Page 121 - IJOCTA-15-3

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.3, pp.493-502 (2025)
https://doi.org/10.36922/IJOCTA025120056

RESEARCH ARTICLE

Investigate the solution of an initial Hilfer fractional value problem

2
1
Amol D. Khandagale , Arif S. Bagwan , Sabri T. M. Thabet 3,4,5* , and Imed Kedim 6
1
Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Chhatrapati
Sambhajinagar, India
2
Department of Applied Sciences Humanities, Pimpri Chinchwad College of Engineering, Nigdi, Pune, India
3
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical
Sciences, Saveetha University, Chennai, Tamil Nadu, India
4
Department of Mathematics, Radfan University College, University of Lahej, Lahej, Yemen
5
Department of Mathematics, College of Science, Korea University, Seongbuk-gu, Seoul, Republic of Korea
6
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz
University, Al-Kharj, Saudi Arabia
kamoldsk@gmail.com, arif.bagwan@gmail.com, th.sabri@yahoo.com, i.kedim@psau.edu.sa
ARTICLE INFO ABSTRACT

Article History:
Received: March 20, 2025 This paper aims to investigate sufficient criteria of the existence solution for a
1st revised: May 7, 2025 new category of nonlinear fractional differential equation under the Hilfer frac-
2nd revised: May 10, 2025 tional derivative. The primary existence results are achieved by using a mod-
Accepted: May 13, 2025 ified version of the Krasnoselskii-Dhage fixed-point theorem in the weighted
Published Online: June 3, 2025 Banach space. Finally, an application is illustrated to test the validity of the
Keywords: findings.
Initial value problem
Hilfer fractional derivative
Fixed point theorem
AMS Classification 2010:
26A33, 34A08, 34A12, 47H09
47H10

1. Introduction theorems and their generalizations. See ref. 22–25
and references therein.
In numerous mathematical studies, the theory of In ref., 26–29 the researchers discussed some
fractional calculus and its applications have been existence and optimal control results for various
thoroughly examined. Many real-world prob- fractional differential equations and inclusions by
lems in science, engineering, and economics are means of fixed point theorems. Similarly, several
now mathematically modeled with the help of scholars have demonstrated a great deal of inter-
fractional calculus. See the books and research est in the theory of fractional differential equa-
works 1–7 and references therein. Fractional dif- tions with linear perturbation. See ref. 30–33 and
ferential equations (FDE) are undoubtedly still a references therein. In ref., 25 Dhage presented a
popular topic among writers. The use of the fixed- novel form of the Kransoselskii-type FPT termed
point theorem approach to obtain the existence the Krasnoselskii-Dhage type FPT by utilizing
and uniqueness results for fractional differential a nonlinear D−contraction condition. In ref., 30
equations is the most favored topic by several Dhage and Lakshmikantham used fixed point the-
scholars. See ref. 8–21 and references therein for orems (FPT) of the Krasnoselskii type to study
the recent development in this area. This mo- the first-order hybrid differential problem with
tivates researchers to study various fixed-point linear perturbations:
*Corresponding Author
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