%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[11pt]{amsart} \usepackage{amsmath} \usepackage{amssymb} \usepackage{graphicx} %\usepackage[initials]{amsrefs} %\usepackage{fancyhdr} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Theorem 1.1, Lemma 1. 2, Proposition 1.3 %%111111111111111111111% Theorem 1.1, Lemma 1. 2, Proposition 1.3 % 1111111111111111111 %Please use the following commands. %sample Theorem 1.1 %begin{theoorem}\label{th1} %.... %\end{theorem} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \newcommand{\x}{(x,x^{*})} \newcommand{\y}{(y,y^{*})} \newcommand{\X}{X\times X^{*}} \newcommand{\Xx}{X\times X} \newcommand{\xX}{X^{*}\times X^{*}} \newcommand{\mn}{\mathbb{N}} \newcommand{\mc}{\mathbb{C}} \newcommand{\mq}{\mathbb{Q}} \newcommand{\mi}{\mathbb{I}} \newcommand{\mk}{\mathbb{K}} \newcommand{\mz}{\mathbb{Z}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\mbr}{\mathbb{R}} \newcommand{\mrp}{\mathbb{R}_{+}} \newcommand{\mrl}{\mathbb{R}^{\ell}} \newcommand{\mrn}{\mathbb{R}^{n}} \newcommand{\mrm}{\mathbb{R}^{m}} \newcommand{\mrlp}{\mathbb{R}^{\ell}_{+}} \newcommand{\pe}{\partial^{\sigma}} \newcommand{\si}{\mathscr{I}} \newcommand{\ic}{{\rm int}~C} \newcommand{\icy}{{\rm int}C(y)} \newcommand{\icx}{{\rm int}C(x)} \newcommand{\icz}{{\rm int}C(z)} \newcommand{\ici}{{\rm int}~C_{i}} \newcommand{\icix}{{\rm int}~C_{i}(x)} \newcommand{\ns}{\not\subseteq} \newcommand{\pcfiy}{\partial^{c}f_{i}(y)} \newcommand{\pcfix}{\partial^{c}f_{i}(x)} \newcommand{\pcfi}{\partial^{c}f_{i}} \newcommand{\pcf}{\partial^{c}f} \newcommand{\pv}{\partial \varphi} %\newcommand{\pe}{\partial^{\sigma}} \newcommand{\rra}{\rightrightarrows} \newcommand{\bz}{{\bf 0}} \newcommand{\co}{\operatorname{co}} \newcommand{\R}{\operatorname{R}} \newcommand{\Dm}{\operatorname{Dom}} \newcommand{\D}{\operatorname{D}} \newcommand{\G}{\operatorname{G}} \newcommand{\inte}{\operatorname{int}} \newcommand{\con}{\operatorname{Cont}} \newcommand{\bxx}{Bx} \newcommand{\ax}{ \langle x,x'\rangle} \newcommand{\px}{(x,x')} \newcommand{\laa}{ \langle } \newcommand{\raa}{\rangle} \newcommand{\lp}{(} \newcommand{\bb}{\beta_{n}} \newcommand{\ve}{\Vert} \newcommand{\aaa}{a_{n}} \newcommand{\w}{w'_{n}} \newcommand{\rp}{)} \newcommand{\vv}{v'_{n}} \newcommand{\n}{\mathbb{N}} \newcommand{\FA}{F_{A}} \newcommand{\xtx}{X\times X'} \newcommand{\f}{(FPV)} \newcommand{\so}{\sum_{i=1}^{n}} \newcommand{\z}{z_{n}} %\newcommand{\bb}{b_{i}} \newcommand{\al}{\alpha_{i}} \newcommand{\roe}{\partial _{\epsilon}} \newcommand{\te}{T^{\epsilon}} \newcommand{\ff}{f:\mathbb{R}^{n}\rightarrow \mathbb{R}\cup \{+\infty \}} \newcommand{\fx}{f:X\rightarrow \mathbb{R}\cup \{+\infty \}} \newcommand{\rinf}{\mathbb{R}\cup \nolinebreak \{+\infty \}} \newcommand{\rn}{\mathbb{R}^{n}} \newcommand{\ett}{\mathbb{E}(T)} \newcommand{\btt}{B^{T}} \newcommand{\all}{\alpha } \newcommand{\be}{\beta } \newcommand{\xx}{2^{X'}} \newcommand{\xxx}{:X\rightarrow 2^{X^{*}}} \newcommand{\st}{S^{T} } \newcommand{\pa}{\partial } \newcommand{\rb}{\mathbb{R} } \newcommand{\rr}{\mathbb{R}_{+} } \newcommand{\rx}{:\mathbb{R}_{+}\times X \rightarrow 2^{X'}} \newcommand{\qa}{q_{A}(} \newcommand{\ctx}{C \times X'} %%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% %Number of mathematical formula %222222222222222%Number of mathematical formula% 222222222222222222222222 %Please use the follwing commands. Not use \eqno and \leqno %\begin{equation}\label{???} %>>> %\end{equation} %\begin{eqnarray}, \begin{align}, etc %Eequation, Consultation %%333333333333333333%Eequation, Consultation% 3333333333333333333 %Please use the follwoing commands. %Please use the following commands of \label, \ref (\eqref), bibitem, cites commands. %Reference %%4444444444444444444444%Reference% 44444444444444444444444444444444444 %Not use Bibtex (Please use (find) the sytle of Reference from web http://wwww.ybook.co.jp/jnca.html) %arrange author's name in alphabetical order \numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{CHARACTERIZATION OF APPROXIMATE MONOTONE OPERATORS} \author[Z. S. Mirsaney]{Zahra Sadat Mirsaney} \address[Z. S. Mirsaney]{University of Isfahan, Iran.} \email{\tt z.mirsaney@sci.ui.ac.ir} \author[M. Rezaei]{Mahboubeh Rezaei} \address[M. Rezaei]{University of Isfahan, Iran.} \email{{\tt mrezaie@sci.ui.ac.ir}} \keywords{$\theta^{'}$-monotone, $\theta$-Fitzpatrick function, convex, $\theta$-subdifferential, maximal $\theta^{'}$-monotone, $\theta$-conjugate. } \subjclass[2010]{47H05, 49J53, 90C25, 42A50} \maketitle \begin{abstract} Results concerning local boundedness of operators have a long history. In 1994, Vesel$\acute{\text{y}} $ connected the concept of approximate monotonicity of an operator with local boundedness of that. It is our desire in this note to characterize an approximate monotone operator. Actually, we show that a well-known property of monotone operators, namely representing by convex functions, remains valid for the larger subclass of operators. In this general framework we establish the similar results by Fitzpatrick. %\cite{fit} Also, celebrated results of Mart$\acute{\text{i}}$nez-Legaz and Th$\acute{\text{e}}$ra % \cite{conv rep}, inspired us to prove that the set of maximal $ \varepsilon$-monotone operators between a normed linear space $ X$ and its continuous dual $ X^{*} $ can be identified as some subset of convex functions on $ \X $. