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The split common null point problem. (English) Zbl 1262.47073

In this paper, the authors study the Split Null Point Problem for set-valued mappings in Hilbert spaces. More precisely, if \(H_1, H_2\) are two Hilbert spaces, \(B_i:H_1\to 2^{H_1}\) for \(1\leq i\leq p\), respectively \(F_j:H_2\to 2^{H_2}\) for \(1\leq j\leq r\), are given set-valued operators and \(A_j:H_1\to H_2\) for \(1\leq j\leq r\) are bounded linear operators, the problem is to find \(x^*\in H_1\) such that \(0\in \bigcap_{i=1}^pB_i(x^*)\) and such that the points \(y_j^*=A_j(x^*)\) solve \(0\in \bigcap_{j=1}^{r}F_j(y_j^*)\). Iterative algorithms are proposed and weak/strong convergence theorems are given.

MSC:

47H04 Set-valued operators
49J40 Variational inequalities
90C25 Convex programming
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