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ON THE CONVERGENCE OF THE ISHIKAWA ITERATES TO A COMMON FIXED POINT FOR A PAIR OF MAPPINGS

ON THE CONVERGENCE OF THE ISHIKAWA ITERATES TO A COMMON FIXED POINT FOR A PAIR OF MAPPINGS DEMONSTRATIO MATHEMATICAVol. XXVIIINo 21995R. A. Rash wanON THE CONVERGENCE OF THE ISHIKAWA ITERATESTO A COMMON FIXED POINT FOR A PAIR OF MAPPINGSIn [5], [6] and [7] it has shown that for a mapping T satisfying certainconditions, if the sequence of Mann iterates converges, then it converges toa fixed point of T. In [4], the author has shown that, for a pair of mappings5 , T satisfying some contractive conditions, if the sequence of Mann iteratesassociated with 5 or T is convergent, then its limit point is a common fixedpoint of S and T.In the present paper we extend the results of [4] for a pair of mappingsS, T in a normed space, using Ishikawa iterates process.Let 5 , T be self-mappings on a Banach space X. We shall consider theIshikawa iterates {a;n} [3] associated with S as:(1) z n + i=(1- an)xn+anSyn,yn= (1 -/3n)xn+PnSxn,n >0,where {«„}, {/?„} satisfyoo0 5: an < fin < 1 for all n, lim 0 n = 0 andn—»oo'a„/3„ = oo.n=lWe shall use the conditions(i) o < « „ , / ? „ < 1 ,(ii) limn^oo p n = 0,(iii) l i m ^ o o a n = h, 0 < h < http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

ON THE CONVERGENCE OF THE ISHIKAWA ITERATES TO A COMMON FIXED POINT FOR A PAIR OF MAPPINGS

Rashwan, R. A.
Demonstratio Mathematica , Volume 28 (2): 4 – Apr 1, 1995
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Publisher
de Gruyter
Copyright
© by R. A. Rashwan
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1995-0206
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XXVIIINo 21995R. A. Rash wanON THE CONVERGENCE OF THE ISHIKAWA ITERATESTO A COMMON FIXED POINT FOR A PAIR OF MAPPINGSIn [5], [6] and [7] it has shown that for a mapping T satisfying certainconditions, if the sequence of Mann iterates converges, then it converges toa fixed point of T. In [4], the author has shown that, for a pair of mappings5 , T satisfying some contractive conditions, if the sequence of Mann iteratesassociated with 5 or T is convergent, then its limit point is a common fixedpoint of S and T.In the present paper we extend the results of [4] for a pair of mappingsS, T in a normed space, using Ishikawa iterates process.Let 5 , T be self-mappings on a Banach space X. We shall consider theIshikawa iterates {a;n} [3] associated with S as:(1) z n + i=(1- an)xn+anSyn,yn= (1 -/3n)xn+PnSxn,n >0,where {«„}, {/?„} satisfyoo0 5: an < fin < 1 for all n, lim 0 n = 0 andn—»oo'a„/3„ = oo.n=lWe shall use the conditions(i) o < « „ , / ? „ < 1 ,(ii) limn^oo p n = 0,(iii) l i m ^ o o a n = h, 0 < h <

Journal

Demonstratio Mathematica – de Gruyter

Published: Apr 1, 1995

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