Properties of the resolvent of a linear Abel integral equation: implications for a complementary fractional equation - Leigh Becker (Electronic Journal of Qualitative Theory of Differential Equations, 2016)
New and known properties of the resolvent of the kernel of linear Abel integral equations of the form \begin{equation} \tag{A$_λ$} x(t) = f(t) - λ \int_0^t (t - s)^{q-1}x(s)\,ds, \end{equation} where $λ > 0$ and $q \in (0,1)$, are assembled and derived here. First, a priori bounds on potential solutions of the resolvent equation \begin{equation} \tag{R$_λ $} R(t) = {λ }t^{q-1} - {λ }\int_0^t (t - s)^{q-1}R(s)\,ds \end{equation} are obtained. Second, it is proven - using these bounds, Banach's contraction mapping principle, new continuation and translation results, and Schaefer's fixed point theorem - that (R$_λ $) has a unique continuous solution on $(0,∞)$, which is called the resolvent in the literature and denoted here by $R(t)$. Third, both known and new properties of $R(t)$ are derived. Fourth, $R(t)$ is shown to be completely monotone and the unique continuous solution of the initial value problem of fractional order $q$: \begin{equation} \tag{I$_λ $} D^{q}x(t) = -λ Γ(q) x(t), \qquad \lim_{t → 0^+} t^{1-q}x(t) = λ , \end{equation} where $D^{q}$ denotes the Riemann-Liouville fractional differential operator. Finally, the resolvent integral function $\int_0^t R(s)\,ds$ is shown to be the unique continuous solution of an integral equation closely related to (R$_λ$). Closed-form expressions for it and $R(t)$ are derived.