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Convergence theorems for multivalued Φ-hemicontractive operators and Φ-strongly accretive operators

Authors
Journal
Computers & Mathematics with Applications
0898-1221
Publisher
Elsevier
Publication Date
Volume
46
Identifiers
DOI: 10.1016/s0898-1221(03)90183-0
Keywords
  • Multivalued φ-Hemicontraction
  • φ-Strongly Accretive Operator
  • Xu'S Ishikawa Iteration
  • Real Uniformly Smooth Banach Spaces

Abstract

Abstract Suppose E is a uniformly smooth Banach space and T : E → 2 E is a multivalued φ-hemicontractive operator with bounded range. Suppose { a n }, { b n }, { c n } and { a n ′}, { b n ′} { c n ′} are real sequences in [0, 1] satisfying the following conditions: (i) a n + b n + c n = a ′ n + b ′ n + c ′ n = 1, for all n ϵ N ; (ii) lim n→∞ b n = lim n→∞b ′ n = lim n→∞ c n = 0 ; (iii) ∑ ∞ n=1 b n = ∞ ; (iv) c n = o(b n) . For arbitrary x i , u 1, v 1 ϵ E, define the sequence {x n} n=1 ∞ by x n+1 = a n x n + b n η n + c n u n , ∃ η n ϵ Ty n, n ϵ N ; y n = a′ nx n + b′ nξ n + c′ nv n ∃ξ n ϵ Tx n, n ϵ N , where { u n } n=1 ∞ {v n} n=1 ∞ are arbitrary bounded sequences in E. Then { x n} n=1 ∞ converges strongly to the unique fixed point of T. Related results deal with the iterative solutions of nonlinear multivalued φ-accretive operator equation f ϵ Tx.

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