This note is motivated by some papers treating the fractional hybrid differential equations involving Riemann-Liouville differential operators of order 0 < α < 1. An existence theorem for this equation is proved under mixed Lipschitz and Carathéodory conditions. Some fundamental fractional differential inequalities which are utilized to prove the existence of extremal solutions are also established. Necessary tools are considered and the comparison principle is proved, which will be useful for further study of qualitative behavior of solutions.

This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition D α f t = lim h ⟶ 0 f t + h e α − 1 t − f t / h , for all t > 0 , and α ∈ 0,1 . If α = 0 , this definition coincides to the classical definition of the first order of the function f .

In this paper, a class of nonlocal impulsive differential equation with conformable fractional derivative is studied. By utilizing the theory of operators semigroup and fractional derivative, a new concept on a solution for our problem is introduced. We used some fixed point theorems such as Banach contraction mapping principle, Schauder’s fixed point theorem, Schaefer’s fixed point theorem, and Krasnoselskii’s fixed point theorem, and we derive many existence and uniqueness results concerning the solution for impulsive nonlocal Cauchy problems. Some concrete applications to partial differential equations are considered. Some concrete applications to partial differential equations are considered.

In this work, we discuss the existence and uniqueness of solution for a boundary value problem for the Langevin equation and inclusion with the Hilfer fractional derivative. First of all, we give some definitions, theorems, and lemmas that are necessary for the understanding of the manuscript. Second of all, we give our first existence result, based on Krasnoselskii’s fixed point, and to deal with the uniqueness result, we use Banach’s contraction principle. Third of all, in the inclusion case, to obtain the existence result, we use the Leray–Schauder alternative. Last but not least, we give an illustrative example.

This paper is motivated by some papers treating the fractional hybrid differential equations with nonlocal conditions and the system of coupled hybrid fractional differential equations; an existence theorem for fractional hybrid differential equations involving Caputo differential operators of order1<α≤2is proved under mixed Lipschitz and Carathéodory conditions. The existence and uniqueness result is elaborated for the system of coupled hybrid fractional differential equations.