en-
The first practical application of fractional calcu- gineering processes under uncertainty. Also, Ar-
lus was made by Abel in 1823. Recently, there qub et al. 10 presents a method to solve fuzzy
has been considerable progress in the field of fractional Volterra and Fredholm equations us-
fractional differential equations, as documented ing kernel functions combined with the Atan-
in numerous monographs. Notable contributions gana–Baleanu–Caputo distributed order deriva-
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include works by Kilbas et al., Miller and Ross, 7 tive.
*Corresponding Author
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