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Journal of Scientific Computing

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01-01-2021

Inertial-Type Algorithm for Solving Split Common Fixed Point Problems in Banach Spaces

Authors: A. Taiwo, L. O. Jolaoso, O. T. Mewomo

Published in: Journal of Scientific Computing | Issue 1/2021

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Abstract

In this paper, motivated by the works of Kohsaka and Takahashi (SIAM J Optim 19:824–835, 2008) and Aoyama et al. (J Nonlinear Convex Anal 10:131–147, 2009) on the class of mappings of firmly nonexpansive type, we explore some properties of firmly nonexpansive-like mappings [or mappings of type (P)] in p-uniformly convex and uniformly smooth Banach spaces. We then study the split common fixed point problems for mappings of type (P) and Bregman weak relatively nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. We propose an inertial-type shrinking projection algorithm for solving the two-set split common fixed point problems and prove a strong convergence theorem. Also, we apply our result to the split monotone inclusion problems and illustrate the behaviour of our algorithm with several numerical examples s. The implementation of the algorithm does not require a prior knowledge of the operator norm. Our results complement many recent results in the literature in this direction. To the best of our knowledge, it seems to be the first to use the inertial technique to solve the split common fixed point problems outside Hilbert spaces.

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Agarwal, R.P., Regan, D.O., Sahu, D.R.: Fixed Point Theory for Lipschitzian-Type Mappings with Applications. Springer, Berlin (2009)

Alakoya, T.O., Jolaoso, L.O., Mewomo, O.T.: Modified inertia subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems. Optimization (2020). https://doi.org/10.1080/02331934.2020.1723586CrossRefMATH

Alber, Y., Butnariu, D.: Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces. J. Optim. Theory Appl. 92(1), 33–61 (1997)MathSciNetCrossRef

Ansari, Q.H., Rehan, A.: Split feasibility and fixed point problems. In: Ansari, Q.H. (ed.) Nonlinear Analysis: Approximation Theory, Optimization and Application, pp. 281–322. Springer, New York (2014)MATH

Aoyama, K., Kohsaka, F., Takahashi, W.: Strong convergence theorems for a family of mappings of type (P) and applications. In: Nonlinear Analysis and Optimization, pp. 1–17. Yokohama Publishers, Yokohama (2009)

Aoyama, K., Kohsaka, F., Takahashi, W.: Three generalizations of firmly nonexpansive mappings: their relations and continuity properties. J. Nonlinear Convex Anal. 10, 131–147 (2009)MathSciNetMATH

Bauschke, H.H., Borwein, J.M., Combettes, P.L.: Bregman monotone optimization algorithms. SIAM J. Control Optim. 42(2), 596–636 (2003)MathSciNetCrossRef

Bello Cruz, J.Y., Shehu, Y.: An iterative method for split inclusion problems without prior knowledge of operator norms. J. Fixed Point Theory Appl. 19, 2017–2036 (2017). https://doi.org/10.1007/s11784-016-0387-8MathSciNetCrossRefMATH

Borwein, J.M., Reich, S., Sabach, S.: A characterization of Bregman firmly nonexpansive operators using a new monotonicity concept. J. Nonlinear Convex Anal. 12(1), 161–184 (2011)MathSciNetMATH

10.

Bryne, C.: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 18(2), 441–453 (2002)CrossRef

11.

Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)MathSciNetMATH

12.

Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)CrossRef

13.

Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8(2), 221–239 (1994)MathSciNetCrossRef

14.

Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071–2084 (2005)MathSciNetCrossRef

15.

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

16.

Chen, J., Wan, Z., Yuan, L., Zheng, Y.: Approximation of fixed points of weak Bregman relatively nonexpansive mappings in Banach spaces. IJMMS (2011). Article ID 420192

17.

Chidume, C.E.: Geometric Properties of Banach Spaces and Nonlinear Iterations, Lecture Notes in Mathematics 1965, vol. 1965. Springer, London (2009)CrossRef

18.

Cholamjiak, P., Thong, D.V., Cho, Y.J.: A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems. Acta Appl. Math. (2019). https://doi.org/10.1007/s10440-019-00297-7CrossRef

19.

