Hybrid methods for a finite family of G-nonexpansive mappings in Hilbert spaces endowed with graphs

Peer review under responsibility of Kalasalingam University.

Figures & data

Fig. 1 Error plots for sequences $\left\{x_n\right\}$ in Tables 1 and 2.

Table 1 Numerical results of iteration (3.1(3.1) $\left\{\begin{array}{cc}\hfill y_{i,n}=& {\alpha }_{i,n}{x}_n+(1-{\alpha }_{i,n})T_i{x}_n,\hfill \\ \hfill C_{i,n}=& \{z\in C:\Vert y_{i,n}-z\Vert \le \Vert x_n-z\Vert \},\hfill \\ \hfill C_n=& \bigcap _{i=1}^N{C}_{i,n},\hfill \\ \hfill Q_n=& \{z\in Q_{n-1}:〈x_1-x_n,z-x_n〉\le 0\},n\ge 2,\hfill \\ \hfill x_{n+1}=& P_{C_n\cap Q_n}{x}_1,n\ge 1,\hfill \end{array}\right.$(3.1) ).

$n$	$y_{1,n}$	$y_{2,n}$	$C_n\cap Q_n$	$x_n$
1	0.48852	0.50966	[0.25483, 3.00000]	0.00000
2	0.64072	0.60174	[0.44777, 3.00000]	0.25483
3	0.73736	0.68992	[0.59257, 3.00000]	0.44777
4	0.80652	0.76353	[0.69955, 3.00000]	0.59257
5	0.85695	0.82151	[0.77825, 3.00000]	0.69955
6	0.89402	0.86604	[0.83613, 3.00000]	0.77825
7	0.92137	0.89978	[0.87875, 3.00000]	0.83613
8	0.94160	0.92516	[0.91018, 3.00000]	0.87875
9	0.95660	0.94418	[0.93339, 3.00000]	0.91018
10	0.96772	0.95839	[0.95055, 3.00000]	0.93339
⋮	⋮	⋮	⋮	⋮
30	0.99991	0.99988	[0.99986, 3.00000]	0.99981

Table 2 Numerical results of iteration (3.10(3.10) $\left\{\begin{array}{cc}\hfill y_n=& {\alpha }_{0,n}{x}_n+\sum _{i=1}^N{\alpha }_{i,n}{T}_i{x}_n,\hfill \\ \hfill C_n=& \{z\in C:\Vert y_n-z\Vert \le \Vert x_n-z\Vert \},\hfill \\ \hfill Q_n=& \{z\in Q_{n-1}:〈x_1-x_n,z-x_n〉\le 0\},n\ge 2,\hfill \\ \hfill x_{n+1}=& P_{C_n\cap Q_n}{x}_1,n\ge 1,\hfill \end{array}\right.$(3.10) ).

$n$	$y_n$	$C_n\cap Q_n$	$x_n$
1	0.20707	[0.10354, 3.00000]	0.00000
2	0.34473	[0.22413, 3.00000]	0.10354
3	0.45763	[0.34088, 3.00000]	0.22413
4	0.55155	[0.44621, 3.00000]	0.34088
5	0.62963	[0.53792, 3.00000]	0.44621
6	0.69442	[0.61617, 3.00000]	0.53792
7	0.74809	[0.68213, 3.00000]	0.61617
8	0.79247	[0.73730, 3.00000]	0.68213
9	0.82913	[0.78321, 3.00000]	0.73730
10	0.85938	[0.82130, 3.00000]	0.78321
⋮	⋮	⋮	⋮
30	0.99731	[0.99655, 3.00000]	0.99579