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.

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We use the cosine family of linear operators to prove the existence, uniqueness, and stability of the integral solution of a nonlocal telegraph equation in frame of the conformable time-fractional derivative. Moreover, we give its implicit fundamental solution in terms of the classical trigonometric functions.

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is paper deals with the existence of mild solutions for the following Cauchy problem:where d α (.)/dt α is the so-called conformable fractional derivative. e linear part A is the infinitesimal generator of a uniformly continuous semigroup (T(t)) t≥0 on a Banach space X, f and g are given functions. e main result is proved by using the Darbo-Sadovskii fixed point theorem without assuming the compactness of the family (T(t)) t>0 and the Lipshitz condition on the nonlocal part g.

In this paper we study a fractional differential equations problem with not instantaneous impulses involving a non-compact semigroup. We present some concepts and facts about the strongly continuous semigroup and the measure of noncompactness. After that we give an existence theorem of our problem using a condensing operator and the measure of noncompactness.

In this paper, we investigate second order evolution differential equation in the frame of sequential conformable derivatives with nonlocal condition. First, we establish Duhamel's formula in terms of a standard cosine family of linear operators. Then, we prove some results concerning the existence, uniqueness, stability, and regularity of mild solution concept. Moreover, we present a concrete application of the main results. MSC: 34A08; 47D09

In this paper we are interested in studying the existence of solutions for a controlled impulsive fractional evolution equations. We use several tools such as fractional calculus, fixed point theorems and the theory of semigroup. We first give some preliminaries and notations, the second part of the work we provide an existence result for our problem and in the final section, we give some examples to show the importance of our results.