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% \section{Introduction} The concept of monotonicity for multivalued operators defined on a Banach space and taking values in its dual has imposed itself \cite{2 teta,3 teta,4 teta,14 teta,15 teta} play an important role in convex analysis, optimization theory, partial differential equations and other fields of mathematics. Recently, it was a challenging problem to represent maximal monotone operators by convex functions \cite{ex r1,fit,conv rep,ex r2}. As we will show the approximate monotone operators are as important as monotone oparators, therefore useful characterization of them is important to the same extent. Fitzpatrick developed the representation of monotone operators on $ X$ in terms of the subdifferentials of convex functions on $X\times X^{*}$ \cite{fit}. Our aim here is to extend this idea to the approximate monotone operators. Also, in \cite{conv rep} Mart$\acute{\text{i}}$nez-Legaz and Th$\acute{\text{e}}$ra proved that the set of maximal monotone operators between a normed linear space $ X$ and its continuous dual $ X^{*} $ can be identified as some subset of the set of lower semicontinuous convex proper functions on $X \times X^{*}$, inspired by this idea we will characterize $ \varepsilon$-monotone operator. According to \cite{teta}, an operator $ T $ is $ \theta^{'} $-monotone if \begin{equation*} \langle u -v, x -y\rangle \geq \theta^{'}(x, y)\Vert x -y\Vert, ~ \forall (x,u),(y,v) \in \G(T). \end{equation*} Via some examples in \cite{teta}, it is shown that the $\theta^{'}$-monotonicity is more general than most of monotonicity properties known in literature.\\ The $ \theta^{'} $-monotone operators are the key ingredient of some branches of mathematics \cite{2 teta,3 teta,9 teta,10 teta,11 teta,12 teta,24 teta}. As we know, the question about local boundedness of an operator is very important in the theory of monotone operators. Classical results due to Rockafellar \cite{2ves,3ves,1ves}, Borwein and Fitzpatrick \cite{4ves}, answer this question. Using a trick from \cite{5ves}, Vesel\'{y} \cite{ves} presented a Banach-Steinhaus type theorem for families of $ \varepsilon $-monotone operators and some useful consequences. Let $ \varepsilon\geq 0$, the $ \varepsilon $-monotone operator $ T\xxx $ defined as follows \begin{center} $ \langle u -v, x -y\rangle \geq -\varepsilon,~ \forall (x,u),(y,v) \in \G(T) $. \end{center} In this case we can say $ T $ is $ \theta^{'} $-monotone in which \begin{equation}\label{epsilon} \theta^{'}(x,y)=\dfrac{-\varepsilon}{\Vert x-y\Vert},~~\theta^{'}(x,x)=0,~~\forall x,y\in D(T):~~x\neq y. \end{equation} We see in \cite{ves}, one of the most important result in the theory of approximate monotone operators, which establishes a connection between $ \varepsilon $-monotonicity of an operator and local boundedness of that. Before introducing this proposition, we state some basic concepts. By $ k(M) $ we denote the convex cone generated by a set $ M\subseteq X$, i.e. \begin{equation*} k(M)=\co (\underset{t\geq 0}\bigcup tM), \end{equation*} in which $\co$ means convex hull of a set. We say that $x\in X $ is a sup-point for a set $ M\subseteq X$ if there exists $ x^{*}\in X^{*}\setminus \lbrace 0\rbrace $ such that $ \langle x^{*},x \rangle\geq \sup\langle M,x^{*}\rangle$, i.e. if $ M$ is contained in a closed halfspace having $ x$ as a boundary point. \begin{proposition}\label{cor4ves} Let $ \varepsilon\geq 0 $, $ T$ be an $ \varepsilon $-monotone operator on $ X$ and $ x\in \overline{\D(T)} $. Let either $k(\D(T)-X) $ be a second category set, or $ X$ be barrelled and $\inte k(\D(T)-X)\neq \emptyset$. \begin{itemize} \item[1)] If x is not a sup-point for $ \D(T) $, then $T $ is locally bounded at $ x$. \item[2)] If $ T$ is maximal $ \varepsilon$-monotone, \end{itemize} then the following assertions are equivalent: \begin{itemize} \item[(i)] $ x$ is not a sup-point for $ \D(T) $ and $x\in \D(T) $; \item[(ii)] $ x$ is not a sup-point for $ \D(T) $; \item[(iii)] $ T$ is locally bounded at $ x$; \item[(iv)] $x\in \D(T) $ and $T(x)$ is bounded. \end{itemize} \end{proposition} \begin{proof} The proof can be found in \cite[Corollary 4]{ves}. \end{proof} As we said the $ \varepsilon $-monotone operators play an important role in the theory of monotone operators. Consequently, we need the clear recognition of $ \varepsilon $-monotone operators.\\ The paper is organized as follows. After preleminaries in Section 2, we establish in section 3 some useful properties regarding the convex functions on $ \X $ and their $ \theta $-subdifferential. In section 4, under reasonable assumptions we show that the convex functions representing maximal $ \varepsilon$-monotone operators satisfy a minimality condition. In section 5, we restrict our attention to prove that the set of maximal $ \varepsilon$-monotone operators between a normed linear space $ X$ and $ X^{*}$ can be identified as some subset of the set of all proper convex functions on $ X\times X^{*}$.\\ \section{Notations and preleminaries} Throughout the paper, we use standard notations except special symbols introduced when they are defined. All spaces considered are Banach. For any space $ X$, we consider its dual space $X^{*} $. \\The norms in $ X$ and $X^{*} $ will be denoted by $ \Vert.\Vert$. We denote by $\langle.,.