  In this paper, we propose new extragradient algorithms for solving a split
equilibrium and nonexpansive mapping SEPNM($C, Q, A, f, g, S, T)$ where $C, Q$
are nonempty closed convex subsets in real Hilbert spaces $\mathcal{H}_1,
\mathcal{H}_2 $ respectively, $A : \mathcal{H}_1 \to \mathcal{H}_2$ is a
bounded linear operator, $f$ is a pseudomonotone bifunction on $C$ and $g$ is a
monotone bifunction on $Q$, $S, T$ are nonexpansive mappings on $C$ and $Q$
respectively. By using extragradient method combining with cutting techniques,
we obtain algorithms for the problem. Under certain conditions on parameters,
the iteration sequences generated by the algorithms are proved to be weakly and
strongly convergent to a solution of this problem.
