?-fractional derivative Caputo Operator Convex Functions Hermite–Hadamard’s Inequality Jenson Integral Inequality Kirchhoff-type equations Laplace Adomian Decomposition Method (LADM) Lyapunov functions Mittag–Leffler synchronization Moser iteration method Navier-Stokes equation Power-mean Inequality Razumikhin method Riemann—Liouville Fractional Integration b-vex functions conformable Laplace transform conformable double Laplace decomposition method conformable fractional derivative conformable partial fractional derivative controllability and observability Gramians delay differential system delays dependence on a parameter distributed delays energy inequality existence and uniqueness fixed point index fixed point theorem on mixed monotone operators fountain theorem fractional p-Laplacian