Open AccessOn fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfer-generalized proportional fractional derivative
Author(s) -
Saima Rashid,
Fahd Jarad,
Khadijah M. Abualnaja
Publication year - 2021
Publication title -
aims mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.329
H-Index - 15
ISSN - 2473-6988
DOI - 10.3934/math.2021635
Subject(s) - mathematics , uniqueness , differentiable function , lipschitz continuity , volterra equations , fuzzy logic , mathematical analysis , operator (biology) , nonlinear system , physics , linguistics , philosophy , biochemistry , chemistry , repressor , quantum mechanics , transcription factor , gene
This investigation communicates with an initial value problem (IVP) of Hilfer-generalized proportional fractional ($ \mathcal{GPF} $) differential equations in the fuzzy framework is deliberated. By means of the Hilfer-$ \mathcal{GPF} $ operator, we employ the methodology of successive approximation under the generalized Lipschitz condition. Based on the proposed derivative, the fractional Volterra-Fredholm integrodifferential equations $ (\mathcal{FVFIE}s) $ via generalized fuzzy Hilfer-$ \mathcal{GPF} $ Hukuhara differentiability ($ \mathcal{HD} $) having fuzzy initial conditions are investigated. Moreover, the existence of the solution is proposed by employing the fixed-point formulation. The uniqueness of the solution is verified. Furthermore, we derived the equivalent form of fuzzy $ \mathcal{FVFIE}s $ which is supposed to demonstrate the convergence of this group of equations. Two appropriate examples are presented for illustrative purposes.