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Investigate the solution of an initial Hilfer fractional value problem
and Conflict of interest
1
g (s, φ (s)) = sin(|φ(s)|).
2
s + 1 The authors declare no conflict of interest.
Hence, we find that
Author contributions
f (s, φ) − f (s, ¯φ) ≤ ln(1 + |φ|) − ln(1 + | ¯φ|)

Conceptualization: Amol D. Khandagale, Arif S.
1 + |φ|

f (s, φ) − f (s, ¯φ) = ln
Bagwan, Sabri T. M. Thabet
1 + | ¯φ|
Formal analysis: Arif S. Bagwan, Sabri T. M.

|φ| − | ¯φ|
= ln 1 + Thabet and Imed Kedim
1 + | ¯φ| Investigation: Arif S. Bagwan, Sabri T. M. Tha-

|φ − ¯φ| bet, Imed Kedim
≤ ln 1 +
1 + | ¯φ| Methodology: All authors
Writing-original draft: Arif S. Bagwan, Sabri T.
≤ ln (1 + |φ − ¯φ|)
M. Thabet
≤ ψ D (|φ − ¯φ|) , Writing-review & editing: Arif S. Bagwan, Sabri
1
for each s ∈ (0, ], where ψ D (s) = ln(1 + s) is T. M. Thabe
2
a D−function, such that 0 < ψ D (0) = 0 and
ψ D (s) < s, that is h(s) = s − ln(1 + s), we have Availability of data
1
′
h (s) = 1 − 1 = s > 0, for s ∈ (0, ]. Also,
1+s 2
Not applicable.
1
|g (s, φ (s)) | ≤ = ξ(s).
2
s + 1
AI tools statement
Additionally, (q − p) 1−κ ≈ 0.84 < 1. Therefore,
all assumptions of Theorem 2 are satisfied, it im- All authors confirm that no AI tools were used in
plies that the IVP (18)–(19) possesses a solution the preparation of this manuscript.
in C 1−κ (J , R).
References
5. Conclusions
1. Hilfer R. Fractional Time Evolution: Applications
In this article, a Hilfer FDE with linear pertur- of Fractional Calculus in Physics. London: World
bations of the second type under a certain initial Scientific; 2000.
condition was discussed for the existence of the 2. Smart DR. Fixed Point Theorems. Cambridge
solution. An equivalent VIE is obtained for the University Press; 1973.
IVP (1)–(2). The main result on the existence of 3. Kilbas AA, Srivastava HM, Trujillo JJ. Theory
and Applications of Fractional Differential Equa-
the solution to IVP was established employing the
tions. Amsterdam:Elsevier; 2006.
Krasnoselskii–Dhage type fixed-point theorem in
4. Miller KS, Ross B. An Introduction to the Frac-
the weighted Banach space. At the end, an appli-
tional Calculus and Differential E quations. New
cation was given to check the validity of the out-
York: John Wiley; 1993.
comes. The initial value problem discussed in this 5. Abdeljawad T, Thabet STM, Kedim I, and Cortez
paper along with the fractional derivative opera- VM. On a new structure of multi-term Hilfer frac-
tor considered in the IVP (1)–(2) generalizes the tional impulsive neutral Levin-Nohel integrodif-
existing results available in the literature. In IVP ferential system with variable time delay. AIMS
(1)–(2) if we put τ = 1, then the results obtained Math. 2024;9(3):7372-7395.
in the paper coincide with the results published 6. Rezapour S, Thabet STM, Rafeeq AS, Kedim
in. 32,33 However, it could be interesting to see the I, Vivas-Cortez M, Aghazadeh N. Topology de-
impact of the inclusion of the control term in IVP gree results on a G-ABC implicit fractional dif-
ferential equation under three-point boundary
(1)–(2). This idea may be an open problem for fu-
conditions. PLoS ONE 2024; 19(7): e0300590.
ture investigations.
https://doi.org/10.1371/journal.pone.0300590
7. Salim A, Thabet STM , Kedim I, Vivas-
Acknowledgments
Cortez M. On the nonlocal hybrid (k, φ)-
This study is supported via funding from Prince Hilfer inverse problem with delay and antici-
Sattam bin Abdulaziz University project number pation. AIMS Math. 2024;9(8): 22859–22882.
https://doi.org/10.3934/math.20241112
(PSAU/2025/R/1446).
8. Ahmad B, Ntouyas SK. Initial value problem
Fundings of fractional order Hadamard-type functional
differential equations. Electron J Differ Equat.
None. 2015;2015:1-9.
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