In this paper, we establish sufficient conditions to approve the existence and uniqueness of solutions of a nonlinear implicit ψ-Hilfer fractional boundary value problem of the cantilever beam model with nonlinear boundary conditions. By using Banach’s fixed point theorem, the uniqueness result is proved. Meanwhile, the existence result is obtained by applying the fixed point theorem of Schaefer. Apart from this, we utilize the arguments related to the nonlinear functional analysis technique to analyze a variety of Ulam’s stability of the proposed problem. Finally, three numerical examples are presented to indicate the effectiveness of our results.

In this paper, we suggest the bare minimum initial capital a firm providing insurance must hold to avoid going bankrupt. A case-separated investment discrete-time surplus process in motor insurance claims serves as the study's model. The 50th, 60th, 70th, and 80th percentiles are used as the dividing line between a claim's standard claim and large claim situations. We also discover a link between an insurance company's initial capital and the likelihood of ruin. The least squares regression method is utilized to calculate the minimum initial capital, and the simulation approach would be used to determine the ruin probability. The results indicate that the least initial capital is better provided by the 70th percentile than the 50th, 60th, and 80th percentiles, respectively.