Let \(\mathcal H\) be a Hilbert space and \(B:{\mathcal H} \rightarrow 2^{\mathcal {H}}\) be a set-valued mapping. Let \(dom(B)\) denote the domain of \(B\), viz., the set of all \(x \in {\mathcal H}\) for which \(Bx \neq \emptyset\). \(B\) is said to be monotone if \(\langle x-y,u-v \rangle \geq 0\) for all \(x,y \in dom(B)\), \(u \in B(x)\), \(v\in B(y)\). A monotone operator is said to be maximal if it is maximal in the sense of graph inclusion. Let \(B\) be a maximal monotone operator and \(r > 0\). The resolvent of \(B\) for \(r\) is the single-valued operator \(J_r:{\mathcal H} \rightarrow dom(B)\) defined by \(J_r:=(I+rB)^{-1}\). Let \({\mathcal H}_1\) and \({\mathcal H}_2\) be real Hilbert spaces. For \(1 \leq i \leq m\) and \( 1 \leq j \leq n\), let \(A_i: {\mathcal H}_1 \rightarrow 2^{{\mathcal H}_1}\) and \(B_j: {\mathcal H}_2 \rightarrow 2^{{\mathcal H}_2}\) be set-valued mappings. Let \(T_j:\mathcal H_1 \rightarrow {\mathcal H}_2\) be bounded linear operators, \(1 \leq j \leq n\). Then the split common null point problem is to find \(z \in {\mathcal H}_1\) such that \(z \in (\cap _{i=1}^m A_i^{-1}\{0\}) \cap (\cap _{j=1}^n T_j^{-1}(B_j^{-1}\{0\}))\).
First, the authors derive certain properties of the resolvent operator. These are used in establishing a strong convergence theorem for finding a solution of the split common null point problem. This solution is characterized as a unique solution of a certain variational inequality problem of a nonlinear operator. Two applications are presented.