Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space

By Eric Uwadiegwu Ofoedu. 2006

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Abstract Let E be a real Banach space. Let K be a nonempty closed and convex subset of E , T : K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence { k n } n ⩾ 0 ⊂ [ 1 , + ∞ ) , lim n → ∞ k n = 1 such that F ( T ) ≠ ∅ . Let { α n } n ⩾ 0 ⊂ [ 0 , 1 ] be such that ∑ n ⩾ 0 α n = ∞ , ∑ n ⩾ 0 α n 2 ∞ and ∑ n ⩾ 0 α n ( k n − 1 ) ∞ . Suppose { x n } n ⩾ 0 is iteratively defined by x n + 1 = ( 1 − α n ) x n + α n T n x n , n ⩾ 0 , and suppose there exists a strictly increasing continuous function ϕ : [ 0 , + ∞ ) → [ 0 , + ∞ ) , ϕ ( 0 ) = 0 such that 〈 T n x − x ∗ , j ( x − x ∗ ) 〉 ⩽ k n ‖ x − x ∗ ‖ 2 − ϕ ( ‖ x − x ∗ ‖ ) , ∀ x ∈ K . It is proved that { x n } n ⩾ 0 converges strongly to x ∗ ∈ F ( T ) . It is also proved that the sequence of iteration { x n } defined by x n + 1 = a n x n + b n T n x n + c n u n , n ⩾ 0 (where { u n } n ⩾ 0 is a bounded sequence in K and { a n } n ⩾ 0 , { b n } n ⩾ 0 , { c n } n ⩾ 0 are sequences in [ 0 , 1 ] satisfying appropriate conditions) converges strongly to a fixed point of T .

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

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