International Journal of Nonlinear Analysis and Applications
Approximating common fixed points of mean nonexpansive mappings in hyperbolic spaces
Document Type : Research Paper
Authors
1 Department of Mathematics and Statistics, University of Port Harcourt, Nigeria
2 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
3 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
Abstract
In this paper, we prove some fixed points properties and demiclosedness principle for mean nonexpansive mapping in uniformly convex hyperbolic spaces. We further propose an iterative scheme for approximating a common fixed point of two mean nonexpansive mappings and establish some strong and $\bigtriangleup$-convergence theorems for these mappings in uniformly convex hyperbolic spaces. The results obtained in this paper extend and generalize corresponding results in uniformly convex Banach spaces, CAT(0) spaces and other related results in literature.
Keywords
[1] M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesnik, 66(2) (2014) 223-234.
[2] T.O. Alakoya, L.O. Jolaoso and O.T. Mewomo, A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications, J. Ind. Manag. Optim. (2020), doi:10.3934/jimo.2020152.
[3] T.O. Alakoya, L.O. Jolaoso and O.T. Mewomo, Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems, Demonstr. Math. 53 (2020)208-–224.
[4] T.O. Alakoya, L.O. Jolaoso and O.T. Mewomo, Modified inertia subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, Optim. (2020), DOI:10.1080/02331934.2020.1723586.
[5] T.O. Alakoya, A. Taiwo, O.T. Mewomo and Y.J. Cho, An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings, Ann. Univ.Ferrara Sez. VII Sci. Mat., DOI: 10.1007/s11565-020-00354-2
[6] K.O. Aremu, H.A. Abass, C. Izuchukwu and O.T. Mewomo, A viscosity-type algorithm for an infinitely countable family of (f, g)-generalized k-strictly pseudononspreading mappings in CAT(0) spaces, Anal. 40(1) (2020) 19–37.
[7] K.O. Aremu, C. Izuchukwu, G.N. Ogwo and O.T. Mewomo, Multi-step Iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces, J. Ind. Manag. Optim. (2020),DOI:10.3934/jimo.2020063.
[8] K.O. Aremu, L.O. Jolaoso, C. Izuchukwu and O.T. Mewomo, Approximation of common solution of finite family of monotone inclusion and fixed point problems for demicontractive mappings in CAT(0) spaces, Ric. Mat. 69(1) (2020) 13–34.
[9] S.S. Chang, G. Wang, L. Wang, Y.K. Tang and Z.L. Ma, △-convergence theorems for multi-valued nonexpansive mappings in hyperbolic spaces, Appl. Math. Comp. 249 (2014) 535–540.
[10] G. Das and J. P. Debata, Fixed points of quasinonexpansive mappings, Indian J. Pure Appl. Math. 17 (1986) 1263–1269.
[11] H. Dehghan, C. Izuchukwu, O.T. Mewomo, D.A. Taba and G.C. Ugwunnadi, Iterative algorithm for a family of monotone inclusion problems in CAT(0) spaces, Quaest. Math. 43(7) (2020) 975-–998.
[12] A. Gibali, L.O. Jolaoso, O.T. Mewomo and A. Taiwo, Fast and simple Bregman projection methods for solving variational inequalities and related problems in Banach spaces, Results Math. 75 (2020) Art. No. 179, 36 pp.
[13] E.C. Godwin, C. Izuchukwu and O.T. Mewomo, An inertial extrapolation method for solving generalized split feasibility problems in real Hilbert spaces, Boll. Unione Mat. Ital. (2020), DOI 10.1007/s40574-020-00.
[14] K. Goebel and W.A. Kirk, Iteration processes for nonexpansive mappings, Topological Meth. Nonlinear Funct. Anal. S. P. Singh, S. Thomeier, and B.Watson, Eds., Vol. 21 of Contemporary Mathematics, 115–123, American Mathematical Society, Providence, RI, USA, 1983.
[15] K. Goebel and S. Reich, Uniform convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekket, New York, 1984.
[16] Z.H. Gu, Ishikawa iterative for mean nonexpansive mappings in uniformly convex Banach spaces, J. Guangzhou Economic Management College, 8 (2006) 86–88.