\rangle$ the duality pairing between $X^{*} $ and $ X$. The space $ X\times X^{*}$ is also a Banach space, with the norm given by \begin{equation*} \Vert (x,x^{*})\Vert=\Vert x\Vert+\Vert x^{*}\Vert,~~~\forall \x \in \X. \end{equation*} Identifying $ (\X)^{*}$ with $ X^{*}\times X $, we adopt the natural duality between $ \X $ and $ X^{*}\times X $ given by $\langle (y^{*},y),\x\rangle=\langle y^{*},x\rangle+\langle x^{*},y\rangle $. %%%%%%%teta %%%%%%%teta %%%%%%%teta From now on $\theta:(X\times X^{*})\times (X\times X^{*})\longrightarrow\mathbb{R},~\sigma: X^{*} \times X^{*} \longrightarrow\mathbb{R}$ and $\theta^{'}:X\times X\longrightarrow\mathbb{R}$ be the given functions that \begin{center} $ \theta((x,x^{*}),(y,y^{*}))=\theta^{'}(x,y)+\sigma (x^{*},y^{*}), $ \end{center} by the properties that $\theta^{'}(x, y) = \theta^{'}(y, x),~\sigma(x^{*}, y^{*}) = \sigma(y^{*}, x^{*})$ and $\theta^{'}(x, x) =0=\sigma(x^{*}, x^{*})$ for all $\x, \y \in \X$.\\ \textbf{ \underline{From now on, the functions $ \theta,~\theta^{'} $ are defined such as \eqref{epsilon}.} }\label{bas}\\ In what follows we introduce the concept of $\theta^{'}$-monotonicity for a multivalued operator $T\xxx$. \\We denote by $\D(T) = \lbrace x \in X : T(x) \neq \emptyset\rbrace$ its domain. The graph of the operator $T$ is the set $\G(T) =\lbrace(x, u)\in X \times X^{*} : u \in T(x)\rbrace$. We need the following definitions which can be found in \cite{teta}. \begin{definition} The operator $T\xxx$ is $\theta^{'}$-monotone, if \begin{equation} \langle u -v, x -y\rangle \geq \theta^{'}(x, y)\Vert x -y\Vert, ~\forall (x,u),(y,v) \in \G(T). \label{def 1.1} \end{equation} Moreover a $ \theta^{'}$-monotone operator $T$ is maximal $ \theta^{'}$-monotone if for every operator $T^{'}\xxx$ which is $ \theta^{'}$-monotone with $\G(T)\subseteq \G(T^{'})$, one has $T=T^{'}$. \end{definition} \begin{definition}\label{def 1.2} We say that the pair $(x,x^{*})$ is $ \theta^{'}$-monotonically related to a subset $M $ of $ X \times X^{*} $ if \begin{equation*} \langle x^{*} -y^{*}, x -y\rangle \geq \theta^{'}(x, y)\Vert x -y\Vert, ~\forall (y,y^{*}) \in M. \end{equation*} \end{definition} The following result can be found in \cite[Proposition 3.4.]{teta}. \begin{proposition}\label{pro 1.3} A $ \theta^{'}$-monotone operator $T\xxx $ is maximal $ \theta^{'}$-monotone if and only if whenever a pair $(x, u) \in X \times X^{*}$ is $ \theta^{'}$-monotonically related to $ \G(T)$, it holds that $u \in T(x)$. \end{proposition} \begin{example} Let $ T:\mathbb{R}\longrightarrow \mathbb{R} $ be a single-valued operator defined as, $T(x)=x $ and suppose that $\theta^{'}(x,y)=\vert x-y\vert$ for all $ x,y\in \mathbb{R} $. By Proposition and Definitions above we conclude that $ T $ is a maximal $\theta^{'}$-monotone. \end{example} Let $ f:X\longrightarrow \mathbb{R}\cup\lbrace+\infty\rbrace $ be given. We denote the domain of $f$ by \begin{center} $ \Dm f:=\lbrace x \in X : f(x)<\infty\rbrace$. \end{center} We say that $f $ is proper if $ \Dm f$ is nonempty, also $ f $ is a convex function if \begin{center} $ f(\lambda x +(1-\lambda)y)\leq \lambda f(x)+(1-\lambda)f(y), $ \end{center} whenever $x, y \in X $ and $0<\lambda<1 $. The notions of the subdifferential of a convex function are fundamental in optimization. The subdifferential of a function $f : X \longrightarrow\mathbb{R}$ is the multivalued operator $\partial f \xxx$ defined by \begin{center} $\partial f(x) = \lbrace v \in X^{*} |f(y) \geq f(x) + \langle y - x,v\rangle,~ \forall y \in X\rbrace.$ \end{center} We recall a new subdifferential concept, the so-called $\theta^{'}$-subdifferential which is introduced at first in \cite{teta}. \begin{definition}\label{def 4.5} Let $f :X \longrightarrow \mathbb{R}\cup \lbrace+\infty\rbrace$ be a proper function. One says that $x^{*} \in X^{*}$ is a $\theta^{'}$-subgradient of $f$ in $x \in \Dm f $, if \begin{center} $ \langle x^{*}, y -x \rangle \leq f(y) -f(x) -\theta^{'}(x, y)\Vert x -y\Vert, ~\forall y \in X$. \end{center} The set \begin{align}\nonumber &\partial_{\theta^{'}}f(x)\\ &= \lbrace x^{*} \in X^{*} : \langle x^{*}, y -x \rangle \leq f(y) -f(x) -\theta^{'} (x, y)\Vert x -y\Vert, ~\forall y \in X\rbrace\label{16} \end{align} is called the $\theta^{'}$-subdifferential of $f$ at $x \in \Dm f $. \end{definition} \begin{remark}\label{rem0} It is easy to check that ${\partial}_{\theta^{'}}f \xxx$ is a $2\theta^{'}$-monotone operator. Let $x^{*} \in {\partial}_{\theta^{'}}f(x)~\text{and}~y^{*} \in {\partial}_{\theta^{'}}f(y)$, then $ \langle x^{*}, y -x \rangle \leq f(y) -f(x) -\theta^{'}(x, y)\Vert x-y\Vert$ and $ \langle y^{*}, x -y \rangle \leq f(x) -f(y) -\theta^{'}(x, y)\Vert y-x\Vert$, which added give us $ \langle y^{*}-x^{*}, y-x \rangle \geq 2\theta^{'}(x, y)\Vert x -y\Vert$. \end{remark} \begin{example} Let us consider the function $f : \mathbb{R}\longrightarrow\mathbb{R}$ defined by \begin{equation*} f (x)=\left\{ \begin{array}{rl} 0~~~,~~~ &x= 0\\ x^{2}+1, ~~~&x\neq0\\ \end{array} \right. \end{equation*} and $ \theta^{'}(x,y)=c\vert x-y\vert $ such that $ c\in \mathbb{R}, c<1 $. By standard calculus we find \begin{align*} \partial_{\theta^{'}}f (0)&=\lbrace x^{*}\in \mathbb{R}:\langle x^{*}, y\rangle\leq y^{2}+1-cy^{2},~ \forall y \in \mathbb{R},~y\neq0\rbrace\\ & (\text{if $ y=0 $ all $ x^{*}\in \mathbb{R} $ satisfy \eqref{16}}) \\ &=\lbrace x^{*}\in \mathbb{R}: 0\leq (1-c) y^{2}+1-x^{*} y,~ \forall y \in \mathbb{R},~y\neq0\rbrace\\ &=\lbrace x^{*}\in \mathbb{R}: 0\geq {x^{*}}^{2}-4(1-c)\rbrace\\ &=\lbrace x^{*}\in \mathbb{R}: \vert x^{*}\vert\leq2\sqrt{1-c}\rbrace. \end{align*} \end{example} We introduce next the very important concept of conjugate functions, originally developed by Fenchel in \cite{84 set}. The Fenchel conjugate of a function $f : X \longrightarrow \mathbb{R}\cup\lbrace +\infty\rbrace$ is the function $f^{*} : X \longrightarrow \mathbb{R}\cup\lbrace +\infty\rbrace$ defined by \begin{equation*} f^{*}(x^{*})=\underset{x\in X}\sup\lbrace \langle x^{*},x \rangle -f(x)\rbrace.