q-Mittag–Leffler Resolvents for Nonlinear Caputo q-Fractional Equations: Well-Posedness, Stability, and the q → 1− Classical Limit

Technical Summary: q-Mittag–Leffler Resolvents for Nonlinear Caputo q-Fractional Equations

Problem Statement
This research addresses the analysis of nonlinear semilinear Caputo $q$-fractional initial-value problems on a finite $q$-time horizon. The specific equation under investigation is:
$${}^{C}D_{q,0}^{\alpha}x(t) = \lambda x(t) + F(t, x(t)), \quad x(0) = x_0, \quad 0 < \alpha < 1$$
where ${}^{C}D_{q,0}^{\alpha}$ denotes the Caputo $q$-fractional derivative. The paper seeks to move beyond the mere definition of $q$-analogues of special functions (such as the $q$-Gamma and $q$-Mittag–Leffler functions) and instead focuses on three structural questions:

1. Interpreting the $q$-Mittag–Leffler function as the resolvent of the evolution equation rather than just a series definition.
2. Establishing explicit conditions for the well-posedness (existence and uniqueness) of the nonlinear problem via contraction arguments.
3. Proving that the nonlinear $q$-problem converges to the classical Caputo fractional problem as the deformation parameter $q \to 1^{-}$, specifically at the level of solutions, not just coefficients.

Methodology
The analysis relies on a fixed-point framework within the Banach space $X_q = C([0, T]_q, \mathbb{R})$ equipped with the sup-norm.

• Resolvent Formulation: The authors reformulate the differential equation into a Volterra-type integral equation using the $q$-fractional integral $I_{q,0}^{\alpha}$. They define a resolvent operator $R_{q,\alpha,\lambda} = (I - \lambda I_{q,0}^{\alpha})^{-1}$. By expanding this via a Neumann series, they demonstrate that the action of this resolvent on the initial datum $x_0$ yields $x_0 qE_{\alpha}(\lambda, t)$, thereby identifying the $q$-Mittag–Leffler function as the natural homogeneous response of the system.
• Well-Posedness: Existence and uniqueness are established using the Banach Fixed-Point Theorem. The authors derive a sufficient condition for the contraction mapping principle to hold:
  $$\kappa_q := (|\lambda| + L) \frac{T^{\alpha}}{\Gamma_q(\alpha + 1)} < 1$$
  where $L$ is the Lipschitz constant of the nonlinear term $F$.
• Stability and Dependence: Continuous dependence on initial data is derived using a $q$-fractional Gronwall inequality, resulting in an estimate bounded by a $q$-Mittag–Leffler function. Stability is discussed via established Lyapunov theory for $q$-fractional systems, linking the decay of solutions to the properties of the $q$-Mittag–Leffler function.
• Convergence Analysis: The core theoretical contribution is a proof of uniform convergence as $q \to 1^{-}$. The authors leverage the known convergence of the Jackson integral to the Riemann integral and the $q$-Gamma function to the classical Gamma function. They construct a perturbation estimate for the fixed-point maps, showing that if the classical problem has a solution and the $q$-problems remain uniformly contractive, the $q$-solutions converge uniformly to the classical solution.

Key Results

1. Resolvent Interpretation: The paper rigorously establishes that the $q$-Mittag–Leffler function $qE_{\alpha}(\lambda, t)$ is the resolvent response of the linear Caputo $q$-fractional operator, separating the linear memory mechanism from nonlinear forcing.
2. Explicit Well-Posedness: A transparent, explicit condition (Equation 25) is provided for the existence and uniqueness of solutions on a finite horizon. This condition is noted as sufficient rather than sharp but is crucial for the subsequent convergence proofs.
3. Solution-Level Convergence: Theorem 6.1 proves that under uniform contractivity and consistency of the integral operators, the unique $q$-solution $x_q$ converges uniformly to the classical Caputo solution $x$ as $q \to 1^{-}$. The error bound is explicitly related to the difference between the $q$-fractional and classical fractional integrals.
4. Numerical Verification:
   • Linear Case: Numerical experiments on a linear relaxation model ($CD_{q,0}^{0.75}x = -0.8x$) confirm the theoretical convergence. The error $\|x_q - x\|_\infty$ decreases as $q \to 1$, with a log-log plot suggesting an approximate first-order dependence on $1-q$ for the tested benchmark.
   • Nonlinear Case: A nonlinear cubic relaxation model ($CD_{q,0}^{0.75}x = -0.7x - 0.6x^3$) was solved using a geometric Jackson grid. The results confirm that the nonlinear $q$-solutions converge to the classical reference solution, validating the theoretical framework beyond the linear regime.

Significance and Claims
The authors explicitly state that the definitions of $q$-Gamma, $q$-Beta, and $q$-Mittag–Leffler functions are not new contributions, nor are the existence of Caputo $q$-problems or their stability properties. Instead, the paper's significance lies in its structural synthesis:

• It reframes the $q$-Mittag–Leffler function as an intrinsic part of the resolvent operator for semilinear problems.
• It provides a unified proof that the entire nonlinear dynamical system, not just its special function components, recovers the classical fractional dynamics in the limit $q \to 1^{-}$.
• It distinguishes the role of the fractional order $\alpha$ (controlling nonlocal memory) from the deformation parameter $q$ (controlling the geometric/quantum structure of differentiation), arguing that $q$ should not be conflated with the fractional order.

The paper concludes that these results replace a sequence of non-original definitions with a precise mathematical question and a reproducible answer regarding the continuity of quantum-deformed fractional dynamics toward their classical limits.