if the transition entropy converges to zero and the entropy does not, then the process converges to a cycle.
They can be identified by the asymptotic behavior of entropy and transition entropy (this works only if no nodes are internal): If both the transition entropy and the entropy converge to zero, then the process converges to a fixed point.
It is proved, when the data are compatible, this fixed point process converges to the Cauchy problem solution.
Taking initial guess x 0 = 0.6 (away from the fixed point), the Picard process converges to a fixed point in 8 iterations, the Mann process converges in 75 iterations, the Ishikawa process converges in 38 iterations, the Noor process converges in 27 iterations, the Agarwal et al. process converges in 6 iterations and the S-iterative process converges in 6 iterations.
For x 0 = 0.8, the Picard process converges to a fixed point in 8 iterations, the Mann process converges in 69 iterations, the Ishikawa process converges in 34 iterations, the Noor process converges in 24 iterations, the Agarwal et al. process converges in 7 iterations and the S-iterative process converges in 6 iterations.
For x 0 = 0.8, the Picard process converges to a fixed point in 4 iterations, the Mann process converges in 36 iterations, the Ishikawa process converges in 22 iterations, the Noor process converges in 5 iterations and the Agarwal et al. as well as the S-iterative processes converge in 3 iterations.
Taking initial guess x 0 = 0.6 (away from the fixed point), the Picard process converges to a fixed point in 5 iterations, the Mann process converges in 52 iterations, the Ishikawa process converges in 40 iterations, the Noor process converges in 57 iterations and the Agarwal et al. as well as the S-iterative processes converge in 4 iterations.
Here, the optimization process converges to an acceptable solution with a certain degree of freedom.
The optimization process converges to a feasible solution s t d≃0 (highlighted in blue in Table 2).
