Mild Solutions of Fractional Integrodifferential Diffusion Equations with Nonlocal Initial Conditions via the Resolvent Family

Abstract

Fractional integrodifferential diffusion equations play a significant role in describing anomalous diffusion phenomena. In this paper, we study the existence and uniqueness of mild solutions to these equations. Firstly, we construct an appropriate resolvent family, through which the related equicontinuity, strong continuity, and compactness properties are studied using the convolution theorem of Laplace transform, the probability density function, the Cauchy integral formula, and the Fubini theorem. Then, we construct a reasonable mild solution for the considered equations. Finally, we obtain some sufficient conditions for the existence and uniqueness of mild solutions to the considered equations by some fixed point theorems.

Fractional Neutral Integro-Differential Equations with Nonlocal Initial Conditions

“…in [19]. Furthermore, the study achieves the same results as those reported in [19] without the use of path integration.…”

“…Then, applying the Lebesgue dominated convergence theorem [28] and assumption (H 2 ), I 6 , I 7 → 0 as n → ∞ [19]. Consequently, this establishes the continuity of the operator G on B k 0 .…”

“…, t > 0, where E γ,η (•) is the Mittag-Leffler function for γ, η > 0 [19]. Set u(t)x = u(x, t), H(t, u(t))x = H(x, t, u(x, t)), and f (t, u(t))x = f (x, t, u(x, t)), where x ∈ [0, 1], t ∈ [0, T], and g : C([0, T], X) → X is given by…”

“…However, due to the simultaneous inclusion of derivative and integral terms in this paper, the single-parameter resolvent family is deemed unsuitable. Mu et al [19] demonstrated the existence of mild solutions for fractional diffusion equations with Dirichlet boundary conditions using the (α, β)-resolvent family, which is also relevant to Equation (4).…”