Abstract
We introduce a modified Ishikawa iterative process for approximating a fixed point of two infinitely nonexpansive self-mappings by using the hybrid method in a Hilbert space and prove that the modified Ishikawa iterative sequence converges strongly to a common fixed point of two infinitely nonexpansive self-mappings.
Highlights
Let C be a nonempty closed convex subset of a Hilbert space H, T a self-mapping of C
Qn {v ∈ C : xn − v, xn − x0 ≤ 0}, xn 1 PCn∩Qn x0, Fixed Point Theory and Applications where PC denotes the metric projection from H onto a closed convex subset C of H
Assume that F ⊂ Qn for all n ≥ q 1, since xn 1 is the projection of xq onto Cn Qn, so xn 1 − z, xq − xn 1 ≥ 0, ∀z ∈ Cn Qn
Summary
Let C be a nonempty closed convex subset of a Hilbert space H, T a self-mapping of C. X0 ∈ C chosen arbitrarity, yn αnxn 1 − αn T xn, Cn v ∈ C : yn − v ≤ xn − v , Qn {v ∈ C : xn − v, xn − x0 ≤ 0}, xn 1 PCn∩Qn x0 , Fixed Point Theory and Applications where PC denotes the metric projection from H onto a closed convex subset C of H. They prove the sequence {xn} generated by that algorithm 1.1 converges strongly to a fixed point of. We will use the notation: for weak convergence and → for strong convergence. ωw xn {x : ∃xnj x} denotes the weak ω-limit set of xn
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