IJPAM: Volume 66, No. 1 (2011)

CONVERGENCE AND STABILITY OF THE ISHIKAWA
ITERATION SCHEMES FOR QUASI-CONTRACTIVE
MAPPINGS IN UNIFORMLY CONVEX BANACH SPACES

Zeqing Liu$^1$, Jeong Sheok Ume$^2$
$^1$Department of Mathematics
Liaoning Normal University
Dalian, Liaoning, 116029, P.R. CHINA
e-mail: [email protected]
$^{2}$Department of Applied Mathematics
Changwon National University
Changwon, 641-773, KOREA
e-mail: [email protected]


Abstract.It is proved that certain Ishikawa iteration schemes with errors introduced by Xu (see J. Math. Anal. Appl., 224 (1998), 91-101) can be used to approximate fixed points of quasi-contractive mappings in $p$-uniformly convex Banach spaces. A related result deals with the stability of the Ishikawa iteration scheme for quasi-contractive mappings in $p$-uniformly convex Banach spaces. Our results not only resolve affirmatively two open questions put forth by Rhoades [17] and Naimpally and Singh [13] and Osilike [16], respectively, in the more general setting, but also extend, improve and unify the corresponding results obtained by Chidume [4], Liu [14], [15], Osilike [16] and others.

Received: July 16, 2010

AMS Subject Classification: 47H10, 40A05

Key Words and Phrases: quasi-contractive mapping, fixed point, stability, Ishikawa iteration sequence with errors, Ishikawa iteration sequence, closed convex bounded subset, $p$-uniformly convex Banach space

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2011
Volume: 66
Issue: 1