Gibali, A., Jolaoso, L.O., Mewomo, O.T., Taiwo, A.: Fast and simple Bregman projection methods for solving variational inequalities and related problems in Banach spaces. Results Math. 75 (2020). Art. No. 179

20.

Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)MATH

21.

Izuchukwu, C., Mebawondu, A.A., Mewomo, O.T.: A new method for solving split variational inequality problems without co-coerciveness. J. Fixed Point Theory Appl. 22(4), 1–23 (2020)MathSciNetCrossRef

22.

Izuchukwu, C., Ogwo, G.N., Mewomo, O.T.: An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions. Optimization (2020). https://doi.org/10.1080/02331934.2020.1808648CrossRef

23.

Izuchukwu, C., Ugwunnadi, G.C., Mewomo, O.T., Khan, A.R., Abbas, M.: Proximal-type algorithms for split minimization problem in p-uniformly convex metric space. Numer. Algorithms 82(3), 909–935 (2019)MathSciNetCrossRef

24.

Jolaoso, L.O., Alakoya, T.O., Taiwo, A., Mewomo, O.T.: A parallel combination extragradient method with Armijo line searching for finding common solution of finite families of equilibrium and fixed point problems. Rend. Circ. Mat. Palermo II 69(3), 711–735 (2020)MathSciNetCrossRef

25.

Jolaoso, L.O., Taiwo, A., Alakoya, T.O., Mewomo, O.T.: Strong convergence theorem for solving pseudo-monotone variational inequality problem using projection method in a reflexive Banach space. J. Optim. Theory Appl. 185(3), 744–766 (2020)MathSciNetCrossRef

26.

Jolaoso, L.O., Taiwo, A., Alakoya, T.O., Mewomo, O.T.: A strong convergence theorem for solving variational inequalities using an inertial viscosity subgradient extragradient algorithm with self adaptive stepsize. Demonstr. Math. 52(1), 183–203 (2019)MathSciNetCrossRef

27.

Jolaoso, L.O., Taiwo, A., Alakoya, T.O., Mewomo, O.T.: A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem. Comput. Appl. Math. 39(1), 38 (2020)MathSciNetCrossRef

28.

Jolaoso, L.O., Alakoya, T.O., Taiwo, A., Mewomo, O.T.: Inertial extragradient method via viscosity approximation approach for solving Equilibrium problem in Hilbert space. Optimization (2020). https://doi.org/10.1080/02331934.2020.1716752CrossRefMATH

29.

Kohsaka, F., Takahashi, W.: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM J. Optim. 19, 824–835 (2008)MathSciNetCrossRef

30.

Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces, vol. II. Springer, Berlin (1979)CrossRef

31.

Lin, L.J., Chen, Y.D., Chuang, C.S.: Solutions for a variational inclusion problem with applications to multiple sets split feasibility problems. Fixed Point Theory Appl. 2013, 333 (2013)MathSciNetCrossRef

32.

Moudafi, A.: A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal. 74(12), 4083–4087 (2011)MathSciNetCrossRef

33.

Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275–283 (2011)MathSciNetCrossRef

34.

Naraghirad, E., Yao, J.C.: Bregman weak relatively non expansive mappings in Banach space. Fixed Point Theory Appl. (2013). https://doi.org/10.1186/1687-1812-2013-141CrossRefMATH

35.

Ogwo, G.N., Izuchukwu, C., Aremu, K.O., Mewomo, O.T.: A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space. Bull. Belg. Math. Soc. Simon Stevin 27(1), 127–152 (2020)MathSciNetCrossRef

36.

Phelps, R.R.: Convex Functions, Monotone Operators, and Differentiability. Lecture Notes in Mathematics, vol. 1364, 2nd edn. Springer, Berlin (1993)

37.

Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 4, 1–17 (1964)CrossRef

38.

Reich, S., Sabach, S.: Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 299-314. Springer, New York (2010)

39.

Sch\(\ddot{o}\)pfer, F.: Iterative regularization method for the solution of the split feasibility problem in Banach spaces. Ph.D thesis, Saabr\(\ddot{u}\)cken (2007)

40.

Schopfer, F., Schuster, T., Louis, A.K.: An iterative regularization method for the solution of the split feasibility problem in Banach spaces. Inverse Probl. 24(5), 55008 (2008)MathSciNetCrossRef

41.

