Period Math Hung DOI 10.1007/s10998-017-0223-y Darbo type fixed and coupled fixed point results and its application to integral equation Hemant K. Nashine 1 · Reza Arab 2 · Ravi P. Agarwal 1 · Ali Shole Haghighi 2 © Akadémiai Kiadó, Budapest, Hungary 2017 Abstract We propose two new notion of contraction mappings involving measure of non- compactness in the frame work of Banach space and derive some basic Darbo type fixed and coupled fixed point results. The results are correlated with the classical Banach fixed point theorems. Further we show the applicability of obtained results to the theory of integral equations following a concrete example which illustrate the application part. Keywords Fixed point · Coupled fixed point · Measure of noncompactness · Functional- integral equations Mathematics Subject Classification 54H25 · 47H10 1 Introduction and preliminaries First we recall some notations, definitions and theorems to obtain all the results of this work. Denote by R the set of real numbers and put R + =[0, +∞). Let ( E , ‖.‖) be a real Banach space with zero element 0. Let B(x , r ) denote the closed ball centered at x with radius r . The symbol B r stands for the ball B(0, r ). For X , a nonempty subset of E , we denote by X and Conv X the closure and the closed convex hull of X , respectively. Moreover, let us denote by M E the family of nonempty bounded subsets of E and by N E its subfamily consisting of all B Reza Arab mathreza.arab@iausari.ac.ir Hemant K. Nashine drhknashine@gmail.com Ali Shole Haghighi ali.sholehaghighi@gmail.com 1 Department of Mathematics, Texas A & M University - Kingsville, Kingsville, TX 78363-8202, USA 2 Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran 123