This paper studies a family of $ (\mu, \wp) $-cotangent fractional integral and differential operators of Riemann–Liouville and Caputo type. A central structural observation is made explicit. After the change of variable $ x = \wp(z)-\wp(c) $ and an exponential conjugation, the proposed operators reduce to classical Riemann–Liouville/Caputo operators of normalized order $ \alpha = \sigma/\mu $, multiplied by an explicit normalization factor. Consequently, the parameters $ (\sigma, \mu, r) $ do not represent three independent degrees of freedom in the kernel shape; up to normalization, their effect is described by the normalized order $ \alpha $ and the attenuation rate $ a = \cot(\pi r/2)/\mu $. We retain $ \sigma $ and $ \mu $ separately because this notation is compatible with the $ \mu $-Gamma calculus and makes the scaling relations and special cases transparent. Within this corrected interpretation, we establish semigroup and composition rules, Riemann–Liouville/Caputo inversion formulas, and an extended Gronwall inequality. The induction step in the Gronwall proof is written with the complete Beta-to-$ \Gamma_\mu $ conversion, and the associated Mittag–Leffler series is identified explicitly with a scaled classical one-parameter Mittag–Leffler function. As a concrete application, we prove an existence result and a Gronwall-based uniqueness and continuous-dependence estimate for a nonlinear Cauchy problem governed by the $ (\mu, \wp) $-cotangent Caputo derivative. In addition to the classical-limit tests, a nontrivial example with $ \mu = 2 $, $ r = 1/2 $, and $ \wp(z) = \log z $ illustrates how the exponential attenuation and normalization jointly affect the bound.
Citation: Ziyad A. Alhussain. Generalized $ (\mu, \wp) $-cotangent fractional operators: Theory, properties, and an extended Gronwall inequality[J]. AIMS Mathematics, 2026, 11(9): 29587-29609. doi: 10.3934/math.20261174
This paper studies a family of $ (\mu, \wp) $-cotangent fractional integral and differential operators of Riemann–Liouville and Caputo type. A central structural observation is made explicit. After the change of variable $ x = \wp(z)-\wp(c) $ and an exponential conjugation, the proposed operators reduce to classical Riemann–Liouville/Caputo operators of normalized order $ \alpha = \sigma/\mu $, multiplied by an explicit normalization factor. Consequently, the parameters $ (\sigma, \mu, r) $ do not represent three independent degrees of freedom in the kernel shape; up to normalization, their effect is described by the normalized order $ \alpha $ and the attenuation rate $ a = \cot(\pi r/2)/\mu $. We retain $ \sigma $ and $ \mu $ separately because this notation is compatible with the $ \mu $-Gamma calculus and makes the scaling relations and special cases transparent. Within this corrected interpretation, we establish semigroup and composition rules, Riemann–Liouville/Caputo inversion formulas, and an extended Gronwall inequality. The induction step in the Gronwall proof is written with the complete Beta-to-$ \Gamma_\mu $ conversion, and the associated Mittag–Leffler series is identified explicitly with a scaled classical one-parameter Mittag–Leffler function. As a concrete application, we prove an existence result and a Gronwall-based uniqueness and continuous-dependence estimate for a nonlinear Cauchy problem governed by the $ (\mu, \wp) $-cotangent Caputo derivative. In addition to the classical-limit tests, a nontrivial example with $ \mu = 2 $, $ r = 1/2 $, and $ \wp(z) = \log z $ illustrates how the exponential attenuation and normalization jointly affect the bound.
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Ziyad A. Alhussain. Generalized $ (\mu, \wp) $-cotangent fractional operators: Theory, properties, and an extended Gronwall inequality[J]. AIMS Mathematics, 2026, 11(9): 29587-29609. doi: 10.3934/math.20261174
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