\label{fen} \end{equation*} An immediate consequence of the definition of the Fenchel conjugate is the famous Fenchel-Young inequality: \begin{proposition}[Fenchel-Young Inequality]\label{4.4.1z} Let $f : X \longrightarrow \mathbb{R}\cup\lbrace+\infty\rbrace$ be a convex function. Suppose that $ x\in \Dm f$ and $ x^{*}\in X^{*} $. Then satisfy the inequality \begin{center} $ f(x)+f^{*}(x^{*})\geq \langle x^{*},x \rangle.$ \end{center} Equality holds if and only if $x^{*}\in \partial f(x)$. \end{proposition} \begin{proof} The proof can be found in \cite[Proposition 4.4.1]{z}. \end{proof} Now we are ready to introduce the $\theta$-conjugate function and inspired by the proof of Proposition \ref{4.4.1z}, we extend this theorem.\\ Let $f$ be a function from $X$ to $\mathbb{R}\cup\lbrace+\infty\rbrace$. Let $y \in X$ be fixed. We define the $\theta$-conjugate function $f^{*}_{y}(\theta,.) : X^{*} \longrightarrow \mathbb{R}\cup\lbrace+\infty\rbrace$ of $ f $ at $ y $ by \begin{equation} f^{*}_{y}(\theta^{'},\xi)=\underset{x\in X}\sup\lbrace \langle \xi,x \rangle -f(x)+\theta^{'}(x,y)\Vert x-y\Vert \rbrace\label{17}. \end{equation} \begin{proposition}\label{ineq} Let $f : X \longrightarrow \mathbb{R}\cup\lbrace+\infty\rbrace$ and $ \theta^{'}:X\times X\longrightarrow \mathbb{R} $. Then for all $ x\in \Dm f$ and $ x^{*}\in X^{*} $ we have \begin{center} $ f(x)+f^{*}_{x}(\theta^{'},x^{*})= \langle x^{*},x \rangle\Longleftrightarrow x^{*}\in \partial_{\theta^{'}}f(x).$ \end{center} \end{proposition} \begin{proof} For all $ x\in \Dm f$ and $ x^{*}\in X^{*} $ we have following inequality \begin{align*} x^{*}\in \partial_{\theta^{'}}f(x)&\Longleftrightarrow \langle x^{*},y-x\rangle\leq f(y)-f(x)-\theta^{'}(x,y)\Vert x-y\Vert,~~\forall y\in X\\ &\Longleftrightarrow f(x)+f^{*}_{x}(\theta^{'},x^{*})\leq \langle x^{*},x \rangle. \end{align*} On the other hand by \eqref{17}, we have \begin{center} $f(x)+f^{*}_{x}(\theta^{'},x^{*})\geq \langle x^{*},x \rangle,$ \end{center} and the proof is complete. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{ convex functions on $ X\times X^{*} $} In this section, let us establish some required properties of convex functions. At first we introduce a special function on $ X $, using the $ \theta $-subdifferential of function on $ \X $. \begin{definition} For each function $f:X\times X^{*}\longrightarrow\mathbb{R}$ let \begin{equation*} T^{\theta}_{f}(x):=\lbrace x^{*}\in X^{*}:(x^{*},x)\in \partial_{\theta}f(x,x^{*}) \rbrace \end{equation*} for each $ x\in X $. \end{definition} %mybe we must defin teta subdiff \begin{proposition}\label{2.2} $ T^{\theta}_{f} $ is a $ \theta^{'} $-monotone operator on $ X$. \end{proposition} \begin{proof} If $x^{*}\in {T^{\theta}_{f}(x)}$ and $y^{*}\in{T^{\theta}_{f}(y)}$. For $ x=y $ we have $$\langle x^{*}-y^{*},x-y\rangle\geq\theta^{'}(x,y)\Vert x-y\Vert$$ On the other hand, by Remark \ref{rem0} for all $ x\neq y $, we conclude that \begin{align*} \langle x^{*}-y^{*},x-y\rangle &=\dfrac{1}{2}\langle (x^{*},x)-(y^{*},y),(x,x^{*})-(y,y^{*}) \rangle\\ & \geq \theta((x,x^{*}),(y,y^{*}))\Vert (x,x^{*})-(y,y^{*})\Vert \\ &=\theta^{'}(x,y)\Vert x-y\Vert. \end{align*} \end{proof} This poses a natural question: is there any convex function $ f $ on $ \X $ such that for each maximal $ \theta^{'} $- monotone operator $ T $, $ T^{\theta}_{f}=T $? We answer this question in Corollary \ref{answer}, in the next section. \begin{example} Let $ g:X\longrightarrow \mathbb{R} $ be a function. If for arbitrary $ z\in X $, the function $ f^{z} $ on $ \X $ defined as follows \begin{center} $f^{z}(x,x^{*})=g(x)+g^{*}_{z}(\theta^{'},x^{*}),~\forall \x \in \X$. \end{center} Then $T^{\theta}_{f^{x}}(x)= \partial_{\theta^{'}}g(x)$ for all $ x\in X $. \end{example} \begin{proof} If $x^{*}\in \partial_{\theta^{'}}g(x)$, then by \eqref{16} and \eqref{17}, we have \begin{align*} \langle x, y^{*}-x^{*}\rangle&\leq -g(x)-g^{*}_{x}(\theta^{'},x^{*})+\langle y^{*},x\rangle\\ &\leq -g^{*}_{x}(\theta^{'},x^{*})+\underset{z\in X}\sup\lbrace\langle y^{*},z\rangle-g(z) +\theta^{'}(x,z)\Vert x-z\Vert\rbrace\\ &=-g^{*}_{x}(\theta^{'},x^{*})+g^{*}_{x}(\theta^{'},y^{*}), \end{align*} so that, $x\in \partial g^{*}_{x}(\theta^{'},x^{*})$. Hence by an elementary computation, we have \begin{align*} &\langle (x^{*},x),\y -\x \rangle\leq g(y)+g^{*}_{x}(\theta^{'},y^{*})\\ &-[g^{*}_{x}(\theta^{'},x^{*})+g(x)]-\theta^{'}(x,y)\Vert x-y\Vert \\ &=f^{x}(y,y^{*})-f^{x}(x,x^{*}) -\theta^{'}(x,y)\Vert x-y\Vert \\ &\leq f^{x}(y,y^{*})-f^{x}(x,x^{*}) -\theta(\x,\y)\Vert \x-\y\Vert , \end{align*} hence $(x^{*},x)\in \partial_{\theta}f^{x}(x,x^{*})$. On the other hand, if $(x^{*},x)\in \partial_{\theta}f^{x}(x,x^{*})$, then for $ u\in X $ we have \begin{align*} \langle x^{*},u\rangle &= \langle (x^{*},x),(u,0)\rangle\leq f^{x}(x+u,x^{*})-f^{x}(x,x^{*})\\ &-\theta((x+u,x^{*}),(x,x^{*}))\Vert(u,0)\Vert\\ &=g(x+u)-g(x)-\theta^{'}(x+u,x)\Vert u\Vert+\underset{0}{\underbrace{ g^{*}_{x}(\theta^{'},x^{*})-g^{*}_{x}(\theta^{'},x^{*})}}, \end{align*} so that $x^{*}\in \partial_{\theta^{'}}g(x)$ as required. \end{proof} We will be interested in the case when $f(x,x^{*})\geq \langle x^{*},x\rangle$ for all $ (x,x^{*})\in X\times X^{*}$. This allows a simple way to guarantee $x^{*}\in T^{\theta}_{f}(x)$. \begin{theorem}\label{2.4} Suppose that $f$ is a convex function on $X\times X^{*}$. If for some $(y,y^{*})\in X\times X^{*}$ we have $f(y,y^{*}) =\langle y^{*},y\rangle$ and $f(x,x^{*})\geq \langle x^{*},x\rangle$ for all $(x,x^{*})$ in some neighbourhood $U$ of $(y,y^{*})$, then $y^{*}\in T^{\theta}_{f}(y)$. \end{theorem} \begin{proof} Let $(z,z^{*})\in X\times X^{*}$ and $1>s > 0$, so that $(y+sz,y^{*}+sz^{*})\in U$. Then %u is absorbing \begin{align*} f(y+z,y^{*}+&z^{*})-f(y,y^{*})\geq \dfrac{f(y+sz,y^{*}+sz^{*})-f(y,y^{*})}{s}\\ &+\dfrac{s(1-s)\theta((y+z,y^{*}+z^{*}), (y,y^{*}))\Vert (y+z,y^{*}+z^{*})- (y,y^{*})\Vert}{s}\\ %by assumption &\geq\dfrac{ \langle y+sz,y^{*}+sz^{*}\rangle-\langle y^{*},y\rangle}{s}\\ &+(1-s)\theta((y+z,y^{*}+z^{*}), (y,y^{*}))\Vert (y+z,y^{*}+z^{*})- (y,y^{*})\Vert\\%-teta is positiv &= \langle z^{*},y\rangle+\langle y^{*},z\rangle+ s \langle z^{*},z\rangle\\ &+(1-s)\theta((y+z,y^{*}+z^{*}), (y,y^{*}))\Vert (y+z,y^{*}+z^{*})- (y,y^{*})\Vert\\ &\text{Letting $ s\longrightarrow 0^{+} $ we have}\\ &=\langle (y^{*},y),(z,z^{*})\rangle\\ &+\theta((y+z,y^{*}+z^{*}), (y,y^{*}))\Vert (y+z,y^{*}+z^{*})- (y,y^{*})\Vert, \end{align*} %by convexity f so that $ (y^{*},y)\in \partial_{\theta}f(y,y^{*}) $, hence $y^{*}\in T^{\theta}_{f}(y)$ as required. \end{proof} Because of interesting applications of the previous theorem in the following sections, we are concerned with obtaining functions satisfying the hypotheses of this theorem. Now denote the $x$-section of $f$ by $f_{x}(x^{*}) := f(x,x^{*})$ for $x\in X$ and $x^{*}\in X^{*}$. \begin{theorem}\label{2.5} If for some $ \x \in \X $ we have $x\in \partial_{\sigma}f_{x}(x^{*})$ and $ (f_{x})^{*}_{x^{*}}(\sigma,x)=0 $, then $f(x,x^{*}) = \langle x^{*},x \rangle$. % such that $ \sigma:X^{*}\times X^{*}\longrightarrow \mathbb{R}$. If % \end{theorem} \begin{proof} By $ (f_{x})^{*}_{x^{*}}(\sigma,x)=0 $, we have \begin{equation}\label{sup1} \underset{z^{*}\in X^{*}}\sup\lbrace \langle x,z^{*}\rangle+\sigma(x^{*},z^{*})\Vert x^{*}-z^{*}\Vert-f_{x}(z^{*})\rangle\rbrace=0, \end{equation} and $x\in \partial_{\sigma}f_{x}(x^{*})$ conclude that \begin{center} $\langle u^{*},x\rangle \leq f_{x}(x^{*}+u^{*})-f_{x}(x^{*})-\sigma(x^{*},x^{*}+u^{*})\Vert u^{*}\Vert,$ \end{center} for all $ u^{*}\in X^{*}$, so we have \begin{equation*} \langle x,x^{*}+u^{*} \rangle -f(x,x^{*}+u^{*})+\sigma(x^{*},x^{*}+u^{*})\Vert u^{*}\Vert\leq \langle x,x^{*}\rangle-f(x,x^{*}). \end{equation*} Taking the supremum over $u^{*}$, by \eqref{sup1} we have $ f(x,x^{*})\leq \langle x,x^{*}\rangle$. However putting $z^{*}=x^{*}$ in \eqref{sup1} we get $ f(x,x^{*})\geq \langle x,x^{*}\rangle$, so $ f(x,x^{*})= \langle x,x^{*}\rangle$. \end{proof} \begin{corollary} Suppose that $f$ is a convex function on $X\times X^{*}$. If $ (f_{x})^{*}_{x^{*}}(\sigma,x)=0 $ for some $ (x,x^{*})\in X\times X^{*}$ and $f(y,y^{*})\geq \langle y^{*},y\rangle$ for all $(y,y^{*}) \in X\times X^{*}$. Then $ x\in \partial_{\sigma}{f_{x}}(x^{*}) $ if and only if $x^{*}\in T^{\theta}_{f}(x)$. \end{corollary} \begin{proof} Combine Theorems \ref{2.4} and \ref{2.5}, we conclude that $x^{*}\in T^{\theta}_{f}(x)$. Conversly, take $x^{*}\in T^{\theta}_{f}(x)$, then for all $ u^{*}\in X^{*} $ \begin{align*} \langle (x^{*},x),(x,x^{*}+ u^{*})-&(x,x^{*})\rangle \leq f_{x}(x^{*}+ u^{*})-f_{x}(x^{*})\\ &-\theta((x,x^{*}+ u^{*}),(x,x^{*}))\Vert (x,x^{*}+ u^{*})-(x,x^{*})\Vert\\ &\Downarrow\\ \langle x, u^{*} \rangle &\leq f_{x}(x^{*}+ u^{*})-f_{x}(x^{*})-\theta((x,x^{*}+ u^{*}),(x,x^{*}))\Vert u^{*}\Vert\\ &=f_{x}(x^{*}+ u^{*})-f_{x}(x^{*})-\sigma(x^{*}+ u^{*},x^{*})\Vert u^{*}\Vert\\ \end{align*} This completes the proof. \end{proof} \section{convex functions from $ \varepsilon $-monotone operators} Fitzpatrick \cite{fit}, showed that the convex functions representing maximal monotone operators satisfy a minimality condition. It is natural to ask whether this result is still valid for the larger class of operators. In this section we use a convex function on $ X\times X^{*}$ for reperesenting an $ \varepsilon $-monotone operator on $ X$ and show that this functions satisfy a minimality condition. \begin{definition} Let $T$ be a $ \theta^{'} $-monotone operator. The $ \theta$-Fitzpatrick function is defined as \begin{equation*} L^{\theta }_{T}(x,x^{*})=\underset{(y,y^{*})\in \G(T)}\sup\lbrace \langle x^{*},y\rangle+\langle y^{*},x-y\rangle+\theta(\x,\y)\Vert \x-\y\Vert\rbrace. \end{equation*} for $x\in X$ and $x^{*}\in X^{*}$. \end{definition} The first result is immediate from the definition. \begin{proposition}\label{conv} Let $ T $ be an $ \varepsilon $-monotone operator as in \eqref{epsilon} and $ \G(T) $ is a subset of $ \X $ satisfies following property, \begin{equation} \lambda \x+(1-\lambda)\y \in \G(T) \Longleftrightarrow \x=\y\in \G(T), \end{equation}\label{pdt} for all $\lambda \in (0,1)$, then the function $ L^{\theta}_{T}$ is convex on $ X\times X^{*}$. \end{proposition} %\begin{proof} %Let $ \x,~\y \in \X $ and $ (z,z^{*})=\lambda \x+(1-\lambda)\y $ for an arbitrary $ \lambda\in \mathbb{R}_{+} $. We claim that for all $ (u,u^{*})\in \G(T) $ %\begin{align*} %&\langle z^{*},u\rangle+\langle u^{*},z-u\rangle+\theta^{'}(z,u)\Vert z-u\Vert\leq\\ %&\lambda(\langle x^{*},u\rangle+\langle u^{*},x-u\rangle+\theta^{'}(x,u)\Vert x-u\Vert)+\\ %&(1-\lambda)(\langle y^{*},u\rangle+\langle u^{*},y-u\rangle+\theta^{'}(x,u)\Vert x-u\Vert). %\end{align*} %If $z\in \D(T) $, then $ x=y=z $ and $\theta^{'}(z,u)\Vert z-u\Vert\leq \lambda \theta^{'}(x,u)\Vert x-u\Vert+(1-\lambda)\theta^{'}(y,u)\Vert y-u\Vert$. %On the other hand, if $z\notin D(T) $ we have two cases:\\ %one case is, $ x\neq y \in D(T) $. Another case is $ x=y \notin D(T) $. %\end{proof} As a start to examining $ \partial_{\theta}{L^{\theta}_{T}}$ we have the following definition and result. \begin{lemma}\label{3.3} Let $ T $ be an $ \varepsilon $-monotone operator. If there exists $ (y,y^{*})\in \G(T) $ such that for $ (x,x^{*})\in X\times X^{*}$ we have \begin{equation*} L^{\theta^{'}}_{T}(x,x^{*})=\langle x^{*},y\rangle+\langle y^{*},x-y\rangle+\theta(\x,\y)\Vert \x-\y\Vert, \end{equation*} then $ (y,y^{*})\in \partial_{\theta}{L^{\theta}_{T}}(x,x^{*}) $. \end{lemma} \begin {proof} For each $ u\in X $ and $ u^{*}\in X^{*} $ we have \begin{align*} &L^{\theta}_{T}(x+u,x^{*}+u^{*})-L^{\theta }_{T}(x,x^{*})\\ &=\underset{(v,v^{*})\in \G(T)}\sup\lbrace \langle x^{*}+u^{*},v\rangle+\langle v^{*},x+u\rangle-\langle v,v^{*}\rangle+\\ &\theta ((x+u,x^{*}+u^{*}),(v,v^{*}))\Vert(x+u,x^{*}+u^{*})-(v,v^{*})\Vert\rbrace-L_{T}(x,x^{*})\\ %(v,v*)=(y,y*) &\geq \langle y,x^{*}+u^{*}\rangle+\langle x+u, y^{*}\rangle-\langle y,y^{*}\rangle\\ &+\theta ((x+u,x^{*}+u^{*}),(y,y^{*}))\Vert(x+u,x^{*}+u^{*})-(y,y^{*})\Vert-\langle x^{*},y\rangle-\\ &\langle y^{*},x-y\rangle-\theta ((x^{*},x),(y,y^{*}))\Vert(x^{*},x)-(y,y^{*})\Vert\\ &\geq \langle (y^{*},y),(u,u^{*})\rangle+\theta((x+u,x^{*}+u^{*}),(x,x^{*}))\Vert (u,u^{*})\Vert \end {align*} so we have $(y^{*},y)\in \partial_{\theta}{L^{\theta}_{T}}(x,x^{*}) $. \end{proof} \begin{theorem}\label{3.4} If $ T $ is a $ \theta^{'} $-monotone operator and $ (x,x^{*})\in \G(T) $, then $ L^{\theta}_{T}(x,x^{*})=\langle x^{*},x \rangle $. \end{theorem} \begin{proof} By $ \theta^{'} $-monotonicity of $T $, for all $(y,y^{*})\in \G(T)$ we have \begin{align*} \langle x,x^{*}\rangle&\geq \langle x^{*},y\rangle+\langle y^{*},x-y\rangle+\theta^{'}(x,y)\Vert x-y\Vert\\ &\geq \langle x^{*},y\rangle+\langle y^{*},x-y\rangle+\theta(\x,\y)\Vert \x-\y\Vert, \end{align*} so that $ L^{\theta}_{T}(x,x^{*})\leq\langle x,x^{*}\rangle $. On the other hand, \begin{equation*} L^{\theta}_{T}(x,x^{*})\geq\langle x,x^{*}\rangle+\langle x-x,x^{*}\rangle+\theta(\x,\x)\Vert \x-\x\Vert=\langle x,x^{*}\rangle. \end{equation*} \end{proof} \begin{theorem}\label{3.6} If $ f(x,x^{*})\geq \langle x,x^{*}\rangle$ for all $ (x,x^{*}) \in X\times X^{*} $, then $ L^{\theta}_{T^{\theta}_{f}}\leq f.$ \end{theorem} \begin{proof} By our assumptions for all $ (y,y^{*})\in \X $ we have \begin{align*} &L^{\theta}_{T^{\theta}_{f}}(y,y^{*})- f(y,y^{*})\\ &=\underset{x^{*}\in T^{\theta}_{f}x}\sup\lbrace \langle y^{*},x\rangle+\langle x^{*},y-x\rangle+\theta(\x,\y)\Vert \x-\y\Vert-f(y,y^{*})\rbrace\\ &=\underset{(x^{*},x)\in \partial_{\theta}f(x,x^{*}) } \sup\lbrace\langle y^{*}-x^{*},x\rangle+\langle x^{*},y-x\rangle\\ &+\langle x^{*},x\rangle+\theta(\x,\y)\Vert \x-\y\Vert-f(y,y^{*})\rbrace\\ &=\underset{(x^{*},x)\in \partial_{\theta}f(x,x^{*}) } \sup\lbrace f(y,y^{*})-f(x,x^{*})-\theta((x,x^{*}),(y,y^{*}))\Vert (x,x^{*})-(y,y^{*})\Vert\\ &+\langle x^{*},x\rangle+\theta(\x,\y)\Vert \x-\y\Vert-f(y,y^{*})\rbrace \\ &\leq 0. \end{align*} Consequently, $ L^{\theta}_{T^{\theta}_{f}}\leq f.$ \end{proof} \begin{theorem}\label{3.7} If $T$ be a $ \theta^{'} $-monotone operator and $ f $ be a convex function such that $ f(y,y^{*})=\langle y^{*},y \rangle $ for all $(y,y^{*})\in \G(T)$ and $ f(x,x^{*})\geq\langle x^{*},x \rangle $ for all $(x,x^{*})\in X\times X^{*}$, then $ L^{\theta}_{T}\leq f.$ \end{theorem} \begin{proof} By Theorem \ref{2.4}, if $y^{*}\in Ty$, then $y^{*}\in T^{\theta}_{f}y$. Thus for all $x\in X$ and $x^{*}\in X^{*}$ we have \begin{align*} &L^{\theta^{'}}_{T}(x,x^{*})\\ &=\underset{(y,y^{*})\in \G(T)}\sup\lbrace \langle x^{*},y\rangle+\langle y^{*},x\rangle-\langle y^{*},y\rangle+ \theta(\x,\y)\Vert \x-\y\Vert\rbrace\\ &\leq \underset{(y,y^{*})\in G(T^{\theta}_{f})}\sup\lbrace \langle x^{*},y\rangle+\langle y^{*},x\rangle-\langle y^{*},y\rangle+ \theta(\x,\y)\Vert \x-\y\Vert\rbrace\\ &=L^{\theta}_{T^{\theta}_{f}}(x,x^{*})\leq f(x,x^{*}), \end{align*} by Theorem \ref{3.6}. \end{proof} \begin{theorem}\label{3.8} If $T$ is a $ \theta^{'} $-monotone operator. Then $T$ is maximal $ \theta^{'} $-monotone if and only if $ L^{\theta}_{T}(x,x^{*})>\langle x^{*},x \rangle$ whenever $x\in X$ and $x^{*}\in X^{*}\setminus T(x)$. \end{theorem} \begin{proof} If $ L^{\theta}_{T}(x,x^{*})\leq \langle x^{*},x \rangle $, then we have \begin{center} $ \langle x^{*},y\rangle+\langle y^{*},x-y\rangle+\theta(\x,\y)\Vert \x-\y\Vert \leq \langle x^{*},x \rangle$ \end{center} for all $(y,y^{*})\in \G(T)$ so \begin{center} $\langle x^{*}-y^{*},x-y \rangle\geq \theta(\x,\y)\Vert \x-\y\Vert.$ \end{center} Generally, if $x=y$, then $\langle x^{*}-y^{*},x-y \rangle= \theta^{'}(x,y)\Vert x-y\Vert=0.$ On the other hand, for all $ y\neq x $ we have \begin{center} $\langle x^{*}-y^{*},x-y \rangle\geq \theta(\x,\y)\Vert \x-\y\Vert= \theta^{'}(x,y)\Vert x-y\Vert.$ \end{center} When $ T$ is maximal $ \theta^{'} $-monotone that implies $x^{*}\in Tx$.