IIARD International Institute of Academic Research and Development

INTERNATIONAL JOURNAL OF APPLIED SCIENCES AND MATHEMATICAL THEORY (IJASMT )

E- ISSN 2489-009X
P- ISSN 2695-1908
VOL. 8 NO. 3 2022
DOI: https://doi.org/10.56201/ijasmt.v8.no3.2022.pg1.15

---

A Modified Inertial Iterative Algorithms for Solving Split Common Fixed Point Problems in real Hilbert Spaces

F.O. Nwawuru and B. E. Chukwuemeka

---

Abstract

We introduce and study an inertial-based iterative algorithm for solving split common fixed point problem involving a certain class of nonlinear mapping in real Hilbert spaces. Under some mild assumptions, we obtain a strong convergence result of the proposed algorithm. Our result improves and extends newly announced results in this area.

keywords:

Metric projection; nonexpansive mappings; parallel algorithm; split common fixed point problem.

---

References:

[1] Censor, Y., Elfving, T. (1994). A multi projection algorithm using Bregman projections in a product space.
Numer. Algorithm. 8(2):221–239. DOI:
10.1007/BF02142692.

Byrne, C. (2002). Iterative oblique projection onto convex sets and the split feasibility
problem. Inverse Probl. 18(2):441–453. DOI: 10.1088/0266-5611/18/2/310.

Byrne, C. (2004). A unified treatment of some iterative algorithms in signal processing
and image reconstruction. Inverse Probl. 20(1):103–120. DOI: 10.1088/0266-
5611/20/1/006.

Censor, Y., Elfving, T., Kopf, N., Bortfeld, T. (2005). The multiple-sets split feasibility
problem and its application. Inverse Probl. 21(6):2071–2084. DOI: 10.1088/0266-
5611/21/6/017.

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

Censor, Y., Gibali, A., Reich, R. (2012). Algorithms for the split variational
inequality problem. Numer. Algorithm. 59(2):301–323. DOI: 10.1007/s11075-011-9490-5.

Xu, H. K. (2006). A variable Krasnosel’skii-Mann algorithm and the multiple-set split
feasibility problem. Inverse Probl. 22(6):2021–2034. DOI: 10.1088/0266-5611/22/6/007.

Xu, H. K. (2010). Iterative methods for the split feasibility problem in infinite
dimensional Hilbert spaces. Inverse Probl. 26(10):105018. DOI: 10.1088/0266-5611/
26/10/105018.

Masad, E., Reich, S. (2007). A note on the multiple-set split convex feasibility problem in
Hilbert space. J. Nonlinear Convex Anal. 8:367–371.

Shehu, Y., Iyiola, O. S., Enyi, C. D. (2016). An iterative algorithm for solving split
feasibility problems and fixed point problems in Banach spaces. Numer. Algorithm.
72(4):835–864. DOI: 10.1007/s11075-015-0069-4.

Eicke, B. (1992). Iteration methods for convexly constrained ill-posed problems in
Hilbert space. Numer. Funct. Anal. Optim. 13:423–429.

Cegielski, A. (2015). General method for solving the split common fixed point problem. J.
Optim. Theory Appl. 165(2):385–404. DOI: 10.1007/s10957-014-0662-z.

Cegielski, A., Al-Musallam, F. (2016). Strong convergence of a hybrid steepest descent
method for the split common fixed point problem. Optimization 65(7):
1463–1476. DOI: 10.1080/02331934.2016.1147038.

Eslamian, M., Eslamian, P. (2016). Strong convergence of a split common fixed
point problem. Numer. Funct. Anal. Optim. 37(10):1248–1266. DOI: 10.1080/
01630563.2016.1200076.

Kraikaew, R., Saejung, S. (2014). On split common fixed point problems. J. Math.
Anal. Appl. 415(2):513–524. DOI: 10.1016/j.jmaa.2014.01.068.

Shehu, Y. (2015). New convergence theorems for split common fixed point problems in
Hilbert spaces. J. Nonlinear Convex Anal. 16:167–181.

Shehu, Y., Cholamjiak, P. (2016). Another look at the split common fixed point
problem for demicontractive operators. Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A
Math. 110(1):201–218. DOI: 10.1007/s13398-015-0231-9.

Tang, Y. C., Liu, L. W. (2016). Several iterative algorithms for solving the split common
fixed point problem of directed operators with applications. Optimization
65(1):53–65. DOI: 10.1080/02331934.2014.984708.

