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• W2129388920 abstract "Abstract In this paper, we introduce a new class of generalized strict pseudocontractions in a real Hilbert space, and we consider a three-step Ishikawa-type iteration method <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>y</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mi>y</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> for finding a common fixed point of a countable family <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>}</mml:mo> </mml:math> of uniformly Lipschitz generalized <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> -strict pseudocontractions. Under mild conditions imposed on the parameter sequences <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>}</mml:mo> </mml:math> , <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>}</mml:mo> </mml:math> and <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>}</mml:mo> </mml:math> , we prove the strong convergence of <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>}</mml:mo> </mml:math> to a common fixed point of a countable family <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>}</mml:mo> </mml:math> of uniformly Lipschitz generalized strict pseudocontractions. On the other hand, we also introduce three-step hybrid viscosity approximation method for finding a common fixed point of a countable family <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>}</mml:mo> </mml:math> of uniformly Lipschitz generalized <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> -strict pseudocontractions with <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> , i.e. , a countable family <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>}</mml:mo> </mml:math> of uniformly Lipschitz pseudocontractions. Under appropriate conditions we derive the strong convergence results for this method. The results presented in this paper improve and extend the corresponding results in the earlier and recent literature. MSC: 47H06, 47H09, 47J20, 47J30." @default.
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• W2129388920 date "2014-03-20" @default.
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• W2129388920 title "Strong convergence of three-step iteration methods for a countable family of generalized strict pseudocontractions in Hilbert spaces" @default.
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