Document Type : Research Paper

Authors

1 Department of Mathematics, University of Uyo, Uyo, Nigeria.

2 Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Nigeria.

Abstract

In this paper, we introduce a three-step implicit iteration scheme with errors for finite families of nonexpansive and uniformly $L$-Lipschitzian asymptotically generalized $\Phi$-hemicontractive mappings in real Banach spaces. Our new implicit iterative scheme properly includes several well known iterative schemes in the literature as its special cases. The results presented in this paper extend, generalize and improve well known results in the existing literature.

Keywords

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[1] Y.I. Alber, C.E. Chidume and H. Zegeye, Regularization of nonlinear ill-posed equations with accretive    operators, Fixed Point Theory and Appl., 1 (2005), pp. 11-33.
[2] R.P. Agarwal, Y.J. Cho, J. Li and N.J. Huang, Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mappings in q-uniformly smooth Banach spaces, J. Math. Anal. Appl., 272 (2002), pp. 435-447.
[3] A. Bnouhachem, M.A. Noor and T.M. Rassias, Three-steps iterative algorithms for mixed variational inequalities, Appl. Math. Comput., 183 (2006), pp. 436-446.
[4] S.S. Chang, Some results for asymptotically pseudocontractive mappings and asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 129 (2001), pp. 845-853.
[5] C.E. Chidume, A.U. Bello, M.E. Okpala and P. Ndambomve, Strong convergence theorem for fixed points    of nearly nniformly L-Lipschitzian Asymptotically    Generalized $Phi$-hemicontractive mappings, Int. J. Math. Anal., 9 (2015), pp. 2555-2569.
[6] C.E. Chidume and C.O. Chidume, Convergence theorems for fixed points of uniformly continuous    generalized $Phi$-hemi-contractive mappings, J. Math. Anal. Appl., 303 (2005), pp. 545-554.
[7] C.E. Chidume and C.O. Chidume, Convergence theorem for zeros of generalized Lipschitz generalized phi-quasi-accretive operators, Proc. Amer. Math. Soc., 134 (2006), pp. 243-251.
[8]    R.C. Chen, Y.S. Song and H. Zhou, Convergence theorems for implicit iteration process for a finite family continuous pseudocontractive mappings, J. Math. Anal. Appl., 314 (2006), pp. 701-706.
[9] Y.J. Cho, H.Y. Zhou and G. Guo, Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl., 47 (2004), pp. 707-717.
[10] P. Chuadchawna, A. Farajzadeh and A. Kaewcharoen, On convergence theorems for two generalized nonexpansive multivalued mappings in hyperbolic spaces, Thai J. Math., 17 (2019), pp. 445-461.
[11] P. Chuadchawna, A. Farajzadeh and A. Kaewcharoen, Fixed-point approximations of generalized nonexpansive mappings via generalized M-iteration process in hyperbolic spaces, Int. J. Math. Math. Sci., (2020), pp. 1-8, article ID 6435043.
[12] P. Chuadchawna, A. Farajzadeh and A. Kaewcharoen, Convergence theorems for total asymptotically nonexpansive single-valued and quasi nonexpansive multi-valued mappings in hyperbolic spaces, J. Appl. Anal., 27 (2021), pp. 129-142.
[13] G. Das and J.P. Debata, Fixed points of Quasi-nonexpansive mappings, Indian J. Pure. Appl. Math., 17 (1986), pp. 1263-1269.
[14] L.C. Deng, P. Cubiotti and J.C. Yao, Approximation of common fixed points of families of nonexpansive mappings, Tai. J. Math., 12 (2008), pp. 487-500.
[15] L.C. Deng, P. Cubiotti and J.C. Yao, An implicit iteration scheme for monotone variational inequalities and fixed point problems, Nonlinear Anal., 69 (2008), pp. 2445-2457.
[16] L.C. Deng, S. Schaible and J.C. Yao, Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings, J. Optim. Theory Appl., 139 (2008), pp. 403-418.
[17] R. Glowinski and P. Le-Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, SIAM, Philadelphia, 1989.
[18] S. Haubruge, V.H. Nguyen and J.J. Strodiot, Convergence analysis and applications of the Glowinski-Le-Tallec splitting method for finding a zero of the sum of two maximal monotone operators, J. Optim. Theory Appl., 97 (1998), pp. 645-673.
[19] F.Gu, Convergence theorems for $phi$-pseudocontractive type mappings in normed linear spaces, Northeast Math. J., 17 (2001), pp. 340-346.
[20] F. Gu, Strong convergence of an implicit iteration process for a finite family of uniformly $L$-Lipschitzian mapping in Banach spaces, J. Ineq. and Appl., doi:10.1155/2010/801961.
