Duality of fractional derivatives: On a hybrid and non-hybrid inclusion problem

The main goal of this paper is to investigate a newly proposed hybrid and hybrid inclusion problem consisting of fractional differential problems involving two different fractional derivatives of order μ, Caputo and Liouville–Riemann operators, with multi-order mixed Riemann–Liouville integro-derivative conditions. Although α is between one and two, we need three boundary value conditions to find the integral equation. The study investigates the results of existence for hybrid, hybrid inclusion, and non-hybrid inclusion problems by employing several analytical approaches, including Dhage’s technique, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>α</m:mi> <m:mo>-</m:mo> <m:mi>ψ</m:mi> </m:mrow> </m:math> {α-ψ} -contractive mappings, fixed points, and endpoints of the product operators. To further illustrate our findings, we present three examples.