Monotone iterative technique for delay differential equations

In proving existence of extremal solutions for delay differential equations, one usually assumes nondecreasing property on the function involved without which the proof breaks down. Our main purpose, in this paper, is to remove the monotone assumption by proving a new comparison result which is required in the analysis and which avoids the standard line of argument. Furthermore, our comparison theorem shows that the choice of initial functions cannot be arbitrary for the results to hold. Using this result we then develop monotone technique to obtain extremal solutions.