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

Abstract

Abstract 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, α - ψ {\alpha-\psi} -contractive mappings, fixed points, and endpoints of the product operators. To further illustrate our findings, we present three examples.