Journal of Advanced Mathematical Modeling

Journal of Advanced Mathematical Modeling

Analysis of the existence of solutions to hybrid nonlinear differential equations including fractional q-derivative

Document Type : Original Paper

Authors
Hamid Reza Sahebi 1
Mohammad Esmael Samei 2
1 Islamic Azad University
2 Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran
10.22055/jamm.2025.49384.2365
Abstract
In this paper, we investigate the existence of solutions for a novel class of nonlinear hybrid differential
equations involving the fractional quantum Caputo derivative. By exploiting the hybrid structure
of the considered equations and the distinctive properties of the fractional quantum derivative, an analytical
framework is established to derive sufficient conditions ensuring the existence of solutions. The proof
is based on Petryshyn’s fixed point theorem, employing the properties of spaces together with compact–
continuous operators. Several illustrative examples are provided to demonstrate the applicability of the
proposed model and to validate the theoretical results. Although explicit numerical values of the solutions
are not computed, the examples confirm that the proposed existence conditions are practically realizable.
The findings of this study serve as a foundation for further research in qualitative and numerical analysis
of fractional differential equations with complex hybrid structures.
Keywords
Quantum calculus
Caputo’s q-derivative
measure of non-compactness
Petryshyn fixed point theorem
nonlinear differential equations
Subjects

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  • Receive Date 19 April 2025
  • Revise Date 25 August 2025
  • Accept Date 02 October 2025