Existence and stability results for nonlinear fractional order Riemann–Liouville Volterra–Stieltjes quadratic integral equations q S. Abbas a , M. Benchohra b,c , M. Rivero d , J.J. Trujillo e,⇑ a Laboratory of Mathematics, University of Saïda, PO Box 138, 20000 Saïda, Algeria b Laboratory of Mathematics, University of Sidi Bel-Abbès, PO Box 89, 22000 Sidi Bel-Abbès, Algeria c Department of Mathematics, Faculty of Science, King Abdulaziz University, 21589 Jeddah, Saudi Arabia d Departamento de Matemáticas, Estadística e I.O., University of La Laguna, 38271 La Laguna, Tenerife, Spain e Departamento de Análisis Matemático, University of La Laguna, 38271 La Laguna, Tenerife, Spain article info Keywords: Volterra–Stieltjes integral equation Fractional integral–differential equations Riemann–Liouville fractional operators Existence and stability of solutions Fixed point abstract Our aim in this paper is to study the existence and the stability of solutions for Riemann–Liouville Volterra–Stieltjes quadratic integral equations of fractional order. Our results are obtained by using some fixed point theorems. Some examples are provided to illustrate the main results. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction Fractional differential and integral equations have recently been applied in various areas of engineering, mathematics, physics and bio-engineering and other applied sciences. There has been a significant development in ordinary and partial fractional differential and integral equations in recent years; see the monographs of Abbas et al. [5], Baleanu et al. [7], Diet- helm [15], Hilfer [17], Kilbas et al. [18], Lakshmikantham et al. [19], Podlubny [20] and Tarasov [28], and the papers by Abbas et al. [1–3,6], Qian et al. [21–23], Vityuk and Golushkov [29]. Recently interesting results of the stability of the solutions of various classes of integral equations of fractional order have obtained by Abbas et al. [4,5], Banas ´ et al. [8–10], Darwish et al. [12], Dhage [13,14] and the references therein. In [8,10], Banas ´ et al. proved some existence results for the following nonlinear Volterra–Stieltjes quadratic integral equation xðtÞ¼ aðtÞþ f ðt; xðtÞÞ Z t 0 uðt; s; xðsÞÞd s gðt; sÞ; t P 0; ð1Þ where a : ½0; 1Þ ! ½0; 1Þ and f : ½0; 1Þ  R ! R are given continuous functions. In this paper we establish some sufficient conditions for the existence of solutions of the following fractional order Riemann–Liouville Volterra–Stieltjes quadratic integral equations of the form http://dx.doi.org/10.1016/j.amc.2014.09.023 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. q This work has been supported in part by the Government of Spain and FEDER (Grants MTM2010-16499 and MTM2013-41704). ⇑ Corresponding author. E-mail addresses: abbas_said_dz@yahoo.fr (S. Abbas), benchohra@yahoo.com (M. Benchohra), mrivero@ull.es (M. Rivero), jtrujill@ullmat.es (J.J. Trujillo). Applied Mathematics and Computation 247 (2014) 319–328 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc