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Convergence Theorems for Two Asymptotically Nonexpansive Non-self Mappings in Uniformly Convex Banach Spaces


Affiliations
1 Department of Mathematics and I. T., Govt. N. P. G. College of Science, Raipur (C. G.), India
     

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Let K be a nonempty closed convex non expansive retract of a uniformly convex Banach space E with P as a non expansive retraction. Let T1, T2: K → E be two asymptotically non expansive non-self mappings with sequences {kn }, {hn } ⊂[1,(∞) such that Σ∞n=1(kn hn -1) < ∞ and F = F(T1) ∩ F(T2) = {x E K : T1x = T2x = x}≠ Φ . Let {xn}∞n=1 be the sequence generated iteratively by xl ∈ K and xn+1 = P(anxn + bnT1 (PT1 )n-1yn + cnln ) ∀n ≥1 yn = P(ān xn + bn Tn (PT2 )n-1xn + cnmn ),∀n ≥1 where {ln }, {mn} are bounded sequences, an+bn +cn = 1 = ān +bn +cn,0 ≤ an +bn +cn, ān +bn +cn ≤ 1, ∀n ∈ N, Σ∞n=1 cn < ∞ and Σ∞n=1 bncn <∞ . If T1 is completely continuous or T1 and T2 satisfy condition (A'), then {xn} converges strongly to a point in F = F(T1) ∩ F(T2). Also if E satisfies Opial's condition or the dual E* of E has the Kedec-Klee property, then {xn} converges weakly to a point in F.

Keywords

Asymptotically Non Expansive Nonself Mappings, Common Fixed Point, the Modified Ishikawa Iterative Sequence with Errors for Non-Self Maps, Uniformly Convex Banach Space, Strong Convergence, Weak Convergence.
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  • Convergence Theorems for Two Asymptotically Nonexpansive Non-self Mappings in Uniformly Convex Banach Spaces

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Authors

G. S. Saluja
Department of Mathematics and I. T., Govt. N. P. G. College of Science, Raipur (C. G.), India

Abstract


Let K be a nonempty closed convex non expansive retract of a uniformly convex Banach space E with P as a non expansive retraction. Let T1, T2: K → E be two asymptotically non expansive non-self mappings with sequences {kn }, {hn } ⊂[1,(∞) such that Σ∞n=1(kn hn -1) < ∞ and F = F(T1) ∩ F(T2) = {x E K : T1x = T2x = x}≠ Φ . Let {xn}∞n=1 be the sequence generated iteratively by xl ∈ K and xn+1 = P(anxn + bnT1 (PT1 )n-1yn + cnln ) ∀n ≥1 yn = P(ān xn + bn Tn (PT2 )n-1xn + cnmn ),∀n ≥1 where {ln }, {mn} are bounded sequences, an+bn +cn = 1 = ān +bn +cn,0 ≤ an +bn +cn, ān +bn +cn ≤ 1, ∀n ∈ N, Σ∞n=1 cn < ∞ and Σ∞n=1 bncn <∞ . If T1 is completely continuous or T1 and T2 satisfy condition (A'), then {xn} converges strongly to a point in F = F(T1) ∩ F(T2). Also if E satisfies Opial's condition or the dual E* of E has the Kedec-Klee property, then {xn} converges weakly to a point in F.

Keywords


Asymptotically Non Expansive Nonself Mappings, Common Fixed Point, the Modified Ishikawa Iterative Sequence with Errors for Non-Self Maps, Uniformly Convex Banach Space, Strong Convergence, Weak Convergence.

References