Ergodic Theorem and Strong Convergence of Averaged Approximants for Non-Lipschitzian Mappings in Banach Spaces (original) (raw)

Approximating fixed points of non-self nonexpansive mappings in Banach spaces

Nonlinear Analysis: Theory, Methods & Applications, 2005

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T : K → E be a nonexpansive non-self map with F (T) : ={x ∈ K : Tx = x} = ∅. Suppose {x n } is generated iteratively by x 1 ∈ K, x n+1 = P ((1 − n)x n + n TP[(1 − n)x n + n Tx n ]), n 1, where { n } and { n } are real sequences in [ , 1 − ] for some ∈ (0, 1). (1) If the dual E * of E has the Kadec-Klee property, then weak convergence of {x n } to some x * ∈ F (T) is proved; (2) If T satisfies condition (A), then strong convergence of {x n } to some x * ∈ F (T) is obtained.

Strong Convergence Theorem for Some Nonexpansive-Type Mappings in Certain Banach Spaces

Thai Journal of Mathematics, 2020

Let E be a uniformly convex and uniformly smooth real Banach space with dual space E^*. A new class of relatively J-nonexpansive maps, T : E → E^* is introduced and studied. A strong convergence theorem for approximating a common J-fixed point of a countable family of relatively Jnonexpansive maps is proved. An example of a countable family of relatively J-nonexpansive maps with a non-empty common J-fixed point is constructed. Finally, a numerical example is presented to show that our algorithm is implementable.

APPROXIMATION OF FIXED POINTS OF ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPPINGS IN BANACH SPACES

Let T be an asymptotically pseudocontractive self-mapping of a nonempty closed convex subset D of a reflexive Banach space X with a Gâteaux differentiable norm. We deal with the problem of strong convergence of almost fixed points xn = µnT n xn + (1 − µn)u to fixed point of T . Next, this result is applied to deal with the strong convergence of explicit iteration process

Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces

Journal of Mathematical Analysis and Applications, 2007

Let E be a real uniformly convex Banach space, K be a closed convex nonempty subset of E which is also a nonexpansive retract with retraction P. Let T 1 , T 2 ,. .. , T m : K → E be asymptotically nonexpansive mappings of K into E with sequences (respectively) {k in } ∞ n=1 satisfying k in → 1 as n → ∞, i = 1, 2,. .. , m, and ∞ n=1 (k in − 1) < ∞. Let {α in } ∞ n=1 be a sequence in [ , 1 − ], ∈ (0, 1), for each i ∈ {1, 2,. .. , m} (respectively). Let {x n } be a sequence generated for m 2 by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x 1 ∈ K, x n+1 = P [(1 − α 1n)x n + α 1n T 1 (P T 1) n−1 y n+m−2 ], y n+m−2 = P [(1 − α 2n)x n + α 2n T 2 (P T 2) n−1 y n+m−3 ],. .. y n = P [(1 − α mn)x n + α mn T m (P T m) n−1 x n ], n 1. Let m i=1 F (T i) = ∅. Strong and weak convergence of the sequence {x n } to a common fixed point of the family {T i } m i=1 are proved. Furthermore, if T 1 , T 2 ,. .. , T m are nonexpansive mappings and the dual E * of E satisfies the Kadec-Klee property, weak convergence theorem is also proved.

Fixed Point Theorem for Weakly Inward Nonself Asymptotically Nonexpansive Mappings in Banach Spaces

2015

In this paper, we established some weak and strong convergence theorems for common fixed points of three nonself asymptotically Banach spaces. Our results extended and improve the result announed by Wang[6] [Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl., 323(2006)550-557.] and WeiQiDeng, Lin Wang and Yi-Juan Chen[13] [Strong and Weak Convergence Theorems for common fixed points of two asymptotically nonexpansive mappings in Banach spaces, International Mathematical Forum, Vol. 7, 2012, no. 9, 407 – 417.] For a smooth banach space E, let us assume that K is a nonempty closed convex subset of with P as a sunny nonexpansive retraction. Let,T1, T2, T3 : K → E be three weakly inward nonself asymptotically nonexpansive mappings with respect to P with three sequences kn i ∁ [1,∞) satisfying (kn i ∞ n=1 −1) < ∞ ,(i=1,2,3) and F(T1)∩ F(T2)∩ F T3 = xεk, T1x = T2x = T3x = x respectively . For any given x1 ∈...