## 3 Some Applications  
### Variational Inequality Problems  
In this part, we apply Theorem 1 to finding the solution sets of the variational inequality corresponding to the Bregman inverse strongly monotone operator. Variational inequalities were introduced by [PERSON] and [PERSON] as a tool for the study of partial differential equations with applications principally drawn from mechanics (see [37]). Note that most of the variational inequality problems contain, as special cases, such recognized problems in mathematical programming as: systems of nonlinear equations, optimization problems, equilibrium problems and complementarity problems. Moreover, these are also related to fixed point problems. **Definition 6**. : ([2]) Let \(f:E\longrightarrow(-\infty,+\infty]\) be a G\(\ddot{\text{a}}\)teaux differentiable function. A mapping \(A:E\longrightarrow 2^{E^{*}}\) satisfying the range condition, _i.e., \(ran(\
abla f-A)\subset ran(\
abla f)\)_ is called _Bregman inverse strongly monotone_ if \(domA\cap int(domf)\
eq\emptyset\) and for any \(x,y\in int(domf)\) and each \(u\in Ax,v\in Ay\),  
\[\langle u-v,\
abla f^{*}(\
abla f(x)-u)-\
abla f^{*}(\
abla f(y)-v)\rangle\geq 0\]  
If \(E=H\) is a real Hilbert space and \(f(x)=\frac{1}{2}\|x\|^{2}\), then we have \(\
abla f=I\), and the Bregman inverse strongly monotone mapping reduces to an inverse strongly monotone mapping. Let \(A:C\longrightarrow E^{*}\) be a Bregman inverse strongly monotone operator, and let \(C\) be a nonempty, closed and convex subset of \(domA\). The _variational inequality problem_ corresponding to \(A\) is to find \(x^{*}\in C\), such that:  
\[\langle Ax^{*},y-x^{*}\rangle\geq 0,\ \ \forall y\in C \tag{23}\]  
The set of solutions of Equation (23) is denoted by \(VI(C,A)\). **Definition 7**. : ([2]) Let \(A:E\longrightarrow 2^{E^{*}}\) be an any operator; the _anti-resolvent_\(A^{f}:E\longrightarrow 2^{E}\) of \(A\) is defined by:  
\[A^{f}=\
abla f^{*}\circ(\
abla f-A)\]  
Observe that \(domA^{f}\subset domA\cap int(domf)\) and \(ranA^{f}\subset int(domf)\). Therefore, we know an operator \(A\) is Bregman inverse strongly monotone if and only if anti-resolvent \(A^{f}\) is a single-valued Bregman firmly nonexpansive mapping (see [38], Lemma 3.5 (c) and (d), p. 2109). From the definition of the anti-resolvent, we obtain the following useful fact, which concerns the variational inequality problem:  
**Lemma 13**. : ([3,29]) _Let \(A:E\longrightarrow E^{*}\) be a Bregman inverse strongly monotone mapping and \(f:E\longrightarrow(-\infty,\infty]\) be a Legendre and totally convex function that satisfies the range condition. If \(C\) is a nonempty, closed and convex subset of \(domA\cap int(domf)\), then:_  
_(1) \(P^{f}_{C}\circ A^{f}\) is Bregman relatively nonexpansive mapping, where \(A^{f}=\
abla f^{*}\circ(\
abla f-A)\);_  
_(2) \(F(P^{f}_{C}\circ A^{f})=VI(C,A)\)._  
From Theorem 1 and Lemma 13, we immediately have the following result:  
_Theorem 2_. : Let us consider a real reflexive Banach space \(E\); let \(f:E\longrightarrow\mathbb{R}\) be a strongly coercive Legendre function, which is bounded, uniformly Fr\(\ddot{\text{c}}\)chet differentiable and totally convex on bounded subsets of \(E\), which satisfies the range condition, \(C\) be a nonempty, closed and convex subset of \(domA\cap int(domf)\), and \(A_{i}:C\longrightarrow E^{*}\)\((i=1,2,...,N)\) be a Bregman inverse strongly monotone function, such that \(\mathcal{F}:=\bigcap_{i=1}^{N}VI(C,A_{i})\
eq\emptyset\). For \(u,x_{0}\in C\), 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{(}P^{f}_{C}\circ A^{f}_{1}x_{n}\big{)})\\ y_{n,2}=\
abla f^{*}(\beta_{n,2}\
abla f(x_{n})+(1-\beta_{n,2})\
abla f\big{(} P^{f}_{C}\circ A^{f}_{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{(} P^{f}_{C}\circ A^{f}_{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}