Shehu, Y., Mewomo, O.T.: Further investigation into split common fixed point problem for demicontractive operators. Acta Math. Sin. (Engl. Ser.) 32, 1357–1376 (2016)MathSciNetCrossRef

42.

Shehu, Y., Vuong, P.T., Cholamjiak, P.: A self-adaptive projection method with an inertial technique for split feasibility problems in Banach spaces with applications to image restorations problems. J. Fixed Point Theory Appl. 21, 50 (2019). https://doi.org/10.1007/s11784-019-0684-0MathSciNetCrossRefMATH

43.

Song, Y.: Iterative methods for fixed point problems and generalized split feasibility problems in Banach spaces. J. Nonlinear Sci. Appl. 11, 198–217 (2018)MathSciNetCrossRef

44.

Su, Y., Wang, Z., Xu, H.: Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings. Nonlinear Anal. 71, 5616–5628 (2009)MathSciNetCrossRef

45.

Suparatulatorn, R., Khemphet, A., Charoensawan, P., Suantai, S., Phudolsitthiphat, N.: Generalized self-adaptive algorithm for solving split common fixed point problem and its application to image restoration problem. Int. J. Comput. Math. 97(7), 1431–1443 (2020). https://doi.org/10.1080/00207160.2019.1622687MathSciNetCrossRef

46.

Taiwo, A., Alakoya, T.O., Mewomo, O.T.: Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces. Numer. Algorithms (2020). https://doi.org/10.1007/s11075-020-00937-2CrossRef

47.

Taiwo, A., Jolaoso, L. O., Mewomo, O. T.: A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces. Comput. Appl. Math. 38(2) (2019). Article 77

48.

Taiwo, A., Jolaoso, L.O., Mewomo, O.T.: Parallel hybrid algorithm for solving pseudomonotone equilibrium and Split Common Fixed point problems. Bull. Malays. Math. Sci. Soc. 43(2), 1893–1918 (2019)MathSciNetCrossRef

49.

Taiwo, A., Jolaoso, L.O., Mewomo, O.T., Gibali, A.: On generalized mixed equilibrium problem with \(\alpha \)-\(\beta \)-\(\mu \) bifunction and \(\mu \)-\(\tau \) monotone mapping. J. Nonlinear Convex Anal. 21(6), 1381–1401 (2020)MathSciNet

50.

Taiwo, A., Owolabi, A.O.-E., Jolaoso, L.O., Mewomo, O.T., Gibali, A.: A new approximation scheme for solving various split inverse problems. Afr. Mat. (2020). https://doi.org/10.1007/s13370-020-00832-yCrossRefMATH

51.

Takahashi, W.: The split common fixed point problem and the shrinking projection method in Banach spaces. J. Convex Anal. 24, 1015–1028 (2017)MathSciNetMATH

52.

Takahashi, W.: The split common fixed point problem for generalized demimetric mappings in two Banach spaces. Optimization (2018). https://doi.org/10.1080/02331934.2018.1522637CrossRefMATH

53.

Tang, J., Chang, S., Wang, L., Wang, X.: On the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces. J. Inequalities Appl. 2015, 305 (2015)MathSciNetCrossRef

54.

Thong, D.V., Hieu, D.V.: An inertial method for solving split common fixed point problems. J. Fixed Point Theory Appl. (2017). https://doi.org/10.1007/s11784-017-0464-7MathSciNetCrossRefMATH

55.

Tuyen, T.M., Thuy, N.T.T., Trang, N.M.: A strong convergence theorem for a parallel iterative method for solving the split common null point problem in Hilbert spaces. J. Optim. Theory Appl. (2019). https://doi.org/10.1007/s10957-019-01523-wMathSciNetCrossRefMATH

56.

Xu, Z.B., Roach, G.F.: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 157(1), 189–210 (1991)MathSciNetCrossRef

Title: Inertial-Type Algorithm for Solving Split Common Fixed Point Problems in Banach Spaces
Authors: A. Taiwo
L. O. Jolaoso
O. T. Mewomo
Publication date: 01-01-2021
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2021
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-020-01385-9

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