## 3 Some Applications
### Zeros of Maximal Monotone Operators  
In this section, we apply Theorem 1 to the problem of finding zero points of maximal monotone operators. This is a very active topic in many fields of pure and applied mathematics. In the real world, many important problems have reformulations that require finding zero points of a maximal monotone operator; for instance, evolution equations, convex minimization problem, economics, finance, image recovery and applied science (see, e.g., [39; 40; 41; 42; 43; 44; 45; 46] and the references therein). Let \(A:E\longrightarrow 2^{E^{*}}\) be a set-valued mapping. We show \(G(A)\) as the _graph_ of \(A\), _i.e._, \(G(A)=\{(x,x^{*})\in E\times E^{*}:x^{*}\in Ax\}\). An operator \(A\) is called _monotone_ if \(\langle x^{*}-y^{*},x-y\rangle\geq 0\) for each \((x,x^{*}),(y,y^{*})\in G(A)\). We call monotone operator \(A\) a _maximal_ if its graph is not contained in the graph of any other monotone operators on \(E\). It is known that if \(A\) is maximal monotone, then the set \(A^{-1}(0^{*})=\{x\in E:0^{*}\in Ax\}\) is closed and convex. The resolvent of \(A\), denoted by \(Res^{f}_{\lambda A}:E\longrightarrow 2^{E}\), is defined as follows [47]:  
\[Res^{f}_{\lambda A}(x)=(\
abla f+\lambda A)^{-1}\circ\
abla f(x)\]  
where \(\lambda>0\). Moreover, from [47], it is known that \(F(Res^{f}_{\lambda A})=A^{-1}(0^{*})\), and \(Res^{f}_{A}\) is single-valued and Bregman firmly nonexpansive. If \(f\) is a Legendre function, which is bounded, uniformly Frechet differentiable on bounded subsets of \(E\), then \(\widehat{F}(Res^{f}_{\lambda A})=F(Res^{f}_{\lambda A})\) (see [48]). It is obvious that if \(\widehat{F}(Res^{f}_{\lambda A})=F(Res^{f}_{\lambda A})\), then a Bregman that is firmly nonexpansive is a Bregman relatively nonexpansive mapping. The Yosida approximation \(A_{\lambda}:E\longrightarrow E,\lambda>0\), is defined by:  
\[A_{\lambda}(x)=\frac{1}{\lambda}\big{(}\
abla f(x)-\
abla f\big{(}Res^{f}_{ \lambda A}\big{)}\big{)}\;\;\text{for all}\;x\in E\;\;\text{and}\;\;\lambda>0\]  
From Proposition 2.7 in [32], we know that \(\big{(}Res^{f}_{\lambda A}(x),A_{\lambda}(x)\big{)}\in G(A)\) and \(0^{*}\in Ax\) if and only if \(0^{*}\in A_{\lambda}(x)\) for all \(x\in E\) and \(\lambda>0\). Take \(C=E\) and \(T_{i}=Res^{f}_{\lambda A_{i}}\), \(\lambda>0\) for each \(i=1,2,...,N\) in Theorem 1; we immediately have the following result:  
Theorem 3: Let us take a real reflexive Banach space \(E\), and let \(f:E\longrightarrow\mathbb{R}\) be a strongly coercive Legendre bounded function, which is uniformly Frechet differentiable and totally convex on bounded subsets of \(E\), and let \(A_{i}:E\longrightarrow 2^{E^{*}}\;(i=1,2,...,N)\) be a finite collection of maximal monotone operators, such that \(\mathcal{F}:=\bigcap_{i=1}^{N}A_{i}^{-1}(0)\
eq\emptyset\). For \(u,x_{0}\in E\), let \(\{x_{n}\}\) be a sequence generated by:  
\[\left\{ \begin{array}{l}y_{n,1}=\
abla f^{*}(\beta_{n,1}\
abla f(x_{n})+(1- \beta_{n,1})\
abla f\big{(}Res^{f}_{\lambda A_{1}}x_{n}\big{)})\\ y_{n,2}=\
abla f^{*}(\beta_{n,2}\
abla f(x_{n})+(1-\beta_{n,2})\
abla f\big{(} Res^{f}_{\lambda A_{2}}y_{n,1}\big{)})\\ \vdots\\ y_{n,N}=\
abla f^{*}(\beta_{n,N}\
abla f(x_{n})+(1-\beta_{n,N})\
abla f\big{(} Res^{f}_{\lambda A_{N}}y_{n,N-1}\big{)})\\ x_{n+1}=\
abla f^{*}(\alpha_{n}\
abla f(u)+(1-\alpha_{n})\
abla f(y_{n,N})),\; \;\forall n\geq 0\end{array} \right. \tag{25}\]  
where \(\lambda>0\). Suppose that \(\{\alpha_{n}\}\) and \(\{\beta_{n,i}\}_{i=1}^{N}\) are as in Theorem 1. Then, \(\{x_{n}\}\) strongly converges to \(p=P^{f}_{\mathcal{F}}(u)\), where \(P^{f}_{\mathcal{F}}\) is the Bregman projection of \(E\) onto \(\mathcal{F}\). **Acknowledgments:** The authors are grateful to the reviewers for careful reading the paper and for the suggestions, which improved the quality of this work. This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU59 No.59000399).