[21] S. Ishikawa, Fixed points by a new iteration method, Proceeding of the America Mathematical society, 4 (1974), pp. 157-150.
[22] S.H. Khan and W. Takahashi, Approximating common fixed points of two asymptotically nonexpansive mappings, Sci. Math. Jpn., 53 (2001), pp. 143-148.
[23] S.H. Khan, I. Yildirim and M. Ozdemir, Some results for finite families of fniformly $L$-Lipschitzian mappings in Banach paces, Thai J. Math., 9 (2011), pp. 319-331.
[24] J.K. Kim, D.R. Sahu and Y.M. Nam, Convergence theorems for fixed points of nearly uniformly $L$-Lipschitzian asymptotically generalized $Phi$-hemicontractive mappings, Nonlinear Anal., 71 (2009), pp. 2833-2838.
[25] G. Lv, A. Rafiq and Z. Xue, Implicit iteration scheme for two phi-hemicontractive operators in arbitrary Banach spaces, Journal of Ineq. and Appl., 2013, 2013:521.
[26] E.U. Ofoedu, Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in a real Banach space, J. Math. Anal. Appl., 321 (2006), pp. 722-728.
[27] M.A. Noor, T.M. Kassias and Z. Huang, Three-step iterations for nonlinear accretive operator equations, J. Math. Anal. Appl., 274 (2001), pp. 59-68.
[28] M.O. Osilike, Iterative solution of nonlinear equations of the $phi$-strongly    accretive type, J. Math. Anal. Appl., 200 (1996), pp. 259-271.
[29] M.O. Osilike and B.G. Akuchu, Common fixed points of finite family of asymptotically pseudocontractive mappings, Fixed Point Theory and Appl., 2004 (2004), pp. 81-88.
[30] M.O. Osilike, S.C. Aniagbosor and B.G. Akuchu, Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces, Panam. Math. J., 12 (2002), pp. 77-88.
[31] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Sci., 4 (2003), pp. 506-510.
[32] A. Rafiq, On convergence of implicit iteration scheme for two hemicontractive mappins, Sci. Int. (Lahore), 24 (2012), pp. 431-434.
[33] A. Rafiq and M. Imdad, Implicit Mann Iteration Scheme for hemicontractive mapping, J. Indian Math. Soc., 81 (2014), pp. 147-153.
[34] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austr. Math. Soc., 43 (1991), pp. 153-159.
[35] J. Schu, Iterative construction of fixed point of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 158 (1991), pp. 407-413.
[36] H.F. Senter and W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44 (1974), pp. 375-380.
[37] N. Shahzad and A. Udomene, Approximating common fixed points of two asymptotically quasinonexpansive mappings in Banach spaces, Fixed Point Theory Appl., (2006), pp. 1-10, article ID 18909.
[38] Z.H. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 286 (2003), pp. 351-358.
[39] S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 311 (2005), pp. 506-517.
[40] W. Takahashi, Iterative methods for approximation of fixed points and their applications, J. Oper. Res. Soc. Jpn., 43 (2000), pp. 87-108.
[41] W. Takahashi and T. Tamura, Limit theorems of operators by convex combinations of nonexpansive retractions in Banach spaces, J. Approx. Theory, 91 (1997), pp. 386-397.
[42] B.S. Thakur, Weak and strong convergence of composite implicit iteration process, Appl. Math. Comput., 190 (2007), pp. 965-973.
[43] B.S. Thakur, Strong Convergence for Asymptotically generalized $Phi$-hemicontractive mappings, ROMAI J., 8 (2012), pp. 165-171.
[44] H.K. Xu and R.G. Ori, An implicit iteration process for nonexpansive mapping, Num. Fun. Anal. Optim., 22 (2001), pp. 767-773.
[45] Y. Yao, Convergence of three-step iterations for asymptotically nonexpansive mappings, Appl. Math. Comput., 187 (2007), pp. 883-892.
[46] L. P. Yang, Convergence theorem of an implicit iteration process for asymptotically pseudocontractive mappings, Bull. of the Iran. Math. Soc., 38 (2012), pp. 699-713.
[47] L.P. Yang and G. Hu, Convergence of implicit iteration process with random errors, Acta Math. Sinica (Chin. Ser.), 51 (2008), pp. 11-22.
[48] L.C. Zeng, On the approximation of fixed points for asymptotically nonexpansive mappings in Banach spaces, Acta Math. Sci., 23 (2003), pp. 31-37.
[49] L.C. Zeng, On the iterative approximation for asymptotically pseudocontractive mappings in uniformly smooth Banach spaces, Chinese Math. Ann., 26 (2005), pp. 283-290.