Abstract
Mann’s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fréchet differentiable norm), but not strongly even in a Hilbert space. Strong convergence is therefore a nontrivial problem. In this paper we provide certain conditions either on the underlying space or on the mapping under investigation so as to guarantee the strong convergence of Mann’s iteration process and its variants.
Highlights
Let X be a real Banach space with norm k · k, let C be a nonempty closed convex subset of X, and let T : C → C be a nonexpansive mapping (i.e., k Tx − Tyk ≤ k x − yk for x, y ∈ C)
We prove strong convergence of Mann’s iteration process in a reflexive Banach space with Opial’s and Kadec-Klee properties when I − T is φ-accretive
Recall that a Banach space X is said to have a weakly continuous duality map if, for some gauge μ, the duality map Jμ : X → X ∗ is continuous when X is endowed with the weak topology and the dual space X ∗ endowed with the weak-star topology
Summary
Let X be a real Banach space with norm k · k, let C be a nonempty closed convex subset of X, and let T : C → C be a nonexpansive mapping (i.e., k Tx − Tyk ≤ k x − yk for x, y ∈ C). Where the initial point x0 ∈ C is arbitrary and (tn ) ⊂ [0, 1] It is known [2] that if X is uniformly convex with a Fréchet differentiable norm, if Fix( T ) is nonempty, and if ∑∞. A regularization method is introduced to approximate a fixed point of T This method implicitly yields a sequence of approximate solutions and we shall prove (in Theorem 4) its strong convergence to a solution of a variational inequality. Combining this regularization method with Mann’s method, we obtain a new iteration process (see (22)). “xn → x” stands for the strong convergence of ( xn ) to x, ωw ( xn ) := { x : ∃ xnk * x } is the set of all weak accumulation points of the sequence ( xn )
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