[17] M. Imdad and S. Dashputre, Fixed point approximation of Picard normal S-iteration process for generalized nonexpansive mappings in hyperbolic spaces, Math. Sci. 10 (2016) 131—138.
[18] C. Izuchukwu, A.A. Mebawondu and O.T. Mewomo, A New Method for solving split variational inequality problems without co-coerciveness, J. Fixed Point Theory Appl. 22(4) (2020) Art. No. 98, 23 pp.
[19] C. Izuchukwu, G.N. Ogwo and O.T. Mewomo, An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions, Optim. (2020), DOI:10.1080/02331934.2020.1808648.
[20] A.R. Khan, H. Fukhar-ud-din and M.A.A. Khan, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl, 2012 (2012, Art. ID 54.
[21] S.H. Khan, Y.J. Cho and M. Abbas, Convergence of common fixed points by a modified iteration process, J. Appl. Math. Comput. 35 (2011) 607–616.
[22] U. Kohlenbach, Some logical metathorems with applications in functional analysis, Trans. Amer. Math. Soc. 357(1) (2005) 89–128.
[23] W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. Theory Methods Appl. Ser. A, Theory Meth. 68(12) (2008) 3689–3696.
[24] T. Kuczumow, An almost convergence and its applications, Ann. Univ. Mariae Curie-Sk lodowska, Sect. A 32 (1978) 79–88.
[2] T.O. Alakoya, L.O. Jolaoso and O.T. Mewomo, A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications, J. Ind. Manag. Optim. (2020), doi:10.3934/jimo.2020152.
[3] T.O. Alakoya, L.O. Jolaoso and O.T. Mewomo, Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems, Demonstr. Math. 53 (2020)208-–224.
[4] T.O. Alakoya, L.O. Jolaoso and O.T. Mewomo, Modified inertia subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, Optim. (2020), DOI:10.1080/02331934.2020.1723586.
[5] T.O. Alakoya, A. Taiwo, O.T. Mewomo and Y.J. Cho, An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings, Ann. Univ.Ferrara Sez. VII Sci. Mat., DOI: 10.1007/s11565-020-00354-2
[6] K.O. Aremu, H.A. Abass, C. Izuchukwu and O.T. Mewomo, A viscosity-type algorithm for an infinitely countable family of (f, g)-generalized k-strictly pseudononspreading mappings in CAT(0) spaces, Anal. 40(1) (2020) 19–37.
[7] K.O. Aremu, C. Izuchukwu, G.N. Ogwo and O.T. Mewomo, Multi-step Iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces, J. Ind. Manag. Optim. (2020),DOI:10.3934/jimo.2020063.
[8] K.O. Aremu, L.O. Jolaoso, C. Izuchukwu and O.T. Mewomo, Approximation of common solution of finite family of monotone inclusion and fixed point problems for demicontractive mappings in CAT(0) spaces, Ric. Mat. 69(1) (2020) 13–34.
[9] S.S. Chang, G. Wang, L. Wang, Y.K. Tang and Z.L. Ma, △-convergence theorems for multi-valued nonexpansive mappings in hyperbolic spaces, Appl. Math. Comp. 249 (2014) 535–540.
[10] G. Das and J. P. Debata, Fixed points of quasinonexpansive mappings, Indian J. Pure Appl. Math. 17 (1986) 1263–1269.
[11] H. Dehghan, C. Izuchukwu, O.T. Mewomo, D.A. Taba and G.C. Ugwunnadi, Iterative algorithm for a family of monotone inclusion problems in CAT(0) spaces, Quaest. Math. 43(7) (2020) 975-–998.
[12] A. Gibali, L.O. Jolaoso, O.T. Mewomo and A. Taiwo, Fast and simple Bregman projection methods for solving variational inequalities and related problems in Banach spaces, Results Math. 75 (2020) Art. No. 179, 36 pp.
[13] E.C. Godwin, C. Izuchukwu and O.T. Mewomo, An inertial extrapolation method for solving generalized split feasibility problems in real Hilbert spaces, Boll. Unione Mat. Ital. (2020), DOI 10.1007/s40574-020-00.
