# Mann type iterative methods for finding a common solution of split feasibility and fixed point problems

Mann type iterative methods for finding a common solution of split feasibility and fixed point... The purpose of this paper is to study and analyze three different kinds of Mann type iterative methods for finding a common element of the solution set Γ of the split feasibility problem and the set Fix(S) of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces. By combining Mann’s iterative method and the extragradient method, we first propose Mann type extragradient-like algorithm for finding an element of the set $${{{\rm Fix}}(S) \cap \Gamma}$$ ; moreover, we derive the weak convergence of the proposed algorithm under appropriate conditions. Second, we combine Mann’s iterative method and the viscosity approximation method to introduce Mann type viscosity algorithm for finding an element of the $${{{\rm Fix}}(S)\cap \Gamma}$$ ; moreover, we derive the strong convergence of the sequences generated by the proposed algorithm to an element of set $${{{\rm Fix}}(S) \cap \Gamma}$$ under mild conditions. Finally, by combining Mann’s iterative method and the relaxed CQ method, we introduce Mann type relaxed CQ algorithm for finding an element of the set $${{{\rm Fix}}(S)\cap \Gamma}$$ . We also establish a weak convergence result for the sequences generated by the proposed Mann type relaxed CQ algorithm under appropriate assumptions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Mann type iterative methods for finding a common solution of split feasibility and fixed point problems

, Volume 16 (3) – Mar 20, 2012
25 pages

/lp/springer_journal/mann-type-iterative-methods-for-finding-a-common-solution-of-split-eOnvBEKqOz
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Operator Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics; Potential Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-012-0174-8
Publisher site
See Article on Publisher Site

### Abstract

The purpose of this paper is to study and analyze three different kinds of Mann type iterative methods for finding a common element of the solution set Γ of the split feasibility problem and the set Fix(S) of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces. By combining Mann’s iterative method and the extragradient method, we first propose Mann type extragradient-like algorithm for finding an element of the set $${{{\rm Fix}}(S) \cap \Gamma}$$ ; moreover, we derive the weak convergence of the proposed algorithm under appropriate conditions. Second, we combine Mann’s iterative method and the viscosity approximation method to introduce Mann type viscosity algorithm for finding an element of the $${{{\rm Fix}}(S)\cap \Gamma}$$ ; moreover, we derive the strong convergence of the sequences generated by the proposed algorithm to an element of set $${{{\rm Fix}}(S) \cap \Gamma}$$ under mild conditions. Finally, by combining Mann’s iterative method and the relaxed CQ method, we introduce Mann type relaxed CQ algorithm for finding an element of the set $${{{\rm Fix}}(S)\cap \Gamma}$$ . We also establish a weak convergence result for the sequences generated by the proposed Mann type relaxed CQ algorithm under appropriate assumptions.

### Journal

PositivitySpringer Journals

Published: Mar 20, 2012

### References

• On relaxed viscosity iterative methods for variational inequalities in Banach spaces
Ceng, L.C.; Ansari, Q.H.; Yao, J.C.

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