I know that maybe is easy . But I depend on it to prove another theorems. I have this theorem : Let $ {V}{\mathrm{,}}{W}\hspace{0.33em}\mathrm{\in}{C}^{1}\left[{\left[{{t}_{0}{\mathrm{,}}{t}_{0}\hspace{0.33em}\mathrm{{+}}{h}}\right]{\mathrm{,}}{R}}\right] $
Be lower and upper solutions of the IVP: $ {x}{\mathrm{'}}\mathrm{{=}}{f}{\mathrm{(}}{t}{\mathrm{,}}{x}{\mathrm{)}}\hspace{0.33em}{\mathrm{;}}{x}{\mathrm{(}}{t}_{0}{\mathrm{)}}\mathrm{{=}}{x}_{0}\hspace{0.33em}\hspace{0.33em}{for}\hspace{0.33em}{t}\mathrm{\geq}{t}_{0} $
$ {V}{\mathrm{'}}\mathrm{\leq}{f}{\mathrm{(}}{t}{\mathrm{,}}{V}{\mathrm{)}}\hspace{0.33em}{\mathrm{;}}{V}{\mathrm{(}}{t}_{0}{\mathrm{)}}\mathrm{\leq}{x}_{0}\hspace{0.33em}\hspace{0.33em}{for}\hspace{0.33em}{t}\mathrm{\geq}{t}_{0} $ $ {W}{\mathrm{'}}\mathrm{\geq}{f}{\mathrm{(}}{t}{\mathrm{,}}{W}{\mathrm{)}}\hspace{0.33em}{\mathrm{;}}{W}{\mathrm{(}}{t}_{0}{\mathrm{)}}\mathrm{\geq}{x}_{0}\hspace{0.33em}{for}\hspace{0.33em}{t}\mathrm{\geq}{t}_{0} $
Uncouppled (natural type ) upper and lower solutions respectively . Suppose that for
$
{x}\mathrm{{>}}{y}
$
satisfies the inequality
$
{f}{\mathrm{(}}{t}{\mathrm{,}}{x}{\mathrm{)}}\mathrm{{-}}{f}{\mathrm{(}}{t}{\mathrm{,}}{y}{\mathrm{)}}\mathrm{\leq}{L}{\mathrm{(}}{x}{\mathrm{,}}{y}{\mathrm{)}}\hspace{0.33em}{\mathrm{;}}{L}\mathrm{{>}}{0}
$
Then
$ {V}{\mathrm{(}}{t}_{0}{\mathrm{)}}\mathrm{\leq}{W}{\mathrm{(}}{t}_{0}{\mathrm{)}}\hspace{0.33em} $ Implies that $ {V}{\mathrm{(}}{t}{\mathrm{)}}\mathrm{\leq}{W}{\mathrm{(}}{t}{\mathrm{)}}\hspace{0.33em} $ $ {t}\mathrm{\in}\left[{{t}_{0}{\mathrm{,}}{t}_{0}\mathrm{{+}}{h}}\right] $ I just want to ask if we add the assumption that the function $ f\mathrm{(}t\mathrm{,}x\mathrm{)} $ is monotone nondecreasing in x .What is the difference ?? What is the difference in the proof ?? And what is the advantage ...? Thanks in advance .....