European Journal of Scientific Research ISSN 1450-216X / 1450-202X Vol. 151 No 3 January, 2019, pp. 320-334 http://www. europeanjournalofscientificresearch.com On Nonlocal Problems for Fractional Integro-Differential Equation in Banach Space Mohammed S. Abdo Research Scholar at Department of Mathematics Dr. Babasaheb Ambedkar Marathwada University Aurangabad 431004 (M.S.), India E-mail: msabdo1977@gmail.com Abdulkafi M. Saeed Associate Professor, Department of Mathematics College of Science, Qassim University, Saudi Arabia E-mail: abdulkafi.ahmed@qu.edu.sa Hanan A. Wahash Research Scholar at Department of Mathematics Dr. Babasaheb Ambedkar Marathwada University Aurangabad 431004 (M.S.), India E-mail: moh_wosabi@hotmail.com Satish K. Panchal Department of Mathematics Dr. Babasaheb Ambedkar Marathwada University Aurangabad 431004 (M.S.), India E-mail: drpanchalsk@gmail.com Abstract The aim of the present paper is to study the Cauchy-type problem for a integro- differential equation of fractional order with nonlocal conditions in Banach spaces. The fractional differential operator is taken in the Caputo sense. New conditions on the nonlinear terms are given to guarantee the equivalence. We shall prove the existence and uniqueness results by means of Banach fixed point and the Krasnoselskii's fixed point theorems. At the end, an illustrative example will be introduced to justify our results. Keywords: Nonlocal problems, Fractional integro-differential equation, fixed point theorem, Banach space 1. Introduction Fractional differential equations are linked with extensive applications such as continuum phenomena mechanics, electrochemistry, biophysics, biotechnology engineering and so forth. For more details see studies of Kilbas et al. [16], Miller and Ross [18] Oldham and Spanier [21], Samko et al. [22], and many other references. The existence and uniqueness of solutions to fractional differential equations have attracted the attention of many scientists and researchers. For instance, (see [1, 2, 3, 4, 5, 11, 12, 13, 15, 17]). Integro-differential equations emerge in many scientific and engineering specialties,