Existence analysis of nonlinear $q$-fractional differential inclusions with $p$-Laplacian operators and an aerospace vibration-modeling illustration

Technical Summary: Existence Analysis of Nonlinear $q$-Fractional Differential Inclusions with $p$-Laplacian Operators

Problem Formulation
This research investigates a coupled system of nonlinear $q$-fractional differential inclusions involving nested $p$-Laplacian operators and nonlocal boundary conditions. The system is defined on the interval $\varrho \in [0, 1]$ and consists of $k$ coupled equations. The core structure involves a sequential layering of operators:

1. Inner Layer: A Caputo $q$-fractional derivative of order $\rho_i$ ($2 < \rho_i < 3$) acting on the state vector $w_i$, linked via the $p$-Laplacian operator $\Phi_p$.
2. Intermediate Layer: The output of the inner layer is transformed by the Jackson $q$-derivative $D_q$ and then subjected to a Riemann–Liouville $q$-fractional derivative of order $\xi_i$ ($3 < \xi_i < 4$), linked via the inverse $p$-Laplacian $\Phi_z$ (where $z$ is the conjugate exponent of $p$).
3. Outer Layer: The resulting function is acted upon by a Caputo $q$-fractional derivative of order $\varsigma_i$ ($2 < \varsigma_i < 3$).

The system is formulated as a differential inclusion:
$$D^{\varsigma_i}_{q,0+} x_i(\varrho) + \vartheta h_i(\varrho, w_i(\varrho)) \in H_i(\varrho, w(\varrho), D_q w(\varrho))$$
where $H_i$ is a multifunction with compact convex values, depending on the full state vector and its $q$-derivative. The problem is subject to three distinct layers of boundary conditions:

• Outer-layer: Nonlocal conditions involving values at $\varrho=0, 1,$ and a specific point $\epsilon$.
• Intermediate-layer: Conditions involving $u_i$ and its derivatives at $0, 1, \epsilon,$ and $\eta$.
• Inner-layer: Conditions involving $w_i$ and its derivatives at $0, 1,$ and an integral constraint over $[0, \eta]$.

Methodology
The authors employ a constructive approach combining $q$-fractional calculus, Green's function theory, and multivalued fixed-point analysis.

1. Layerwise Decomposition: The complex nested system is decoupled into three sequential boundary value problems. The authors derive explicit Green's function representations ($G^{(1)}_i, G^{(2)}_i, G^{(3)}_i$) for each layer, solving the outer problem first to determine the forcing term for the intermediate layer, and so on.
2. Integral Formulation: The differential inclusion is converted into an equivalent integral inclusion using the derived Green's functions. This allows the problem to be recast as a fixed-point problem for a multivalued operator $L$ acting on a product function space $(V_q)^k$.
3. Fixed-Point Theory: The existence of a solution is established using the Bohnenblust–Karlin fixed-point theorem (a generalization of Schauder's theorem for multifunctions). The authors verify that the operator $L$ is upper semi-continuous, has compact convex values, and maps a bounded, closed, convex set into itself. This verification relies on specific growth conditions (Hypotheses H1–H4) imposed on the multifunction $H_i$ and the perturbation term $h_i$.

Key Results

• Existence Theorem: The primary mathematical contribution is Theorem 3.4, which provides sufficient conditions for the existence of at least one solution to the coupled system in the space $(V_q)^k$. The conditions require the multifunction to be $L^1$-Carathéodory and satisfy specific growth bounds relative to the norms of the Green's functions and the conjugate exponents of the $p$-Laplacian.
• Illustrative Example: A scalar case ($k=1$) is presented with specific parameter values ($q=1/2, p=z=2$, etc.) to demonstrate the applicability of the existence theorem. The authors verify that the hypotheses hold for this specific setup and provide numerical visualizations of the Green's functions and selection profiles.
• Aerospace Vibration Modeling: The paper includes a practical illustration modeling the vibration isolation of an Inertial Measurement Unit (IMU) on a multirotor UAV. A base-excited single-degree-of-freedom model is used where the damping term is replaced by a Caputo $q$-fractional derivative. This term is intended to capture memory-dependent behavior in viscoelastic mounting layers.
  • Numerical Comparison: The model compares three configurations: rigid mounting, a classical viscous isolator, and the proposed $q$-fractional isolator.
  • Findings: The results show that while the classical viscous isolator provides the highest reduction in Root Mean Square (RMS) transmitted acceleration for the selected parameters, the $q$-fractional isolator offers a significant intermediate reduction compared to rigid mounting. The authors interpret this as evidence that the $q$-fractional term can effectively model memory-dependent damping, though they do not claim it is superior to viscous damping for the specific parameters tested.

Significance and Claims
The paper claims its main contribution is the rigorous existence result for the proposed nonlinear $q$-fractional differential inclusion system, which features a unique combination of four characteristics:

1. A $k$-dimensional coupled structure.
2. A nested operator structure involving Caputo and Riemann–Liouville $q$-derivatives linked by nonlinear $p$-Laplacian transformations.
3. Dependence of the multifunction on the full state vector and its $q$-derivative.
4. Nonlocal boundary conditions acting at different operator layers.

The authors emphasize that the mathematical framework developed (layerwise Green's functions and fixed-point formulation in product spaces) is necessary to handle these simultaneous features. The aerospace illustration serves to connect the abstract existence theory to a physical context involving memory-dependent damping, demonstrating how the theoretical framework can be instantiated in engineering modeling. The paper concludes modestly, noting that future work could address uniqueness, stability, numerical approximation, and control-oriented extensions, but does not make claims regarding the optimality of the fractional parameters in the vibration model.