Abstract
In this paper, we discuss three modified Halpern iterations as follows: I II III and obtained the strong convergence results of the iterations (I)-(III) for a k-strictly pseudocontractive mapping, where satisfies the conditions: (C1) and (C2) , respectively. The results presented in this work improve the corresponding ones announced by many other authors.
Highlights
Let H be a real Hilbert space with the inner product ·, · and the norm · and let C be a nonempty closed convex subset of H.Recall that a mapping T with domain D(T) and range R(T) in the Hilbert space H is called strongly pseudo-contractive if, for all x, y ∈ D(T), there exists k ∈ (, ) such thatTx – Ty, x – y ≤ k x – y, ( . )while T is said to be pseudo-contractive if ( . ) holds for k =
The results presented in this work improve the corresponding ones announced by many other authors
The purpose of this paper is to present a significant answer to the above open question
Summary
Obtained the strong convergence results of the iterations (I)-(III) for a k-strictly pseudocontractive mapping, where {αn} satisfies the conditions: (C1) limn→∞ αn = 0 and (C2) ) converges strongly to a fixed point of T in a Hilbert space, where {αn} satisfies the following conditions: (C ) (C ) (C ) The strong convergence of Halpern’s iteration to a fixed point of T has been proved in Banach spaces; see, e.g., [ – ].
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