Stability Analysis of Linear and Nonlinear Abstract Cauchy Problems With Resolvent Kernels Using a Novel Hybrid Fractional Derivative Approach

Consider an injective closed linear operator Λ defined on a Banach space Ψ and introduce the modified Atangana–Baleanu–Caputo (mABC) fractional derivative, denoted by . We demonstrate the existence of a unique solution to the abstract Cauchy problem Additionally, we provide the unique solution to the first‐order abstract Cauchy problem u ′ (t) = M γ u(t) + F γ (t), t ≥ 0; u(0) = − Λ −1 f(0), where the family of bounded linear operators M γ constitutes a Yosida approximation of Λ and F γ ⟶f(t) as γ ⟶1. Furthermore, we investigate the conditions under which 1/(1 − γ ) ∈ Y ( Λ ), ensuring that the spectrum of Λ lies outside a closed disk of radius 1/2(1 − γ ) . Under these conditions, the solution to the Cauchy problem for the mABC fractional derivative converges to zero as t⟶∞. We also establish the existence and uniqueness of mild solutions to the semilinear problem with initial conditions in the set ℧≔{y ∈ Ω ( Λ ): y = Λ −1 f(0, y)}. Stability results are obtained, and the theoretical results are applied to establish the existence of mild solutions for two partial integro‐differential processes.