Malaya Journal of Matematik,, Vol. 7, No. 4, 709-715, 2019 https://doi.org/10.26637/MJM0704/0014 Remarks on the fractional abstract differential equation with nonlocal conditions Mohammed Benyoub 1 * Samir Benaissa 2 and Kacem Belghaba 3 Abstract In this paper, we study the existence and uniqueness of a solution to an initial value problem for a class of nonlinear fractional involving Riemann-Liouville derivative with nonlocal initial conditions in Banach spaces. We prove our main result by introducing a regular measure of noncompactness in the weighted space of continuous functions and using fixed point theory. Our result improve and complement several earlier related works. An example is given to illustrate the applications of the abstract result. Keywords Riemann-Liouville fractional derivative, Riemann-Liouville fractional integral, nonlocal initial conditions, point fixed, measure of noncompactness. AMS Subject Classification 26A33, 34A08, 34K30, 47H08, 47H10. 1,2 Department of Mathematics, University Djillali Liab ` es of Sidi Bel-Abb ` es, B.P.89, 22000, Sidi Bel-Abb ` es, Algeria. 3 Department of Mathematics, Laboratory of Mathematics and its applications, University of Oran1 A.B, 31000 Oran, Algeria. *Corresponding author: 1 mohamedbenyoub64@yahoo.com; 2 benaissamir@yahoo.fr ; 3 belghaba@yahoo.fr Article History: Received 18 July 2019; Accepted 27 October 2019 c 2019 MJM. Contents 1 Introduction ....................................... 709 2 Preliminaries ...................................... 710 3 Main Results ...................................... 711 4 An example ........................................ 714 References ........................................ 714 1. Introduction Recently, fractional differential equations have attracted considerable interest in both mathematics and applications, since they have been proved to be valuable tools in modeling many physical phenomena. There has been significant devel- opment in fractional differential equations in recent years, see the monographs of Samko et al.[27], Kilbas et al.[20], Miller and Ross [22], Podlubny [26], and the references therein. The definitions of Riemann-Liouville fractional derivatives or integrals initial conditions play an important role, in some practical problems. Heymans and Podlubny [19], have demon- strated that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives or integrals on the field of the viscoelasticity, and such initial conditions are more appropriate than physically interpretable initial conditions. In [14], Gaston et al. studied fractional order differential equations with Caputo derivative D q x(t )= f (t , x(t )); t ∈ [0, b];0 < α < 1, with nonlocal condition x(0)+ g(x)= x 0 . As indicated in Deng’s pioneering paper [11], the nonlocal condition x(0)+ g(0)= x 0 can be applied in physics with better effect than the classical Cauchy problem with initial condition x(0)= x 0 . For instance the author used g(x)= Σ p i=1 c i x(t i ), where c i = 1, 2, ···, p are given constants and 0 < t 1 < t 2 < ··· < t p ≤ T . To describe the diffusion phenomenon of a small amount in a transparent tube. In this case, the Cauchy problem allows the additional measurements at t i , i = 1, 2, ···, p. In this work we consider the following Cauchy problem for the nonlocal initial conditions fractional differential equation L D α 0 + x(t )= f (t , x(t )); t ∈ J ′ :=(0, b], (1.1) (I 1−α 0 + x)(0)+ g(x)= x 0 , (1.2)