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

Tang, Yu-Chao; Peng, Ji-Gen; Liu, Li-Wei

Math. Model. Anal. 17, No. 4, 457-466 (2012).

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).

Reviewer: Peter Zabreiko (Minsk)

MSC:

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

Keywords:

split feasibility problem; demicontractive mappings; common fixed point; weak convergence