[14] K. Goebel and W.A. Kirk, Iteration processes for nonexpansive mappings, Topological Meth. Nonlinear Funct. Anal. S. P. Singh, S. Thomeier, and B.Watson, Eds., Vol. 21 of Contemporary Mathematics, 115–123, American Mathematical Society, Providence, RI, USA, 1983.
[15] K. Goebel and S. Reich, Uniform convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekket, New York, 1984.
[16] Z.H. Gu, Ishikawa iterative for mean nonexpansive mappings in uniformly convex Banach spaces, J. Guangzhou Economic Management College, 8 (2006) 86–88.
[17] M. Imdad and S. Dashputre, Fixed point approximation of Picard normal S-iteration process for generalized nonexpansive mappings in hyperbolic spaces, Math. Sci. 10 (2016) 131—138.
[18] C. Izuchukwu, A.A. Mebawondu and O.T. Mewomo, A New Method for solving split variational inequality problems without co-coerciveness, J. Fixed Point Theory Appl. 22(4) (2020) Art. No. 98, 23 pp.
[19] C. Izuchukwu, G.N. Ogwo and O.T. Mewomo, An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions, Optim. (2020), DOI:10.1080/02331934.2020.1808648.
[20] A.R. Khan, H. Fukhar-ud-din and M.A.A. Khan, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl, 2012 (2012, Art. ID 54.
[21] S.H. Khan, Y.J. Cho and M. Abbas, Convergence of common fixed points by a modified iteration process, J. Appl. Math. Comput. 35 (2011) 607–616.
[22] U. Kohlenbach, Some logical metathorems with applications in functional analysis, Trans. Amer. Math. Soc. 357(1) (2005) 89–128.
[23] W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. Theory Methods Appl. Ser. A, Theory Meth. 68(12) (2008) 3689–3696.
[24] T. Kuczumow, An almost convergence and its applications, Ann. Univ. Mariae Curie-Sk lodowska, Sect. A 32 (1978) 79–88.
[25] L. Leustean, Nonexpansive iteration in uniformly convex W-hyperbolic spaces, Nonlinear Anal. Optim. 51(3 (2010) 193–209.
[26] A.A. Mebawondu, C. Izuchukwu, K.O. Aremu and O.T. Mewomo, On some fixed point results for (alpha, β)- Berinde-φ-Contraction mapppings with applications, Int. J. Nonlinear Anal. Appl. 11(2) (2020) 363–378.
[27] A.A. Mebawondu, C. Izuchukwu, K.O Aremu and O.T. Mewomo, Some fixed point results for a generalized TAC-Suzuki-Berinde type F-contractions in b-metric spaces, Appl. Math. E-Notes 19 (2019) 629–653.
[28] A. A. Mebawondu, O.T. Mewomo, Some fixed point results for TAC-Suzuki contractive mappings, Commun. Korean Math. Soc. 34(4) (2019) 1201–1222.
[29] A. A. Mebawondu and O.T. Mewomo, Suzuki-type fixed point results in Gb-metric spaces, Asian-Eur. J. Math. (2020), DOI:10.1142/S1793557121500704.
[30] K. Nakprasit, Mean nonexpansive mappings and Suzuki-generalized nonexpansive mappings, J. Nonlinear Anal. Optim. 1(1) (2010) 93–96.
[31] G. N. Ogwo, C. Izuchukwu, K.O. Aremu and O.T. Mewomo, A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space, Bull. Belg. Math. Soc. Simon Stevin 27 (2020) 127–152.
[32] G. N. Ogwo, C. Izuchukwu, K.O. Aremu and O.T. Mewomo, On θ-generalized demimetric mappings and monotone operators in Hadamard spaces, Demonstr. Math. 53(1) (2020) 95—111.
[33] O.K. Oyewole, H.A. Abass and O.T. Mewomo, A Strong convergence algorithm for a fixed point constrainted split null point problem, Rend. Circ. Mat. Palermo II, (2020), DOI:10.1007/s12215-020-00505-6.
