Transmutation operators for the Sturm--Liouville equation in impedance form with an integrable potential

In this work, we study a family of transmutation operators for the Sturm–Liouville operator in impedance form with an integrable potential satisfying a smallness condition. Specifically, we consider transmutations associated with the pair $${\mathbf{\text{L}}}_r=\frac{1}{r^2(x)}\frac{d}{dx}{r}^2(x)\frac{d}{dx}\text{and}\mathbf{\text{M}}=\frac{d^2}{d{x}^2}\mathrm{.}$$ We extend existing results by showing that a particular transmutation operator on $W^{3,1}(-a,a)$ can be represented in integral form, where the kernel function satisfies a corresponding Goursat problem. We establish the existence of this kernel using two distinct approaches: the classical method of successive approximations and an application of Picard’s theorem.