# Some convergence results for split common and fixed point problems in Hilbert space

## DOI:

https://doi.org/10.62918/hjma.v1i1.5## Keywords:

Fixed Point, Split Common Fixed Point, Non Linear Mappings, Weak and Strong Convergence, Hilbert Space## Abstract

In this survey article, we present an introduction of split feasibility problems, multisets split feasibility problems and split common fixed point problems. Parallel and cyclic algorithms for solving the split common fixed point problems for finite family of strictly pseudocontractive mappings in Hilbert spaces are presented, weak and strong convergence theorems are being proved. Application of the split common fixed point problems and some numerical examples are also presented.

## References

G. L. Acedo, H. K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Analysis: Theory, Methods and Applications, 67(7) (2007), 2258– 2271.

Q. H. Ansari, A. Rehan, Split feasibility and fixed point problems, Nonlinear analysis, (2014), 281–322.

A. Auwalu, L. B. Mohammed, and A. Saliu, Synchronal and cyclic algorithms for fixed point problems and variational inequality problems in Banach spaces, Fixed Point Theory and Applications, 2013(1) (2013), 1–24.

H. H. Bauschke, J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM review, 38(3) (1996), 367–426.

F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, Journal of Mathematical Analysis and Applications, 20(2) (1967), 197–228.

L. M. Bulama, A. Kılı¸cman, On synchronal algorithm for fixed point and variational inequality problems in hilbert spaces, SpringerPlus, 5(1) (2016), 1–13.

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse problems, 18(2) (2002), 441–453.

C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse problems, 20(1) (2003), 103–120.

Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numerical Algorithms, 8(2) (1994), 221–239.

Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21(6) (2005), 2071.

Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Physics in Medicine & Biology, 51(10) (2006), 2353-–2365.

Y. Censor, A. Segal, The split common fixed point problem for directed operators, J. Convex Anal, 16(2) (2009), 587–600.

P. L. Combettes, The convex feasibility problem in image recovery, Advances in imaging and electron physics, 95(25) (1996), 155–270.

Q. L.Dong, H. B. Yuan, Y. J. Cho, and T. M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings, Optimization Letters, 12(1) (2018), 87–102.

R. Kraikaew, S. Saejung, On split common fixed point problems, Journal of Mathematical Analysis and Applications, 415(2) (2014), 513–524.

G. Marino, H. K. Xu, Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces, Journal of Mathematical Analysis and Applications, 329(1) (2007), 336–346.

L. B. Mohammed, Strong convergence of an algorithm about quasi-nonexpansive mappings for the split common fixed-pint problem in Hilbert space, International Journal of Innovative Research and Studies, 2(8) (2013), 298–306.

L. B. Mohammed, A Note On The Splitcommon Fixed-Point Problem For strongly Quasi-Nonexpansive Operator In Hilbert Space, International Journal of Innovative Research and Studies, 2(8) (2013), 424-–434.

L. B. Mohammed, A. Auwalu, and S. Afis, Strong convergence for the split feasibility problem in real Hilbert Space, Math. Theo. Model, 3(7) (2013), 2224–5804.

L. B. Mohammed, A. Kılı¸cman, Strong convergence for the split common fixed-point problem for total quasi-asymptotically nonexpansive mappings in Hilbert space, In Abstract and Applied Analysis (Vol. 2015),(2015, January), Hindawi.

A. Moudafi, A note on the split common fixed-point problem for quasi-nonexpansive operators, J. Nonlinear Anal., 74 (2011), 4083-–4087.

A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Analysis: Theory, Methods & Applications, 79 (2013), 117–121.

B. Qu, N. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, 21(5) (2005), 1655.

H. Stark, (Ed.), Image recovery: theory and application, Acadamic Press Ornaldo, Fla, USA, (1987).

X. R. Wang, S. S. Chang, L. Wang, and Y. H. Zhao, Split feasibility problems for total quasi-asymptotically nonexpansive mappings, Fixed Point Theory and Applications, 2012(1) (2012), 1–11.

F. Wang, H. K. Xu, Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Analysis: Theory, Methods & Applications, 74(12) (2011), 4105–4111.

H. K. Xu, Iterative algorithms for nonlinear operators, Journal of the London Mathematical Society, 66(01) (2002), 240–256.

H. Zhou, Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert spaces, Nonlinear Analysis: Theory, Methods & Applications, 69(2) (2008), 456–462.

H. Zhou, Convergence theorems for λ−strict pseudo-contraction in 2-uniformly smooth Banach spaces, Nonlinear Anal., 69 (2008), 3160.

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*Hilbert Journal of Mathematical Analysis*,

*1*(1), 30–58. https://doi.org/10.62918/hjma.v1i1.5

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