[34] K.O. Oyewole, C. Izuchukwu, C.C. Okeke and O.T. Mewomo, Inertial approximation method for split variational inclusion problem in Banach spaces, Int. J. Nonlinear Anal. Appl. 11(2) (2020) 285–304.
[35] A. Ouahab, A. Mbarki, J. Masude and M. Rahmoune, A fixed point theorem for mean nonexpansive mappings semigroups in uniformly convex Banach spaces, J. Math. Anal. 6(3) (2012) 101–109.
[36] S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990) 537–558.
[37] C. Suanoom, C. Klin-eam, Remark on fundamentally non-expansive mappings in hyperbolic spaces, J. Nonlinear Sci. Appl. 9 (2016) 1952–1956.
[38] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088–1095.
[39] A. Taiwo, T.O. Alakoya and O.T. Mewomo, Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces, Numer. Algorithms, (2020), DOI: 10.1007/s11075-020 00937-2.
[40] A. Taiwo, T.O. Alakoya and O.T. Mewomo, Strong convergence theorem for fixed points of relatively nonexpansive multi-valued mappings and equilibrium problems in Banach spaces, Asian-Eur. J. Math. (2020), https://doi.org/10.1142/S1793557121501370.
[41] A. Taiwo, L.O. Jolaoso and O.T. Mewomo, Inertial-type algorithm for solving split common fixed-point problem in Banach spaces, J. Sci. Comput. DOI: 10.1007/s10915-020-01385-9.
[42] A. Taiwo, L.O. Jolaoso, O.T. Mewomo and A. Gibali, On generalized mixed equilibrium problem with α-β-µ bifunction and µ-τ monotone mapping, J. Nonlinear Convex Anal., 21(6) (2020) 1381–1401.
[43] A. Taiwo, A. O.-E. Owolabi, L.O. Jolaoso, O.T. Mewomo and A. Gibali, A new approximation scheme for solving various split inverse problems, Afr. Mat., (2020), DOI:https://doi.org/10.1007/s13370-020-00832-y.
[44] W.A. Takahashi, A convexity in metric space and nonexpansive mappings, I. Kodai Math. Sem. Rep., 22 (1970), 142-149.
[45] C.-X. Wu and L.-J. Zhang, Fixed points for mean nonexpansive mappings, Acta Math. Appl. Sin. 23(3) (2007) 489–494.
[46] Y.S. Yang and Y.A. Cui, Viscosity approximation methods for mean non-expansive mappings in Banach spaces, Appl. Math. Sci. 2(13) (2008) 627–638.
[47] S.S. Zhang, About fixed point theorem for mean nonexpansive mapping in Banach spaces, J. Sichuan Univ. 2 (1975) 67–68.
[48] J. Zhou and Y. Cui, Fixed point theorems for mean nonexpansive mappings in CAT(0) spaces, Numer. Funct. Anal. Opt. 36(9) (2014) 1224–1238.
[49] Z. Zuo, Fixed-point theorems for mean nonexpansive mappings in Banach spaces, Abstr. Appl. Anal. 2015 (2015) Art. ID 746291.
[26] A.A. Mebawondu, C. Izuchukwu, K.O. Aremu and O.T. Mewomo, On some fixed point results for (alpha, β)- Berinde-φ-Contraction mapppings with applications, Int. J. Nonlinear Anal. Appl. 11(2) (2020) 363–378.
[27] A.A. Mebawondu, C. Izuchukwu, K.O Aremu and O.T. Mewomo, Some fixed point results for a generalized TAC-Suzuki-Berinde type F-contractions in b-metric spaces, Appl. Math. E-Notes 19 (2019) 629–653.
[28] A. A. Mebawondu, O.T. Mewomo, Some fixed point results for TAC-Suzuki contractive mappings, Commun. Korean Math. Soc. 34(4) (2019) 1201–1222.
[29] A. A. Mebawondu and O.T. Mewomo, Suzuki-type fixed point results in Gb-metric spaces, Asian-Eur. J. Math. (2020), DOI:10.1142/S1793557121500704.
