Abstract—In this paper, by using the monotone iterative method and Daher's fixed point theorem and an inequality of noncompact measure of Monch and Von Harten, we study the existence of maximum solution and minimal solution and the iterative approximation of the following initial value problem for O.D.E with delay in Banach spaces    u   ’ =   f  (t,  u (t),   ut), ut0 = ψ0 and give a sufficient condition of the existence of solutions of the above problem. The obtained results improve and generalize the corresponding results. 

Index Terms—Banach space, measure of noncompactness, differential equation, normal cone, 2000 Mathematics Subject Classification 34G20, 47H07.

Jinjun Fan, Xiaodi Li are with School of Mathematical Sciences, Shandong Normal University, Ji'nan, 250014, Shandong, China.
Haydar Akca is with the Department of Mathematics, College of Arts and Science faculty, Abu Dhabi University, UAE.

Cite: Jinjun Fan, Xiaodi Li, Haydar Akca, "Existence of Extremal Solutions of Differential Equations with Delay in Banach Spaces," International Journal of Applied Physics and Mathematics vol. 5, no. 2, pp. 115-125, 2015.