Optim Lett (2013) 7:1251–1260 DOI 10.1007/s11590-012-0501-9 ORIGINAL PAPER Existence results for coupled systems of quadratic integral equations of fractional orders A. M. A. El-Sayed · H. H. G. Hashem Received: 21 December 2011 / Accepted: 16 May 2012 / Published online: 30 May 2012 © Springer-Verlag 2012 Abstract We are concerning here with the existence of at least one positive contin- uous solution of coupled systems of quadratic integral equations of fractional orders. Then an existence theorem for a coupled system of Cauchy problems will be proved. Keywords Quadratic integral equation of fractional-order · Coupled system · Tychonov fixed-point theorem · Cauchy problems 1 Introduction It is well known that integral equations have many useful applications in describing numerous events and problems of real world, and the theory of integral equations is rapidly developing with the help of several tools of functional analysis, topology and fixed point theory. For details, we refer to [1–8] and [13–22]. Quadratic integral equations (QIEs) are often applicable in the theory of radiative transfer, kinetic theory of gases, in the theory of neutron transport and in the traf- fic theory. The quadratic integral equations can be very often encountered in many applications. The quadratic integral equations have been studied in several papers and monographs (see for examples [1–8] and [13–22]). A. M. A. El-Sayed · H. H. G. Hashem Faculty of Science, Alexandria University, Alexandria, Egypt e-mail: amasayed@hotmail.com H. H. G. Hashem (B ) Faculty of Science, Qassim University, Buraidah, Saudi Arabia e-mail: hindhghashem@gmail.com; hendhghashem@yahoo.com 123