Bulletin of the
Korean Mathematical Society BKMS

In this paper, we introduce and study a system of nonlinear implicit variational inclusions (SNIVI) in real Banach spaces: determine elements $x^{\ast}, \ y^{\ast}, \ z^{\ast} \in E$ such that $$\theta \in \alpha T(y^{\ast}) + g(x^{\ast}) - g(y^{\ast}) + A(g(x^{\ast})) \hspace{5mm} {\rm for} \ \alpha > 0, $$ $$\theta \in \beta T(z^{\ast}) + g(y^{\ast}) - g(z^{\ast}) + A(g(y^{\ast})) \hspace{5mm} {\rm for} \ \beta > 0,$$ $$\theta \in \gamma T(x^{\ast}) + g(z^{\ast}) - g(x^{\ast}) + A(g(z^{\ast})) \hspace{5mm} {\rm for} \ \gamma > 0, $$ where $T, g : E \to E$, $\theta$ is zero element in Banach space $E$, and $A : E \to 2^E$ be $m$-accretive mapping. By using resolvent operator technique for $m$-accretive mapping in real Banach spaces, we construct some new iterative algorithms for solving this system of nonlinear implicit variational inclusions. The convergence of iterative algorithms be proved in $q$-uniformly smooth Banach spaces and in real Banach spaces, respectively.

Keywords: system of nonlinear implicit variational inclusion, resolvent operator, $m$-accretive mapping, approximation-solvability, iterative algorithms

MSC numbers: 49J40, 47J20