### Inertial residual algorithm for fixed points of finite family of strictly pseudocontractive mappings in Banach spaces

Document Type : Research Paper

Authors

1 Department of Mathematical Sciences, Bayero University, Kano, Nigeria

2 Department of Science and Technology Education, Bayero University, Kano, Nigeria

3 Department of Mathematics, Federal College of Eduction, Katsina, Nigeria

4 Department of Mathematics, University of Eswatini, Private Bag 4, Kwaluseni, Eswatini

5 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 94, Pretoria 0204, South Africa

Abstract

In this paper, we introduce a new iteration method called inertial residual algorithm for finding a common fixed point of finite family of strictly pseudocontractive mappings in a real uniformly smooth Banach spaces. We also establish weak and strong convergence theorems for the scheme. Finally, we give numerical experiment to explain the proposed method. Our results generalize and improve many recent results in the literature.

Keywords

[1] R.P. Agarwal, D. O’Regan, and D.R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Topol. Fixed Point Theory Appl., Springer, New York, NY, USA, 2009.
[2] H. Attouch and J. Peypouquent, The rate of convergence of Nesterov’s accelarated forward - backward method
is actually faster than 1
k2 ., SIAMJ. Optim. 26 (2016), 1824–1834.[3] A. Beck and M. Teboulle, A fast iterative shrinkage thresholding algorithm for linear inverse problem, SIAMJ.
Imaging Sci. 2 (2009), 183–202.
[4] R. I. Bot, E. R. Csetnek and C. Hendrich, Inertial Douglas-Rachford splitting for monotone inclusion problems,
Appl. Math. Comput. 256 (2015), 472–487.
[5] H.H. Bauschke and P.L Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer,
Berlin, 2011.
[6] 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.
[7] S. Banach, Th¶eorie des operations lineires, Warsaw, 1932.
[8] S. Banach, Sur les op´erations dans les ensembles abstraits et leur application aux ´equations int´egrales, Fund.
Math. 3 (1922), 133–181.
[9] R. Cacciopoli, Un teorem generale sull’esistenza di elementi uniti in una trans-formazione funzionale, Rend.
Accad. Naz. Lincei 13 (1931), 498–502.
[10] L. C. Ceng, A. Petrusel, C.F. Wen, J.C. Yao, Inertial-like subgradient extragradient methods for variational
inequalities and fixed points of asymptotically nonexpansive and strictly pseudocontractive mappings, Math. 7
(2019) no. 9, Article Number: 860.
[11] L.C. Ceng, A. Petrusel, X. Qin, J.C. Yao, Two inertial subgradient extragradient algorithms for variational
inequalities with fixed-point constraints, Optim. 70 (2021), no. 5-6, 1337–1358.
[12] P. Chen, J. Huang and X. Zhang, A primal-dual fixed point algorithm for convex separable minimization with
application to image restoration, Inverse Probl. 29 (2013).
[13] W. L. Cruiz, A Residual Algorithm for Finding a Fixed Point of a Nonexpansive Mapping, Fixed Point Theory
and Appl., (2018), Doi: 10.1007/2018/05964.
[14] P. Cholamjiak and S. Suantai, Weak convergence theorems for a countable family of strict pseudocontractions
in Banach spaces, Fixed Point Theory Appl. 2010 (2010), 1–16
[15] P. Cholamjiak and S. Suantai, Strong convergence theorems for a countable family of strict pseudocontractions
in q−uniformly Banach spaces, Comput. Math. Appl. 62 (2011), no. 2, 787–796.
[16] P. Cholamjiak and S. Suantai, Weak and strong convergence theorems for a countable family of strict pseudo-contractions in Banach spaces, Optim. 62 (2013), no. 2, 255–270.
[17] 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.
[18] I. Cioranescu, Geometry of Banach spaces,duality mappings and nonlinear problems, Kluwer, Dordrecht, 1990.
[19] C.E. Chidume, Geometric properties of Banach spaces and nonlinear iterations, Springer, London, UK, 2009.
[20] Q.L. Dong, H.B. Yuan, C.Y. Je and Th.M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm
for nonexpansive mappings, Optim. Lett. 12 (2016), doi.org/10.1007/s11590-016-1102-9
[21] Q.L. Dong and H.B. Yuan, Accelerated Mann and CQ algorithms for finding a fixed point of a nonexpansive
mapping, Fixed Point Theory Appl. 125 (2015), Doi: 10.1186/s13663-015-0374-6
[22] Q.L. Dong, C.Y. Je and Th.M. Rassias, General inertial Mann algorithms and their convergence analysis for
nonexpansive mappings, Appl. Nonlinear Anal. 134 (2018), Doi:10.1007/978-3-319-89815-5-7.
[23] B. Halpern, Fixed points of nonexpansive maps, Bull. Am. Math. Soc., 73 (1967), 957-961.
[24] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer Math. Soc. 44 (1974), 147–150.
[25] M.H. Harbau, Inertial hybrid self-adaptive subgradient extragradient method for fixed point of quasi-ϕnonexpansive multivalued mappings and equilibrium problem, Adv. Theory Nonlinear Anal. Appl., 5 (2021), no.
4, 507–522.
[26] T. Kato, Nonlinear semigroups and Evolution equations, J. Mtah. Soc. Japan 19, (1967), 508–520.
[27] B. Leemon, Residual Algorithms: Reinforcement Learning with Function Approximatin, U.S. Air Force technical
Report, Department of Computer Science, U.S. Air Force Academy, CO 80840-6234, (1995).
[28] H. Liduka, Iterative algorithm for triple-hierarchical constrained nonconvex optimization problem and its application to network bandwidth allocation, SIAM J. Optim. 22 (2012), 862–878.
[29] H. Liduka, Fixed point optimization algorithms for distributed optimization in networked systems, SIAM J. Optim.
23 (2013), 1–26.
[30] D.A. Lorenz and T. Pock, An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging
Vis. 51 (2015), 311–325.
[31] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510.[32] P.E. Mange, Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math. 219 (2008), 223–
236.
[33] C.A. Micchelli, L. Shen and Y. Xu, Proximity algorithms for image models; denoising, Inverse Probl. 27 (2011),
no. 4.
[34] J. Olaleru and G. Okeke, Convergence theorems on asymptotically demicontractive and hemicontractive mappings
in the intermediate sense, Fixed Point Theory Appl. 2013 (2013), 352.
[35] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer.
Math. Soc. 73 (1967), 591–597.
[36] B.T. Polyak, Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math.
Phys. 4 (1964), no. 5, 1–17
[37] E. Picard, M´emoire sur la th´eorie des ´equations aux d´eriv´ees partielles et la m´ethode des approximations successives, J. Math. Pures Appl. 6 (1890), 145–210.
[38] K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration
process, J. Math. Anal. Appl. 178 (1993), 301–308.
[39] Y. Yao, G. Marino and L. Muglia, A modified Korpelevich’s method convergent to the minimum-norm solution
of a variational inequality, Optim. 63 (2012), no. 4, 1–11.
[40] S. Yekini, An iterative approximation of fixed points of strictly pseudocontracvtive mappings in Banach spaces,
MATEMATHYKN BECHNK 2 (2015), 79–91.
[41] H. Zhou, Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in
Banach spaces, Nonlinear Anal. 68 (2008), 2977–2983.
###### Volume 13, Issue 2July 2022Pages 2257-2269
• Receive Date: 30 April 2021
• Revise Date: 29 November 2021
• Accept Date: 03 December 2021