AB–Mittag–Leffler Resolvent Family

• The AB–Mittag–Leffler resolvent family is an operator-valued construct defined via two-parameter Mittag–Leffler functions, serving as a rigorous basis for AB-type fractional evolution equations.
• It employs Laplace-domain representations and sectorial operator theory to establish sharp decay, stability, and regularity estimates under precise parameter constraints.
• The framework offers a robust alternative to traditional Caputo and Volterra methods for solving fractional Cauchy problems with memory effects and nonlocal dynamics.

The AB–Mittag–Leffler resolvent family is an operator-valued construction introduced to provide a rigorous analytic basis for evolution equations driven by the Atangana–Baleanu (AB) fractional derivative, distinguished by its non-singular Mittag–Leffler kernel. This approach extends the classical semigroup and Volterra frameworks to cover equations where standard Bernstein-function or completely monotone methodologies do not apply, establishing a robust foundation for the study of fractional Cauchy problems in Banach spaces with memory effects and nonlocality (Wakrim, 25 Jan 2026).

1. Analytical Preliminaries and Mittag–Leffler Functions

The AB–Mittag–Leffler resolvent family is grounded in the theory of two-parameter Mittag–Leffler functions and the associated AB kernels. For $\alpha>0$ and $\beta\in\mathbb{R}$, the Mittag–Leffler function is defined as

$$E_{\alpha,\beta}(z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma(\alpha k + \beta)}, \qquad z\in\mathbb{C},$$

which is entire and exhibits the decay $|E_{\alpha,\beta}(-z)| \leq C/(1+z)$ for $z\geq0$.

The AB kernel is given by

$$k_{\alpha,\beta}(t) = t^{\beta-1} E_{\alpha,\beta}(-c\, t^\alpha), \qquad t>0,\ 0<\alpha\leq1,\ \beta\geq1,\ c>0,$$

with Laplace symbol

$$\widehat{k}_{\alpha,\beta}(s) = \frac{s^{\alpha-\beta}}{s^\alpha + c}, \qquad s \notin (-\infty,0].$$

This kernel is everywhere finite at $t=0$, and notably fails to correspond to a Bernstein function, distinguishing the AB framework from the Caputo–Volterra tradition.

A sectorial operator $A$ acting on a Banach space $X$ is assumed, with spectrum in $\{\operatorname{Re}z \leq -\omega\}$ and admitting a resolvent estimate of the form

$$\|(zI-A)^{-1}\| \leq M_\varphi/|z+\omega|, \quad z \notin -\omega+\Sigma_\varphi,$$

for $\varphi$ in $(\theta, \pi)$ with sectorial angle $\theta<\pi/2$.

2. Construction of the Fractional Resolvent in the Laplace Domain

The AB–Mittag–Leffler resolvent is defined in the Laplace domain via

$$R_{\alpha,\beta}(s;A) := \frac{s^{\alpha-\beta}}{s^\alpha+c} (s^{\alpha-1}I-A)^{-1}, \qquad s \in \Gamma_\gamma,$$

where $\Gamma_\gamma$ is a left-sectorial Bromwich contour and $A$ is sectorial. Motivated by the Laplace transform of the AB–Caputo fractional derivative, this operator family serves as the fundamental solution operator for associated evolution problems.

The two-regime resolvent bound asserts that, for $0<\alpha\leq1$ and $\beta\geq1$,

$$\|R_{\alpha,\beta}(s;A)\|\leq C_\gamma \times \begin{cases} |s|^{1-\beta}, & |s|\leq1, \ |s|^{-\beta}, & |s|\geq1, \end{cases}$$

uniformly for $s\in\Gamma_\gamma$.

3. Definition, Laplace Representation, and Generation of the AB–Mittag–Leffler Resolvent Family

Under the integrability condition $\beta<1+\alpha$, the operator-valued contour integral

$$S_{\alpha,\beta}(t) := \frac{1}{2\pi i} \int_{\Gamma_\gamma} e^{st} R_{\alpha,\beta}(s;A)\,ds, \qquad t>0,$$

is absolutely convergent and yields a bounded family of operators on $X$. The function $t\mapsto S_{\alpha,\beta}(t)u_0$ is Laplace-transformable for every $u_0\in X$:

$$\mathcal{L}[S_{\alpha,\beta}(\cdot)u_0](s) = R_{\alpha,\beta}(s;A)u_0, \qquad \operatorname{Re}(s)>0.$$