[30] K. Nakprasit, Mean nonexpansive mappings and Suzuki-generalized nonexpansive mappings, J. Nonlinear Anal. Optim. 1(1) (2010) 93–96.
[31] G. N. Ogwo, C. Izuchukwu, K.O. Aremu and O.T. Mewomo, A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space, Bull. Belg. Math. Soc. Simon Stevin 27 (2020) 127–152.
[32] G. N. Ogwo, C. Izuchukwu, K.O. Aremu and O.T. Mewomo, On θ-generalized demimetric mappings and monotone operators in Hadamard spaces, Demonstr. Math. 53(1) (2020) 95—111.
[33] O.K. Oyewole, H.A. Abass and O.T. Mewomo, A Strong convergence algorithm for a fixed point constrainted split null point problem, Rend. Circ. Mat. Palermo II, (2020), DOI:10.1007/s12215-020-00505-6.
[34] K.O. Oyewole, C. Izuchukwu, C.C. Okeke and O.T. Mewomo, Inertial approximation method for split variational inclusion problem in Banach spaces, Int. J. Nonlinear Anal. Appl. 11(2) (2020) 285–304.
[35] A. Ouahab, A. Mbarki, J. Masude and M. Rahmoune, A fixed point theorem for mean nonexpansive mappings semigroups in uniformly convex Banach spaces, J. Math. Anal. 6(3) (2012) 101–109.
[36] S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990) 537–558.
[37] C. Suanoom, C. Klin-eam, Remark on fundamentally non-expansive mappings in hyperbolic spaces, J. Nonlinear Sci. Appl. 9 (2016) 1952–1956.
[38] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088–1095.
[39] A. Taiwo, T.O. Alakoya and O.T. Mewomo, Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces, Numer. Algorithms, (2020), DOI: 10.1007/s11075-020 00937-2.
[40] A. Taiwo, T.O. Alakoya and O.T. Mewomo, Strong convergence theorem for fixed points of relatively nonexpansive multi-valued mappings and equilibrium problems in Banach spaces, Asian-Eur. J. Math. (2020), https://doi.org/10.1142/S1793557121501370.
[41] A. Taiwo, L.O. Jolaoso and O.T. Mewomo, Inertial-type algorithm for solving split common fixed-point problem in Banach spaces, J. Sci. Comput. DOI: 10.1007/s10915-020-01385-9.
[42] A. Taiwo, L.O. Jolaoso, O.T. Mewomo and A. Gibali, On generalized mixed equilibrium problem with α-β-µ bifunction and µ-τ monotone mapping, J. Nonlinear Convex Anal., 21(6) (2020) 1381–1401.
[43] A. Taiwo, A. O.-E. Owolabi, L.O. Jolaoso, O.T. Mewomo and A. Gibali, A new approximation scheme for solving various split inverse problems, Afr. Mat., (2020), DOI:https://doi.org/10.1007/s13370-020-00832-y.
[44] W.A. Takahashi, A convexity in metric space and nonexpansive mappings, I. Kodai Math. Sem. Rep., 22 (1970), 142-149.
[45] C.-X. Wu and L.-J. Zhang, Fixed points for mean nonexpansive mappings, Acta Math. Appl. Sin. 23(3) (2007) 489–494.
[46] Y.S. Yang and Y.A. Cui, Viscosity approximation methods for mean non-expansive mappings in Banach spaces, Appl. Math. Sci. 2(13) (2008) 627–638.
[47] S.S. Zhang, About fixed point theorem for mean nonexpansive mapping in Banach spaces, J. Sichuan Univ. 2 (1975) 67–68.
[48] J. Zhou and Y. Cui, Fixed point theorems for mean nonexpansive mappings in CAT(0) spaces, Numer. Funct. Anal. Opt. 36(9) (2014) 1224–1238.
[49] Z. Zuo, Fixed-point theorems for mean nonexpansive mappings in Banach spaces, Abstr. Appl. Anal. 2015 (2015) Art. ID 746291.
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Volume 12, Issue 1
May 2021Pages 231-244
- Receive Date: 12 November 2020
- Revise Date: 22 December 2020
- Accept Date: 13 January 2021
- First Publish Date: 13 January 2021