A

nonempty set H with a hyper operation "[omicron]" and a constant 0 is called a hyper BCK-algebra (See [16]), if it satisfies the following conditions: for any x,y,z [member of] H,

Under the assumptions that {[[xi].sub.k]} is bounded, i.e., [parallel][[xi].sub.k][parallel] [less than or equal to] N, [for all]n [greater than or equal to] 1, [[lambda].sub.n] [member of] [a, b] [subset] (0, 2/[N.sup.2]), [for all]n [greater than or equal to] 1 and the set [OMEGA] := {u [member of] Fix(T) : f ([y.sup.n], u) [less than or equal to] 0, n [greater than or equal to] 1} is

nonempty, the authors proved that the sequence {[x.sup.n]} generated by the above algorithm converges to a solution of (1.1).

Let CB(X) be the collection of all

nonempty closed bounded subsets of X.

(2) A

nonempty subset A of an ordered AG-groupoid S is called a left (right) ideal of S if we have the following:

Let C be a

nonempty closed convex subset of a real Hilbert space H and let S : C [right arrow] C be a nonexpansive mapping with Fix (S) [not equal to] 0.

It is clear that [X.sup.Par.sub.u] [subset] [X.sup.wPar.sub.u] for every positive integer m, every

nonempty set X, and every function u = ([u.sub.1],..., [u.sub.m]) : X [??] [R.sup.m].

Let ([OMEGA], [SIGMA]) be a measurable space and C be a

nonempty closed convex sunset of a separable Banach space E.

(6) If {[X.sub.n]} is a sequence of

nonempty, bounded (in norm), weakly closed subsets of [W.sup.n,1]([[0, T].sup.N]) and [X.sub.1] [subset] [X.sub.2] [subset] ...

where [alpha], [[alpha].sub.n] [member of] (0,1) and [x.sub.1] is an any given element in a

nonempty closed convex subset E [subset not equal to] X.

If x [member of] H and A is a

nonempty subset of H, then [A.sub.x] = {(y, z) [member of] H x H | x [less than or equal to] y [omicron] z}.

In [9], Schirmer observed that for a given self map f : [S.sup.n] [right arrow] [S.sup.n] of an n-sphere, n [greater than or equal to] 2, any closed

nonempty proper subset A of [S.sup.n] can be realized as the fixed point set of a map g [member of] [f] with Fix(g) = A.