Considering first-order periodic boundary value problems on time scales,the authors have obtained the sufficient conditions for the existence of extremal solutions by employing the approaches for upper and lower solutions and monotone iterative method.

By using the monotone iterative method and Mnch fixed points theorem,some existence and uniqueness theorems of solutions are obtained.

The supposition is that f satisfies Nuguma condition and Lipschitz condition,the method of lower and upper solutions and monotone iterative technique to obtain the existence of solutions between lower and upper solutions and maximal and minimal solutions is generalized.

A class of third-order two-point boundary value problem u(t)+f(t,u′(t),u(t))=0, 0≤t≤1u(0)=u′(0)=u′(1)=0is studied by monotone iterative technique.

By improving classical monotone iterative technique,an elementary approximation process and correspondent error estimate are given for classical Emden equations in unit ball.

Sufficient conditions are obtained for each solution to positive a prior bounds as t→∞ under some hypotheses and to tend to a positive constant as t→∞ in some special cases by using the method of lower and upper solutions and monotone iterative techniques.

The monotone iterative techniques is used to investigate the existence of extremal solution of periodic boundary value problems(PBVP) for neutral delay differential equation.

We make use of monotone iterative techniques and obtain its extremal solutions under weak conditions and give a corresponding application for fourth-order problems.