The generation theorem states that for $A$ sectorial with $0<\alpha\leq1$ and $1\leq\beta<1+\alpha$, the function $u(t)=S_{\alpha,\beta}(t)u_0$ is the unique mild solution of the AB–Cauchy problem:

$${^AB}D_t^{\alpha,\beta}u(t) = Au(t),\quad u(0)=u_0,$$

where ${^AB}D_t^{\alpha,\beta}$ is the Caputo-type AB derivative given by

$${^AB}D_t^{\alpha,\beta}u(t) := \int_0^t k_{\alpha,\beta}(t-s)\,u'(s)\,ds.$$

4. Equivalent Representations and Series Decomposition

An equivalent contour formula expresses $S_{\alpha,\beta}(t)$ as

$$S_{\alpha,\beta}(t) = \frac{1}{2\pi i} \int_{|\arg\lambda| = \varphi} e^{\lambda t} \lambda^{\beta-1} (\lambda^\alpha I + A)^{-1} d\lambda,$$

for $\varphi\in(0,\pi/\alpha)$ with $\lambda^\alpha$ lying in the resolvent set of $-A$. This is a Hille–Phillips–type representation.

By comparison with the Dunford–Taylor functional calculus for the operator-valued Mittag–Leffler function,

$$E_{\alpha,\beta}(-t^\alpha A) = \sum_{k=0}^\infty \frac{(-1)^k t^{\alpha k}}{\Gamma(\alpha k + \beta)} A^k,$$

one obtains the decomposition

$$S_{\alpha,\beta}(t) = E_{\alpha,\beta}(-t^\alpha A) + K(t),$$

with $K(t)$ bounded and subject to the same decay estimates as $E_{\alpha,\beta}(-t^\alpha A)$.

5. Stability, Regularity, and Sharp Fractional-Domain Estimates

The AB–Mittag–Leffler resolvent family satisfies sharp stability of Mittag–Leffler (ML) type: there exist $C,c_1>0$ with

$$\| S_{\alpha,\beta}(t) \| \leq C E_{\alpha,\beta}(-c_1 t^\alpha)$$

for all $t>0$. For large $t$, the estimate $E_{\alpha,\beta}(-z) \sim (1+z)^{-1}$ ensures rapid decay, and the bound is optimal.

Fractional-domain and weighted stability properties include, for $0\leq\gamma<1$,

$$\|A^\gamma S_{\alpha,\beta}(t)\| \leq C_\gamma t^{-\alpha\gamma}, \qquad t>0,$$

and near $t=0$,

$$\|S_{\alpha,\beta}(t)\| = O(t^{\beta-1}),\qquad \|S_{\alpha,\beta}(t)A^\gamma\| = O(t^{\beta-1-\alpha\gamma}),\qquad t\to0^+.$$

For time-derivative regularity, for each $\delta\in(0,1)$, a constant $C_\delta$ exists such that

$$\|\partial_t S_{\alpha,\beta}(t)\| \leq C_\delta t^{\alpha-1-\delta}.$$

6. Comparison with Caputo and Volterra Theories

In the Caputo setting, the fundamental kernel is $t^{-\alpha}/\Gamma(1-\alpha)$, with Laplace symbol $s^{-\alpha}$ (a Bernstein function). The AB symbol $s^{\alpha-\beta}/(s^\alpha + c)$ is not Bernstein, thus precluding the use of classical Volterra–Prüss calculus based on complete monotonicity. Nonetheless, the Laplace-domain approach—via sectorial functional calculus—inherits the existence and uniqueness of mild solutions, sharp smoothing, and decay properties known from both analytic semigroups and classical ML fractional resolvents.

These results rigorously embed AB-type evolution equations in a mature functional-analytic framework, enabling their study with tools analogous to those for Caputo and Volterra models.

7. Parameter Constraints and Theoretical Completeness

The parameter ranges $0<\alpha\leq1$, $\beta\geq1$, and $\beta<1+\alpha$ are necessary and sufficient for the well-posedness and sharp estimates of the AB–Mittag–Leffler resolvent family. Under these, the operator family $S_{\alpha,\beta}(t)$ exists, is bounded, and yields solutions with full regularity theory:

$\sup_{t>0} \|S_{\alpha,\beta}(t)\| < \infty.$

The condition $\beta<1+\alpha$ is sharp, ensuring convergence of the fundamental contour integral. All principal decay, stability, and regularity rates for the AB–Mittag–Leffler resolvent family are optimal and inherit the best-known results from the Caputo–Volterra settings (Wakrim, 25 Jan 2026).