Existence Results for Resonant Functional Boundary Value Problems with Generalized Weighted Fractional Derivatives

In this article, we deduce the existence of a solution to the weighted fractional differential equation with functional boundary data involving an ω-weighted fractional derivative with Riemann–Liouville settings, D0+α,ψ,ω of order α∈]n−1,n[, on certain weighted Banach spaces when the nonlinear term contains the proportional delay term and fractional derivatives of order (0,1). After carefully defining a few weighted spaces and building a few weighted projection operators, we use Mawhin’s coincidence theory to derive a number of existence results at resonance. Furthermore, our method generalizes some prior results because numerous fractional differential operators are specific instances of the operator D0+α,ψ,ω and represent functional boundary conditions in a highly generic way. Lastly, we illustrate and support our theoretical results with an example.