Science and Education Publishing Turkish Journal of Analysis and Number Theory 2333-1232 3 6 2015 12 29 The Solutions of Initial Value Problems for Second-order Integro-differential Equations with Delayed Arguments in Banach Spaces 154 159 EN Tingting Guan School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, P. R. China TJANT2015363 10.12691/tjant-3-6-3 2015 8 16 2015 11 24 2015 12 27 By using the partial order method and some new comparison results, the maximal or minimal solution of the initial value problem for nonlinear second order integro-differential equations with delayed arguments in Banach spaces are investigated. In this paper, we require only a lower solution or an upper solution and some weaker conditions presented here, and we extend and improve some recent results (see [1-11]).