## A cyclic algorithm for the split common fixed point problem of demicontractive mappings in Hilbert spaces.(English)Zbl 1267.47104

The article deals with the following split common fixed point problem: $\text{Find a point} \;x^* \in \bigcap_{i=1}^p \text{Fix} (U_i), \;\text{such that} \;Ax^* \in \bigcap_{j=1}^r \text{Fix} (T_j),(1)$ where $$S$$ is a bounded linear between Hilbert spaces $$H_1$$ and $$H_2$$, $$U_i$$, $$i = 1,\dots,p$$, are $$\beta_i$$-demicontractive and $$T_j$$, $$j = 1,\ldots,r$$ $$\mu_j$$-demicontractive continuous operators. (Recall that an operator $$V$$ is $$k$$-demicontractive ($$k \in (0,1)$$) if $$\|Tx - q\|^2 \leq \|x - q\|^2 + k\|x - Vx\|^2$$ for all $$x \in H$$ and $$q \in \text{Fix}(T)$$.) It is studied the following algorithm: $x_{n+1} = (1 - \alpha_n)u_n + \alpha_nU_{i(n)}(u_n), \quad x_0 \in H_1, \;n \geq 0,(2)$ where $$u_n = x_n + \gamma A^*(T_{j(n)} - I)Ax_n$$, $$i(n) = n (\text{mod}\, p) + 1$$, $$j(n) = n (\text{mod}\, r) + 1$$, $$\gamma \in (0,\lambda^{-1}(1 - \mu)$$ with $$\lambda = \|A^*A\|$$, $$\delta < \alpha_n < 1 - \beta- \delta$$, $$\mu = \max \;\{\mu_j:\;1 \leq j \leq r\}$$, $$\beta = \max \;\{\beta_j:\;1 \leq j \leq r\}$$, $$\delta$$ is positive and small enough. It is assumed that the operators $$I - I_i$$, $$i = 1,\dots,p$$, and $$I - T_j$$, $$j = 1,\ldots,r$$, are demiclosed at 0 and it is proved, provided that the solution set $$\Omega$$ of (1) is nonempty, that the sequence $$(x_n)$$ generated by (2) converges weakly to a solution of (1).

### MSC:

 47J25 Iterative procedures involving nonlinear operators 49J53 Set-valued and variational analysis 47H10 Fixed-point theorems
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