Boundary value problems for hybrid differential equations

This note is motived from some papers treating  the hybrid differential equations. An existence theorem for this equation is proved. Some fundamental differential inequalities for hybrid differential equatins  are also established which are utilized to prove the existence of extremal solutions. Necessary tools are considered and the comparison principle is proved which will be useful for further study of qualitative behavior of solutions

Existence of solutions for hybrid differential equation with fractional order

In this paper, we study the existence of solutions for the following fractional hybrid differential equations involving Riemann-Liouville differential operators of order . An existence theorem for fractional hybrid differential equations is proved under mixed Lipschitz and Carathéodory conditions and using the Dhage point fixe theorem. Keywords: Quadratic perturbations;  Riemann-Liouville derivative;  Hybrid differential equation