Padcharoen, A., Kumam, P., Cho, Y. J. (2019). Split common fixed point problems
for demicontractive operators. Numer. Algorithm. 82(1):297–320. DOI: 10.1007/
s11075-018-0605-0.

Reich, S., Tuyen, T. M. (2019). A new algorithm for solving the split common null
point problem in Hilbert spaces. Numer. Algorithm. https://doi.org/10.1007/s11075-019-00703-z

Reich, S., Tuyen, T. M. (2019). Iterative methods for solving the generalized split
common null point problem in Hilbert spaces. Optimization:1–26. DOI: 10.1080/
02331934.2019.1655562.

Reich, S., Tuyen, T. M., & Trang, N. M. (2019). Parallel Iterative Methods for Solving the
Split Common Fixed Point Problem in Hilbert Spaces. Numerical Functional Analysis and Optimization, 1–28. doi:10.1080/01630563.2019.1681000

Moudafi, A. (2000). Viscosity approximation methods for fixed-points problems. J.
Math. Anal. Appl. 241(1):46–55. DOI: 10.1006/jmaa.1999.6615
Halpern, B. (1967). Fixed points of nonexpanding maps. Bull. Amer. Math. Soc.
73(6):597–957. DOI: 10.1090/S0002-9904-1967-11864-0.

G. 1. MINi'Y, Monotone (nonlinear) operators in Hilbert space, Duke Math. 1., 29 (1962), pp.341-346.

Bauschke, H. H., Combettes, P. L. (2011). Convex Analysis and Monotone Operator
Theory in Hilbert Spaces. New York: Springer.

Goebel, K., Kirk, W. A. (1990). Topics in Metric Fixed Point Theory. Cambridge
Stud. Adv. Math 28. Cambridge: Cambridge University Press.

Mainge, P. E. (2008). Strong convergence of projected subgradient methods for
nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16(7-8):899–912.
DOI: 10.1007/s11228-008-0102-z.

Xu, H. K. (2006). Strong convergence of an iterative method for nonexpansive and
accretive operators. J. Math. Anal. Appl. 314(2):631–643. DOI: 10.1016/j.jmaa.2005.


Hicks TL, Kubicek JD (1977) On the Mann iteration process in a Hilbert space, J. Math.
Anal. Appl. 59 pp. 498{504.
Patrick, L: Combettes, Jean-Christophe Pesquet: Deep Neural Network Structures Solving
Variational Inequalities. Optimization and Control (math.OC). arXiv:1808.07526
[math.OC] (or arXiv:1808.07526v2 [math.OC] for this version) (2019)
https://doi.org/10.48550/arXiv.1808.07526.

Heaton, H, Wu Fung, S, Gibali, A. et al: Feasibility-based fixed point networks. Fixed
Point Theory Algorithms Sci Eng 2021, 21 (2021). https://doi.org/10.1186/s13663-021-
00706-3.

Combettes. PL, Pesquet,JC, "Fixed Point Strategies in Data Science," in IEEE
Transactions on Signal Processing, vol. 69, pp. 3878-3905, 2021, https://doi:
10.1109/TSP.2021.3069677
Jung, A: A Fixed-Point of View on Gradient Methods for Big Data. Frontiers in Applied
Mathematics and Statistics, 3. doi:10.3389/fams.2017.00018.
Censor Y, Gibali A, Reich S. Algorithms for the split variational inequality problem.
Numer Algorithms. 2012;59:301–323.

Gibali A. A new split inverse problem and application to least intensity feasible solutions.
Pure Appl Funct Anal. 2017;2(2):243–258.

Polyak BT (1964) Some methods of speeding up the convergence of iteration methods.
Comput. Math. Math. Phys., 4, 1(17)

Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear
inverse problems. SIAM J. Imaging Sci. 2, 183{202.

Nesterov Y (1983) A method of solving a convex programming problem with convergence
rate O(1/k2). Soviet Math. Doklady 27, 372{376.

Qin X, Wang L, Yao JC (2020)
Inertial splitting method for maximal monotone
mappings. J. Nonlinear Convex Anal. in press.

Luo YL, Tan B (2020) A self-adaptive inertial extragradient algorithm for solving
pseudo-monotone variational inequality in Hilbert spaces. J. Nonlinear Convex Anal. in pres.

DOWNLOAD PDF

---

Back