On a Convergence Theorem for the General Noor Iteration Process in Uniformly Smooth Banach Spaces

Alfred Olufemi Bosede, Bakre Omolara Fatimah, Ashiribo Senapon Wusu


In this paper, two different classes of mappings namely, uniformly continuous asymptotically nonexpansive and uniformly continuous asymptotically demicontractive mappings are considered on the general modified Noor iteration process with errors and proved to converge strongly to the fixed point of uniformly continuous asymptotically demicontractive mappings in uniformly smooth Banach spaces. The new result can be viewed as an improvement to a multitude of results in the fixed point theory especially those of Xu and Noor [8], Owojori and Imoru [5] and also the results of Owojori [6].


47H10; 54H25

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C. E. Chidume and M. O. Osilike, Fixed point iterations for strictly hemicontractive maps in uniformly smooth Banach spaces, Numerical Functional Analysis and Optimization 15(7-8) (1994), 779 – 790, DOI: 10.1080/01630569408816593.

K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proceedings of American Mathematical Society 35 (1972), 171 – 174, DOI: 10.2307/2038462.

K. Goebel, W. Kirk and T. Shimi, A fixed point theorem in uniformly convex spaces, Bollettino dell’Unione Matematica Italiana 4(7) (1973), 67 – 75.

M. O. Osilike, Some stability results for fixed point iteration procedures, Journal of Nigerian Mathematical Society 14/15 (1995), 17 – 29.

O. O. Owojori and C. O. Imoru, On a general Ishikawa fixed point iteration process for continuous hemicontractive maps in Hilbert spaces, Advanced Studies in Contemporary Mathematics 4(1) (2001), 1 – 15, DOI: /URL missing.

O. O. Owojori and C. O. Imoru, On generalized fixed point iterations for asymptotically nonexpansive maps in arbitrary Banach spaces, Proceedings Jangjeon Mathematical Society 6(1) (2003), 49 – 58, DOI: /URL missing.

X. L. Weng, Fixed point iteration for local strictly pseudocontractive mapping, Proceedings of the American Mathematical Society 113 (1991), 727 – 731, DOI: 10.1090/S0002-9939-1991-1086345-8.

B. Xu and M. A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications 267 (2002), 444 – 453, DOI: 10.1006/jmaa.2001.7649.

H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Analysis: Theory, Methods & Applications 16(20) (1991), 1127 – 1138, DOI: 10.1016/0362-546X(91)90200-K.

DOI: http://dx.doi.org/10.26713%2Fjims.v12i2.1333

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