Abstract
In this paper, we present a new algorithm for solving the split common null point and the common fixed point problems between Banach spaces and Hilbert spaces, to find a point which belongs to the common zero point set of a finite family of maximal monotone operators and the common fixed point set of a finite family of demicontractive mappings in a Hilbert space such that its image under a linear transformation belongs to the common zero point set of another finite family of maximal monotone operators in a Banach space. We prove that the sequence generated by the proposed algorithm converges strongly to a common solution of the split common null point and the common fixed point problems, which is also an unique solution of a variational inequality as an optimality condition for a minimization problem.
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References
Alsulami, S.M., Hussain, N., Takahashi, W.: Weakly convergent iterative method for the split common null point problem in Banach spaces. J. Nonlinear Convex Anal. 11, 2333–2342 (2015)
Aoyama, K., Kohsaka, F., Takahashi, W.: Three generalizations of firmly nonexpansive mappings: their relations and continuous properties. J. Nonlinear Convex Anal. 10, 131–147 (2009)
Brezis, H.: Operateurs Maximaux Monotones et Semi-Groups de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam (1973)
Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20, 197–228 (1967)
Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)
Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)
Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)
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)
Censor, Y., Elfving, T.: A multiprojection algorithms using Bregman projection in a product space. Numer. Algorithm 8, 221–239 (1994)
Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications. Inverse Probl. 21, 2071–2084 (2005)
Chidume, C.E., Maruster, S.: Iterative methods for the computation of fixed points of demicontractive mappings. J. Comput. Appl. Math. 234, 861–882 (2010)
Combettes, P.L.: The convex feasibility problem in image recovery. In: Hawkes, P. (ed.) Advances in Imaging and Electron Physics, vol. 95, pp. 155–270. Academic Press, New York (1996)
Eslamian, M.: General algorithms for split common fixed point problem of demicontractive mappings. Optimization 65, 443–465 (2016)
Hicks, T.L., Kubicek, J.R.: On the Mann iterative process in Hilbert spaces. J. Math. Anal. Appl. 59, 498–504 (1977)
Maingé, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16, 899–912 (2008)
Marino, G., Xu, H.K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006)
Marino, G., Xu, H.K.: Weak and strong convergence theorems for strictly pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329, 336–349 (2007)
Masad, E., Reich, S.: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8, 367–371 (2007)
Moudafi, A.: The split common fixed-point problem for demicontractive mappings. Inverse Probl. 26, 055007 (2010)
Takahashi, W.: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama (2009)
Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
Takahashi, W.: Convex Analysis and Approximation of Fixed points. Yokohama Publishers, Yokohama (2000). (Japanese)
Takahashi, W.: The split feasibility problem in Banach spaces. J. Nonlinear Convex Anal. 15, 1349–1355 (2014)
Takahashi, W.: The split common null point problem in Banach spaces. Archiv der Mathematik. 104, 357–365 (2015)
Takahashi, W.: The split common null point problem and the shrinking projection method in Banach spaces. Optimization 65, 281–287 (2016)
Takahashi, S., Takahashi, W., Toyoda, M.: Strong convergence theorems for maximal monotone operators with nonlinear mapping in Hilbert spaces. J. Optim. Theory Appl. 147, 27–41 (2010)
Takahashi, W., Yao, J.C.: Strong convergence theorems by hybrid methods for the split common null point problem in Banach spaces. Fixed Point Theory Appl. 2015, 87 (2015)
Vinh, N.T., Hoai, P.T.: Some subgradient extragradient type algorithms for solving split feasibility and fixed point problems. Math. Methods Appl. Sci. (2016). doi:10.1002/mma.3826
Xu, H.K.: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 65, 109–113 (2002)
Xu, H.K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010)
Zegeye, H., Shahzad, N.: Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings. Comput. Math. Appl. 62, 4007–4014 (2011)
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Eslamian, M., Eskandani, G.Z. & Raeisi, M. Split Common Null Point and Common Fixed Point Problems Between Banach Spaces and Hilbert Spaces. Mediterr. J. Math. 14, 119 (2017). https://doi.org/10.1007/s00009-017-0922-x
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DOI: https://doi.org/10.1007/s00009-017-0922-x