Boolean vector spaces?I

Fixed point theorems for a class of maps in normed Boolean vector spaces

“…In addition, most of these results are in finite dimensional spaces. Recently, Rao and Pant [3] obtained some fixed and common fixed point theorems for asymptotically regular maps on finite dimensional normed Boolean vector spaces (for details of Boolean vector spaces, we refer to Subrahmanyam [4,5]). The purpose of this article is to obtain some coincidence and common fixed point theorems in infinite dimensional normed Boolean vector spaces for certain classes of maps without using continuity conditions.…”

“…Definition 1.4. [4]. Let V = (V, +) be an additive abelian group and B = (B, +, ., ) a Boolean algebra.…”

“…[4]. Let B be any Boolean algebra and V be the additive group of the corresponding Boolean ring; then V is a B − vector space if we define: For a ∈ B and x ∈ V, ax = the (Boolean) product of a and x in B .…”

“…Example 1.6. [4]. Let R be any Ring with unity element 1 and let B denotes the set of all the central idempotents of R; then it is known that (B, ∪, ∩, ) is a Boolean algebra, where, by definition, a ∪ b = a + b -ab, a ∩ b = ab and a' = 1 -a.…”

“…Definition 1.9. [4]. If {x n } is a sequence of elements of V, we say that x n x (x ∈ Further, S and T have a unique common fixed point provided that SSu = Su and S and T commute at the coincidence point.…”