Strong convergence theorems for multivalued $\alpha$-demicontractive and $\alpha$-hemicontractive mappings
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DOI:
https://doi.org/10.26637/MJM0701/0001Abstract
In this paper, we introduce multivalued α -demicontractive and α -hemicontractive mappings and prove strong convergence theorems using Mann and Ishikawa iteration process in Hilbert spaces. We present some numerical examples which emphasize the results proved in the paper. Our theorem and corollaries extend the results of Isiogugu et al. [8] and Chidume et al. [4] in the setting of more general class of multivalued mappings.
Keywords:
Multivalued $$\alpha -demicontractive, Multivalued $\alpha$-hemicontractive, Fixed point, Mann iteration, Ishikawa iteration, Strong convergence, Hilbert spaceMathematics Subject Classification:
Mathematics- Pages: 1-6
- Date Published: 01-01-2019
- Vol. 7 No. 01 (2019): Malaya Journal of Matematik (MJM)
F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, $J$. Math. Anal. Appl, 20(1967), 197-228.
C. E. Chidume and S. A. Mutangadura, An example of the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc, 129(2001), 2359-2363.
C. E. Chidume and S. Maruster, Iterative methods for the computation of fixed points of demicontractive mappings, J. Comput. Appl. Math., 234(3)(2010), 861-882.
C. E. Chidume and M. E. Okpala, On a general class of multivalued strictly pseudocontractive mapping, Journal of Nonlinear Analysis and Optimization, Theory and Applications, 5(2)(2014), 123-134.
T. L. Hicks and J. D. Kubicek, On the Mann iteration process in a Hilbert space, J. Math. Anal. Appl., 59(1977), $498-504$
S. Ishikawa, Fixed points by a new iteration, Proc. Amer. Math. Soc., 44(1974), 147-150.
F. O. Isiogugu, Demiclosedness principle and approximation theorems for certain classes of multivalued mappings in Hilbert spaces, Fixed Point Theory and Applications, $61(2013)$.
F. O. Isiogugu and M.O. Osilike, Convergence theorems for new classes of multivalued hemicontractive-type mappings, Fixed Point Theory and Applications, 93 (2014).
W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4(1953), 506-510.
J. T. Markin, A fixed point theorem for set valued mappings, Bull. Amer. Math. Soc., 74(1968), 639-640.
S. Maruster, The solution by iteration of nonlinear equations in Hilbert spaces, Proc. Amer. Math. Soc., $63(1)(1977), 767-773$.
L. Maruster and S. Maruster, Strong convergence of the Mann iteration for $alpha$-demicontractive mappings, Mathematical and Computer Modelling, 54(2011), 2486-2492.
S. B. Nadler Jr, Multivalued contraction mappings, Pac. J. Math., 30(1969), 475-488.
S. A. Naimpally and K.L. Singh, Extensions of some fixed point theorems of Rhoades, J. Math. Anal. Appl., 96(1983), 437-446.
M. O. Osilike and A.C. Onah, Strong convergence of the Ishikawa iteration for Lipschitz $alpha$-hemicontractive mappings, Analele Universitatii de Vest, Timisoara Seria Matematica Informatica LIII, 1(2015), 151-161.
A. Rafiq, On the Mann iteration in Hilbert spaces, Nonlinear Analysis, 66(2007), 2230-2236.
Y. Song and Y.J. Cho, Some notes on Ishikawa iteration for multivalued mappings, Bull. Korean Math. Soc., 48(3)(2011), 575-584.
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