\\ Conversely, if $T$ is not maximal $ \theta^{'} $-monotone, then there are $x\in X$ and $x^{*}\in X^{*}\setminus T(x)$ such that \begin{center} $\langle x^{*}-y^{*},x-y \rangle\geq \theta^{'}(x,y)\Vert x-y\Vert\geq\theta(\x,\y)\Vert \x-\y\Vert$ \end{center} for all $(y,y^{*})\in \G(T)$. It follows that $ L^{\theta}_{T}(x,x^{*})\leq \langle x^{*},x \rangle$.%-teta positiv \end{proof} \begin{corollary}\label{3.9} If $T$ be a maximal $ \theta^{'} $-monotone operator, Then $ L^{\theta}_{T}(x,x^{*})\geq \langle x,x^{*}\rangle $ for all $x\in X$ and $x^{*}\in X^{*}$, and $L^{\theta}_{T}(x,x^{*})=\langle x^{*},x \rangle$ if and only if $x^{*}\in Tx $. \end{corollary} \begin{proof} Use Theorems \ref{3.4} and \ref{3.8}. \end{proof} \begin{theorem}\label{final} If $T$ be a maximal $ \theta^{'} $-monotone such that $ \G(T) $ satisfying \eqref{pdt}, then $L^{\theta}_{T}$ is minimal element of following family \begin{align*} &{\mathcal{H}}^{\theta^{'}}(T)=\\ &\lbrace f:X\times X^{*}\longrightarrow \mathbb{R}: f ~\text{is}~\text{convex}~,~ f(x,x^{*})\left\{\begin{array}{rl}=\langle x^{*},x\rangle & \text{if}~(x,x^{*})\in \G(T),\\ \geq\langle x^{*},x\rangle & ~\text{otherwise} \end{array}\right. \rbrace. \end{align*} \end{theorem} \begin{proof} we have $ L^{\theta^{'}}_{T}\leq f$ for any such function $ f$ by Theorem \ref{3.7}. However, $ L^{\theta^{'}}_{T}$ has the required properties by Proposition \ref{conv} and Corollary \ref{3.9}. \end{proof} \begin{corollary}\label{answer} Under assumptions of the previous theorem, $ T^{\theta}_{L^{\theta}_{T}}=T $. \end{corollary} \begin{proof} The result follows from Proposition \ref{2.2}, Theorems \ref{2.4} and \ref{final}. \end{proof} \section{corresponding approximate monotone operators and subset of functions on $ \X $} As we see in \cite{conv rep}, the set of maximal monotone operators between a normed linear space $ X$ and its continuous dual $ X^{*} $ can be identified as some subset of the set $\Gamma(\X) $ of all lower semicontinuous convex proper functions on $\X $.\\ Using the identification mentioned above , we want to identify a family of maximal $ \varepsilon $-monotone operators to a subset of convex functions on $ \X $. The set of all $\theta^{'}$-maximal monotone operators is denoted by $\mathfrak{M}_{\theta^{'}}(X)$, in fact, all of $ \varepsilon$-monotone operators.\\ Considering $\varphi: X\times X^{*}\longrightarrow \overline{\mathbb{R}}=\left[ -\infty,+\infty\right]$, which is convex and proper, i.e., $\varphi$ is not identically equal to $+\infty$ and define \begin{equation*} b(\varphi) = \lbrace (x, x^{*}) \in X\times X^{*}: \varphi(x,x^{*})\leq \langle x,x^{*} \rangle \rbrace. \end{equation*} Note $\delta_{b(\varphi)}$ the indicator function of $b(\varphi)$, i.e. \begin{flalign*} \delta_{b(\varphi)}(x,x^{*})= \left\{ \begin{array}{cc} 0&\text{if}~(x,x^{*})\in b(\varphi), \\ \\ +\infty &\text{if}~ (x,x^{*}) \notin b(\varphi). \end{array} \right. \end{flalign*} By \eqref{17} we define $\varphi^{*}_{(x,x^{*})}(\theta,(x^{*},x))$ as follow \begin{equation*} \underset{(y,y^{*})\in X\times X^{*}}\sup\lbrace \langle (x^{*},x),\y \rangle -\varphi(y,y^{*}) +\theta(\x,\y)\Vert \x-\y\Vert\rbrace. \end{equation*} Let $\Phi_{\theta}(X)$ be the set of all convex proper function $\varphi: X\times X^{*}\longrightarrow \overline{\mathbb{R}}=\left[ -\infty,+\infty\right]$ such that \begin{equation} \varphi(x,x^{*})=(\varphi +\delta_{b(\varphi)} )^ {*}_{(x,x^{*})}(\theta,(x^{*},x)),~~~\forall (x,x^{*})\in X\times X^{*}.\label{21} \end{equation} \begin{proposition}\label{pro 6.1} Suppose that $\varphi\in \Phi_{\theta}(X)$, then \begin{equation} \varphi(x,x^{*})\geq \langle x,x^{*} \rangle,~~~~~ \forall~(x,x^{*})\in X\times X^{*}.\label{star} \end{equation} \end{proposition} \begin{proof} If $(x, x^{*}) \notin b(\varphi)$, then obviously inequality \eqref{star} holds in this case. If $(x, x^{*}) \in b(\varphi)$. Then \begin{align*} \varphi(x,x^{*})&=\dfrac{\varphi(x,x^{*})+(\varphi +\delta_{b(\varphi)} )^{*}_{(x,x^{*})}(\theta,(x^{*},x))}{2}\\ &\geq\dfrac{\langle (x^{*},x),(x,x^{*})\rangle}{2}\\ &\geq \langle x^{*},x\rangle \end{align*} \end{proof} %(x,x^{*})\notin {b(\varphi)} clear %(x,x^{*})\in {b(\varphi)}\Longrightarrow \varphi(x,x^{*})=1/2(\varphi(x,x^{*})+\varphi(x,x^{*})) %=1/2(\varphi(x,x^{*})+(\varphi -\theta(x,.)\Vert x-.\Vert)^{*}(x^{*},x))=1/2((\varphi-\theta(x,.)\Vert x-.\Vert)(x,x^{*})+(\varphi(.,.) -\theta(x,.)\Vert x-.\Vert)^{*}(x^{*},x))\geq(x,x^{*}) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem} If $T \in \mathfrak{M}_{\theta^{'}}(X)$ and $ \G(T) $ satisfy \eqref{pdt}. Then $L^{\theta}_{T}\in \Phi_{\theta}(X)$. Moreover, the mapping \begin{equation*} \mathfrak{M}_{\theta^{'}}(X) \ni T\longmapsto L^{\theta}_{T}\in \Phi_{\theta}(X) \end{equation*} is a bijection, with inverse \begin{equation*} \Phi_{\theta}(X)\ni \varphi\longmapsto T_{\varphi}\in \mathfrak{M}_{\theta^{'}}(X) \end{equation*} given by $ T{\varphi}(x) = \lbrace x^{*} \in X^{*}:\varphi(x, x^{*}) = \langle x, x^{*} \rangle \rbrace$. \end{theorem} \begin{proof} Clearly, by Theorem \ref{final}, we conclude that $b(L^{\theta}_{T})=\G(T)$. For each $(x, x^{*}) \in X\times X^{*}$, one has \begin{align*} &L^{\theta}_{T}(x, x^{*})= \\ &\underset{(y,y^{*})\in b(L^{\theta}_{T})}\sup\lbrace \langle y, x^{*} \rangle + \langle x,y^{*} \rangle - \langle y,y^{*} \rangle +\theta(\x,\y)\Vert \x - \y \Vert\rbrace=\\ &\underset{(y,y^{*})\in b(L^{\theta}_{T})}\sup\lbrace \langle (x^{*},x),(y,y^{*}) \rangle -L^{\theta}_{T}(y, y^{*})+\theta(\x,\y)\Vert \x-\y\Vert \rbrace\\ %=\underset{(y,y^{*})\in b(\varphi^{\theta}_{T})}\sup\lbrace \langle (x,x^{*}),(y,y^{*}) \rangle -\varphi^{\theta}_{T}(y, y^{*})-\delta_{b(\varphi^{\theta}_{T})}(y, y^{*})+\theta(\x , \y)\Vert \x - \y\Vert \rbrace\\ &=(L^{\theta}_{T} +\delta_{b(L^{\theta}_{T})})^{*}_{\x}(\theta,(x^{*},x)) \end{align*} Hence $L^{\theta}_{T}\in \Phi_{\theta}(X)$. Now, we shall prove that, for $\varphi\in \Phi_{\theta}(X)$, one has $T_{\varphi}\in \mathfrak{M}_{\theta^{' }}(X)$. Let $x^{*}\in T_{\varphi}(x)$ and $y^{*}\in T_{\varphi}(y)$. Clearly for $ x=y $ \begin{equation*} \langle x -y, x^{*} - y^{*} \rangle \geq \theta^{'}(x,y)\Vert x-y\Vert. \end{equation*} On the other hand, for all $ x\neq y $ we have \begin{align*} &\langle x -y, x^{*} - y^{*} \rangle = \langle x, x^{*} \rangle + \langle y, y^{*} \rangle - \langle x, y^{*} \rangle - \langle y, x^{*} \rangle \\ &=\varphi(x, x^{*})+\varphi(y, y^{*})- \langle x, y^{*} \rangle - \langle y, x^{*} \rangle \\ &= \varphi(x, x^{*})+ (\varphi+\delta_{b(\varphi)})^{*}_{(y,y^{*})}(\theta,(y^{*},y)) - \langle x, y^{*} \rangle - \langle y, x^{*} \rangle \\ &=(\delta_{b(\varphi)}+\varphi)(x,x^{*})+(\delta_{b(\varphi)}+\varphi)^{*}_{(y,y^{*})}(\theta,(y^{*},y))- \langle x, y^{*} \rangle - \langle y, x^{*} \rangle \end{align*} Hence, by \eqref{ineq} we have \begin{center} $ \langle x -y, x^{*} - y^{*} \rangle \geq \theta(\x,\y)\Vert \x-\y\Vert=\theta^{'}(x,y)\Vert x-y\Vert,$ \end{center} establishing the $\theta^{'}$-monotonicity of $T_{\varphi}$.\\ In order to prove that it is maximal, fix $(x, x^{*}) \in X \times X^{*}$ and suppose that \begin{equation*} \langle x -y, x^{*} - y^{*} \rangle \geq \theta^{'}(x,y)\Vert x-y\Vert\geq\theta(\x,\y)\Vert \x-\y\Vert, \end{equation*} for all $(y, y^{*})\in G(T_{\varphi})$, then we have \begin{align*} &\langle x,x^{*} \rangle \geq\\ & \underset{(y, y^{*})\in G(T^{\theta}_{\varphi})}\sup\lbrace \langle x, y^{*} \rangle + \langle y, x^{*} \rangle - \langle y, y^{*} \rangle +\theta(\x,\y)\Vert \x-\y\Vert\rbrace\\ &=\underset{\varphi(y, y^{*}) = \langle y, y^{*} \rangle }\sup\lbrace \langle x, y^{*} \rangle + \langle y, x^{*} \rangle -\varphi(y, y^{*})+\theta(\x,\y)\Vert \x-\y\Vert\rbrace\\ %%%(y, y^{*})\inG(T^{\theta}_{\varphi})\Longrightarrow\varphi(y, y^{*}) = \langle y, y^{*}~\varphi(y, y^{*}) \leq \langle y, y^{*} &=\underset{\varphi(y, y^{*}) \leq \langle y, y^{*} \rangle }\sup\lbrace \langle x, y^{*} \rangle + \langle y, x^{*} \rangle -\varphi(y, y^{*}) +\theta(\x,\y)\Vert \x-\y\Vert\rbrace\\ &=\underset{(y, y^{*}) \in b(\varphi) }\sup\lbrace \langle x, y^{*} \rangle + \langle y, x^{*} \rangle -\varphi(y, y^{*}) +\theta(\x,\y)\Vert \x-\y\Vert\rbrace\\ %&=\underset{ (y, y^{*}) \in X\times X^{*} }\sup\lbrace \langle x, y^{*} \rangle + \langle y, x^{*} \rangle -\varphi(y, y^{*})-\delta_{\varphi(b)}(y) +\theta(\x,\y)\Vert \x-\y\Vert\rbrace\\ %\varphi\in\varphi\in \Phi_{\theta}(X) &=(\varphi +\delta_{b(\varphi)})^{*}_{(x,x^{*})}(\theta,(x^{*},x))=\varphi(x,x^{*})\\ &\geq \langle x,x^{*} \rangle ~~~\text{by Proposition \ref{pro 6.1}}. \end{align*} Hence, $\langle x^{*},x\rangle=\varphi(x,x^{*})$ i.e. $(x,x^{*})\in \G(T_{\varphi})$. This proves that $\G(T_{\varphi})$ is maximal $ \theta^{'} $-monotone according to the Proposition \ref{pro 1.3}. It only remains to prove that, for $T \in \mathfrak{M}_{\theta^{'}}(X)$, one has $T_{L^{\theta}_{T}}=T$ and for $\varphi\in \Phi_{\theta}(X)$, one has $L^{\theta}_{T_{\varphi}}=\varphi$.\\ To prove the first equality, let $x \in X$. We have \begin{align*} T_{L^{\theta}_{T}}(x)&=\lbrace x^{*}\in X^{*}:L^{\theta}_{T}(x,x^{*})= \langle x, x^{*} \rangle\rbrace\\ %%by 6.1 &=\lbrace x^{*}\in X^{*}:L^{\theta}_{T}(x,x^{*})\leq \langle x, x^{*} \rangle \rbrace \\ &=\lbrace x^{*}\in X^{*}: (x,x^{*})\in b(L^{\theta}_{T})\rbrace\\ &=\lbrace x^{*}\in X^{*}: (x,x^{*})\in \G(T)\rbrace\\ &=T(x). \end{align*} To prove the second equality, let $(x,x^{*})\in X\times X^{*}$. We have \begin{align*} &L^{\theta}_{T_{\varphi}}(x,x^{*})\\ &=\underset{(y,y^{*})\in G(T_{\varphi})}\sup\lbrace \langle x,y^{*} \rangle + \langle y,x^{*} \rangle - \langle y,y^{*} \rangle +\theta(\x,\y)\Vert \x-\y\Vert\rbrace \\ &=\underset{\varphi(y, y^{*}) = \langle y, y^{*} \rangle }\sup\lbrace \langle x,y^{*} \rangle + \langle y,x^{*} \rangle - \langle y,y^{*} \rangle +\theta(\x,\y)\Vert \x-\y\Vert\rbrace \\ &=\underset{\varphi(y,y^{*}) \leq \langle y,y^{*} \rangle }\sup\lbrace \langle x,y^{*} \rangle + \langle y,x^{*} \rangle -\varphi(y,y^{*})+\theta(\x,\y)\Vert \x-\y\Vert\rbrace\\ &=\underset{(y,y^{*}) \in b(\varphi)}\sup\lbrace \langle x,y^{*} \rangle + \langle y,x^{*} \rangle -\varphi(y,y^{*})+\theta(\x,\y)\Vert \x-\y\Vert\rbrace\\ %&=\underset{(y,y^{*}) \in X\times X^{*}}\sup\lbrace \langle x,y^{*} \rangle + \langle y,x^{*} \rangle -\varphi(y,y^{*})+\theta(\x,\y)\Vert \x-\y\Vert-\delta_{b(\varphi)}(y,y^{*})\rbrace\\ &=(\varphi+\delta_{b(\varphi)})^{*}_{(x,x^{*})}(\theta,(x^{*},x))=\varphi(x,x^{*}) \end{align*} \end{proof} \begin{corollary} Let $T$ be a $\theta^{'}$-monotone operator and $ \G(T) $ satisfying \eqref{pdt}. $T$ is maximal $\theta^{'}$-monotone if and only if there exists (unique) $\varphi\in \Phi_{\theta}(X)$ such that \begin{center} $T(x)= \lbrace x^{*}\in X^{*}: (x^{*},x)=\varphi(x,x^{*})\rbrace,~~ \forall x\in X$. \end{center} \end{corollary} \begin{thebibliography}{99} \bibitem{ex r1} H. 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