Document Type : Research Paper


1 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa.

2 DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa.

3 Department of Computer Sciences and Mathematics, Mountain Top University, Prayer City, Ogun State, Nigeira.


In this paper, we introduce a new type of modified generalized $\alpha$-nonexpansive mapping and establish some fixed point properties and demiclosedness principle for this class of mappings in the framework of  uniformly convex Banach spaces. We further  propose a new iterative method for approximating a common fixed point of two modified generalized $\alpha$-nonexpansive mappings and present some weak and strong convergence theorems for these mappings in uniformly convex Banach spaces. In addition, we apply our result to solve a  convex-constrained minimization problem, variational inequality and split feasibility problem and present some numerical experiments in infinite dimensional spaces to establish the applicability and efficiency of our proposed algorithm. The obtained results in this paper improve and extend   some related results in the literature.


[1] H.A. Abass, A.A. Mebawondu and O.T. Mewomo, Some results for a new three iteration scheme in Banach spaces, Bull. Transilv. Univ. Brașov, Ser. III, Math. Inform. Phys., 11 (2), pp. 1-18.
[2] M. Abass and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesn., 66 (2) (2014), pp. 223-234.
[3] M. Abbas, Z. Kadelburg and D.R. Sahu, Fixed point theorems for Lipschitzian type mappings in CAT(0) spaces, Math. Comput. Modelling, 55 (3-4) (2012), pp. 1418-1427.
[4] R.P. Agarwal, D. O'Regan and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Convex Anal., 8 (1) (2007), pp. 61-79.
[5] V. Berinde, On the approximation of fixed points of weak contractive mappings, Carpathian J. Math., 19 (2003), pp. 7-22.
[6] V. Berinde, Picard iteration converges faster than Mann iteration for a class of quasicontractive operators, Fixed Point Theory Appl., 2 (2004), pp. 97-105.
[7] F.E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. USA., 54 (1965), pp. 1041-1044.
[8] C. Byrne , Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl., 18 (2) (2002), pp. 441–453.
[9] Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (2) (1994), pp. 221-239.
[10] Y. Censor, X.A. Motova and A. Segal, Perturbed projections and subgradient projections for the multiple-set split feasibility problem, J Math. Anal. Appl., 327 (2007), pp. 1224-1256.
[11] Y. Censor, T. Elfving, N. Kopt and T. Bortfeld, The multiple-sets split feasibility problem and its applications, Inverse Probl., 21 (2005), pp. 2071-2084.
[12] P. Cholamijak and S. Suantai, Iterative variational inequalities and fixed point problem of nonexpansive semigroups, J. Glob. Optim., 57 (2013), pp. 1277-1297.
[13] M. Ertürk, F. Gürsoy, and N. Şimşek, S-iterative algorithm for solving variational inequalities, Int. J. Comput. Math., 98 (3) (2021), pp. 435-448.
[14] M. Ertürk, F. Gürsoy, Q. Ansari and V. Karakaya, Picard type iterative method with application to minimization problems and split feasibility problems, J. Nonlinear Convex Anal., 21 (4) (2020), pp. 1-20.
[15] P. Chuadchawna, A. Farajzadeh and A. Kaewcharoen, Fixed-point approximation of generalized nonexpansive mappings via generalized $M$-iteration in hyperbolic spaces, International Journal of Mathematical Science, (2020), ID 6435043, pp. 1-8.
[16] M. Feng, L. Shi and R. Chen, A new three-step iterative algorithm for solving the split feasibility problem, Univ. Politeh. Buch. Ser. A., 81 (1) (2019), pp. 93-102.
[17] F. Gianness, Vector variational inequalities and vector equilibria, Mathematical Theories, 38, Kluwer Academic publisher, Dordrecht, (2000).
[18] F. Gursou and V. Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, arXiv:1403.2546v2, (2014), pp 1-20.
[19] F. Gursou, M. Ertürk, and M. Abbas Picard-type iterative algorithm for general variational inequalities and nonexpansive mappings, Numer. Algorithms, 83 (2020), pp. 867-883. 
[20] S. Ishikawa, Fixed points by new iteration method, Proc. Am.
Math. Soc., 149 (1974), pp. 147-150.
[21] N. Kadioglu and I. Yildirim, Approximating fixed points of nonexpansive mappings by faster iteration process, arXiv:1402. 6530v1 [math.FA], (2014), pp. 1-20.
[22] V. Karakaya, K. Dogan, F. Gursoy and M. Erturk Fixed point of a new three step iteration algorithm under contractive like operators over normed space, Abstr. Appl. Anal., 2013, Article ID 560258, pp. 1-25.
[23] V. Karakaya, Y. Atalan and K. Dogan, On fixed point result for a three steps iteration process in Banach space, Fixed Point Theory, 18 (2) (2017), pp. 625-640.
[24] T. Kawasaki and W. Takahashi, A strong convergence theorem for countable families of nonlinear nonself mappings in Hilbert spaces and applications, J. Nonlinear Convex Anal., 19 (2008), pp. 543-560.
[25] M.A. Krasnosel'skii, Two remarks on the method of successive approximations, Usp. Mat. Nauk., 10 (1955), pp. 123-127.
[26] W.R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc., 4 (1953), pp. 506-510. 
[27] P.E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912.
[28] A.A. Mebawondu and O.T. Mewomo Fixed point results for a new three steps iteration process, Annals of the University of Craiova-Mathematics and Computer Science Series, 46 (2) (2019), pp. 298-319.
[29] A.A. Mebawondu L.O. Jolaoso, H.A. Abass, and O.K. Narain Generalized relaxed inertial method with regularization for solving split feasibility problems in real Hilbert spaces, Asian-Eur. J. Math., 15 (06) (2022), pp. 1-25.
[30] M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), pp. 217-229.
[31] Z. Opial, Weak convergence of the sequence of the successive approximation for nonexapansive mapping, Bull. Am. Math. Soc., 73 (4) (1967), pp. 591-597.
[32] O. K. Oyewole, H. A. Abass, A. A. Mebawondu, and K. O. Aremu, A Tseng extragradient method for solving variational inequality problems in Banach spaces, Numer. Algorithms, 89 (2) (2022), pp. 769-789
[33] R. Pant and R. Shukla, Approximating fixed point of generalized $\alpha$-nonexpansive mapping in Banach space, Numer. Funct. Anal. Optim., 38 (2) (2017), pp. 248-266.
[34] W. Phuengrattana, and S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval J. Comput. Appl. Math., 235 (2011), pp. 3006-3014.
[35] D. R. Sahu, Application of the S-iteration process to constrained minimization problem and split feasibility problems, Fixed Point Theory, 12 (2011), pp. 187-204.
[36] G. Stampacchia, Formes bilinearies coercivities sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), pp. 4413-4416.
[37] S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 311 (2) (2005), pp. 506-517.
[38] B. S. Thakur, D. Thakur and M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings, App. Math. Comp., 275 (2016), pp. 147–155
[39] K. Ullah and M. Arshad, Numerical reckoning fixed points for Suzuki generalized nonexpansive mappings via new iteration process, Filomat 32 (1) (2018), pp. 187-196.
[40] X. L. Weng, Fixed point iteration for local strictly pseudocontractive mappings, Proc. Amer. Math. Soc., 113 (1991), pp. 727-731.
[41] N. C. Wong, D. R. Sahu and J. C. Yao, Solving variational inequalities involving nonexpansive type mapping, Nonlinear Anal., 69 (2008), pp. 4732-4753.