Abstract
The purpose of this is to introduce and study total asymptotically strict pseudocontractive semigroup, asymptotically strict pseudocontractive semigroup etc. the strong convergence theorems of the explicit iteration process for the new semigroups in arbitrary Banach spaces are established. The results presented in the paper extend and improve some recent results announced by many authors.
Highlights
1 Introduction and preliminaries Let E be a real Banach space, E* be the dual space of E, C is a nonempty closed convex subset of E, R+ is the set of nonnegative real numbers and J : E ® 2E* is the normalized duality mapping defined by
We first recall some definitions: A one parameter family := {T(t) : t ≥ 0} of self mappings of C is said a nonexpansive semigroup, if the following conditions are satisfied: (i) T (t1 + t2)x = T (t1)T (t2)x, for any t1, t2 Î R+ and x Î C; (ii) T (0)x = x, for each x Î C; (iii) for each x Î C, t ↦ T (t)x is continuous; (iv) for any t ≥ 0, T (t) is nonexpansive mapping on C, that is for any x, y Î C
[12] Let E be any real Banach space, E* be the dual space of E and
Summary
Definition 1.1 A one parameter family := {T(t) : t ≥ 0} of self mapping of C satisfies conditions (i)-(iii), it is said (e) total asymptotically strict pseudocontractive semigroup, if there exists bounded function l : [0, ∞) ® (0, ∞) and sequences {μn} ⊂ [0, ∞)and {ξn} ⊂ [0, ∞) with μn ® 0 and ξn ® 0 as n ® ∞. (f) asymptotically strict pseudocontractive semigroup, if there exists a bounded function l : [0, ∞) ® (0, ∞) and a sequence {kn} ⊂ [1, ∞) with kn ® 1 as n ® ∞, for any given x, y Î C, there exists j(x - y) Î J(x - y) such that
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