Abstract
AbstractThe convex feasibility problem (CFP) of finding a point in the nonempty intersection "Equation missing" is considered, where "Equation missing" is an integer and the "Equation missing"'s are assumed to be convex closed subsets of a Banach space "Equation missing". By using hybrid iterative methods, we prove theorems on the strong convergence to a common fixed point for a finite family of relatively nonexpansive mappings. Then, we apply our results for solving convex feasibility problems in Banach spaces.
Highlights
We are concerned with the convex feasibility problem CFPN finding an x ∈ Ci, i1 where N 1 is an integer, and C1, . . . , CN are intersecting closed convex subsets of a Banach space E
There is a considerable investigation on CFP in the framework of Hilbert spaces which captures applications in various disciplines such as image restoration
In computer tomography with limited data, in which an unknown image has to be reconstructed from a priori knowledge and from measured results, each piece of information gives a constraint which in turn, gives rise to a convex set Ci to which the unknown image should belong see 7
Summary
N finding an x ∈ Ci, i1 where N 1 is an integer, and C1, . . . , CN are intersecting closed convex subsets of a Banach space E. Our purpose in the present paper is to obtain an analogous result for a finite family of relatively nonexpansive mappings in Banach spaces. Plubtieng and Ungchittrakool 22 studied the strong convergence to a common fixed point of two relatively nonexpansive mappings of the sequence {xn} generated by x0 x ∈ C, yn J−1 αnJxn 1 − αn Jzn , zn J−1 βn[1] Jxn βn[2] JT xn βn[3] JSxn , xn 1 PHn∩Wn x, n 0, 1, 2, . In 2008, Plubtieng and Ungchittrakool 25 established strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. I → for strong convergence and for weak convergence; ii ωw xn {x : ∃xnr x} denotes the weak ω-limit set of {xn}
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