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Essay On Monotone Mappings

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‎Here‎, ‎we prove that for a monotone mapping that fixes the origin‎, ‎the \text{NCP} has always a solution‎. ‎If‎, ‎moreover‎, ‎the mapping is strictly monotone‎, ‎then zero is the unique solution‎. ‎These results‎ ‎are stronger than known results in this direction for two reasons‎: ‎firstly‎, ‎there is no condition‎ ‎on the nature of the cone and secondly‎, ‎no feasibility assumptions are made‎. ‎We start by mentioning a following lemma‎. ‎\begin{lem} \label{lem 2.1}‎ ‎Let $ F‎ : ‎H \longrightarrow H $ be a continuous and monotone mapping on the nonempty, closed, convex set ‎$ K ‎\subseteq ‎H‎ $‎‎. ‎Then there is a $ z_{r} \in K_{r} $ such that‎ ‎\begin{align*}‎ ‎\langle ‎z‎ - ‎z_{r}, F(z_{r}) \rangle \geqslant 0‎ ‎\end{align*}‎ …show more content…

‎Let $ z_{r} \in K_{r} $ be the point such that‎ ‎\begin{align*}‎ ‎\langle ‎z‎ - ‎z_{r}, ‎F(z_{r}) \rangle \geqslant 0‎ ‎\end{align*}‎ ‎for all $ z \in K_{r} $‎. ‎Then $ z_{r} $ is a solution of the $ \text{NCP} (F‎, ‎K) $‎. ‎\end{thm}‎ ‎\begin{proof}‎ ‎Since‎ ‎\begin{align*}‎ ‎\langle ‎z‎ - ‎z_{r}, ‎F(z_{r}) \rangle \geqslant 0 \quad for all z \in K_{r}‎, ‎\end{align*}‎ ‎it follows by taking $ z = 0 $ that‎ ‎\begin{align} \label{dodo}‎ ‎\langle ‎z_{r}, ‎F(z_{r}) \rangle \leqslant 0‎. ‎\end{align}‎ ‎Let $ t \in [0‎, ‎1] = I $‎. ‎We then have from Lemma \ref{lem 2.2} that‎ ‎\begin{align*}‎ ‎\langle ‎t z_{r}, ‎F(tz_{r}) \rangle \leqslant \langle ‎z_{r}, ‎F(z_{r}) \rangle \leqslant \langle‎ ‎z, ‎F(z_{r}) \rangle‎ ‎\end{align*}‎ ‎for all $ z \in K_{r} $‎. ‎It is also evident that the second inequality above holds for all $ z \in K $‎. ‎Thus we have‎ ‎\begin{align*}‎ - ‎\langle ‎z, F(z_{r}) \rangle \leqslant‎ - ‎\langle ‎z_{r}, F(z_{r}) \rangle \leqslant‎ - ‎\langle‎ ‎t z_{r}, ‎F(tz_{r}) \rangle‎. ‎\end{align*}‎ ‎Since $‎ - ‎\langle ‎z_{r}, F(z_{r})‎ \rangle \geqslant 0 $ by virtue of \eqref{dodo}‎, ‎it follows that‎ ‎\begin{align*}‎ - ‎\langle ‎t z_{r}, F(tz_{r}) \rangle \geqslant 0‎, ‎\end{align*}‎ ‎so that we can apply Cauchy‎ - ‎Schwartz inequality to get‎ ‎\begin{align} \label{dose}‎ - ‎\langle ‎z, F(z_{r}) \rangle \leqslant‎ -

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