Riv. Mat. Univ. Parma, to appear

Strong convergence of split equality variational inequality and fixed point problem

Pages:
Accepted in revised form: 4 November 2016
Mathematics Subject Classification (2010): 47J25, 47N10, 65J15, 90C25.
Keywords: Split equality problem, fixed point, quasi-nonexpansive mapping, variational inequality.
[a]:University of Science and Technology of Mazandaran, Department of Mathematics, Box: 48518-78195, Behshahr, Iran and Institute for Research in Fundamental Science, School of Mathematics, (IPM) P.O.Box:19395-5746, Tehran, Iran

Abstract: The main purpose of this paper is to introduce a new algorithm for finding a solution of split equality variational inequality problem for monotone and Lipschitz continuous operators and common fixed points of a finite family of quasi-nonexpansive mappings in the setting of infinite dimensional Hilbert spaces. Under suitable conditions, we prove that the sequence generated by the proposed new algorithm converges strongly to a solution of the split equality variational inequality and fixed point problem in Hilbert spaces. Our results improve and generalize some recent results in the literature.

References

[1] B. P. N. Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization. 62, 271-283 (2013). Scopus
[2] K. Aoyama, S. Iemoto, F. Kohsaka, W. Takahashi, Fixed point and ergodic theorems for $$\lambda-$$ hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal. 11, 335-343 (2010). MR2682871
[3] H. Attouch, J.Bolte, P. Redont, A.Soubeyran, Alternating proximal algorithms for weakly coupled minimization problems. Applications to dynamical games and PDEs. J. Convex Anal. 15, 485-506 (2008). MR2431407
[4] H. Attouch, A.Cabot, F.Frankel, J.Peypouquet, Alternating proximal algorithms for constrained variational inequalities. Application to domain decomposition for PDE’s. Nonlinear Anal. 74, 7455-7473 (2011). MR2833727
[5] D.P. Bertsekas, E.M. Gafni, Projection methods for variational inequalities with applications to the traffic assignment problem. Math. Progr. Study. 17, 139-159 (1982). MR0654697
[6] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem.Inverse Problem. 18, 441-453 (2002). MR1910248
[7] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problem. 20, 103-120 (2004). MR2044608
[8] Y. Censor, T. Bortfeld, B. Martin, A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353-2365 (2006). Scopus
[9] Y. Censor, T. Elfving, A multiprojection algorithms using Bragman projection in a product space. Numerical Algorithms. 8, 221-239 (1994). MR1309222
[10] Y. Censor, A.Gibali, S. Reich, Algorithms for the split variational inequality problem. Numerical Algorithms. 59 301-323 (2012). MR2873136
[11] Y. Censor, A. Segal, The split common fixed point problem for directed operators. J. Convex Anal. 16, 587-600 (2009). MR2559961
[12] S.S. Chang, R.P. Agarwal, Strong convergence theorems of general split equality problems for quasi-nonexpansive mappings. Journal of Inequalities and Applications, 2014, 2014:367. MR3359099
[13] S.S. Chang, J. Quan, J. Liu, Feasible iterative algorithms and strong convergence theorems for bi-level fixed point problems. J. Nonlinear Sci. Appl. 9, 1515-1528 (2016). MR3452653
[14] C.E.Chidume, S.A.Mutangadura, An example of the Mann iteration method for Lipschitz pseudocontractions. Proc. Amer. Math. Soc. 129, 2359- 2363. (2001) MR1823919
[15] J. Deephoa, J. M. Moreno, P. Kumam, A viscosity of Cesaro mean approximation method for split generalized equilibrium, variational inequality and fixed point problems, J. Nonlinear Sci. Appl. 9, 1475-1496 (2016). Scopus
[16] Q. L. Dong, S. He, J. Zhao, Solving the split equality problem without prior knowledge of operator norms, Optimization. 64, 1887-1906 (2015). MR3361157
[17] M. Eslamian, Hybrid method for equilibrium problems and fixed point problems of finite families of nonexpansive semigroups. Rev. R. Accad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 107, 299-307 (2013). MR3199712
[18] M. Eslamian, General algorithms for split common fixed point problem of demicontractive mappings. Optimization. 65, 443-465 (2016). MR3438119
[19] M. Eslamian, A.Latif, Strong convergence and split common fixed point problem for set-valued operators. J. Nonlinear Convex Anal. 17, 967-986 (2016). MR3520698
[20] M. Eslamian, J.Vahidi, Split Common Fixed Point Problem of Nonexpansive Semigroup. Mediterr. J. Math. 13, 1177-1195 (2016). MR3513163
[21] H. Iiduka, I. Yamada, A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping. SIAM J. Optim. 19, 1881-1893(2009). MR2486054
[22] D.Kinderlehrer, G. Stampaccia, An Iteration to Variational Inequalities and Their Applications. Academic Press, New York (1990).
[23] F. Kohsaka, W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Arch. Math. (Basel). 91, 166-177 (2008). MR2430800
[24] F. Kohsaka, W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM J. Optim. 19, 824-835 (2008). MR2448915
[25] R. Kraikaew, S.Saejung, On split common fixed point problems. J. Math. Anal. Appl. 415, 513-524 (2014). MR3178275
[26] G. Lopez, V. Martìn-Marquez, F. Wang, H. K. Xu, Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Problem. 27, 085004 (2012). MR2948743
[27] D.A. Lorenz, F.S. Schopfer, S. Wenger, The Linearized Bregman Method via Split Feasibility Problems: Analysis and Generalizations. SIAM J. Imajing science. 7, 1237-1262 (2014). MR3215060
[28] P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization} Set-Valued Analysis, 16, 899-912 (2008). MR2466027
[29] P. E. Mainge, A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499-1515 (2008). MR2407025
[30] A. Moudafi, A relaxed alternating CQ algorithm for convex feasibility problems. Nonlinear Anal. 79, 117-121 (2013). MR3005031
[31] A. Moudafi, Alternating CQ-algorithm for convex feasibility and split fixed-point problems. J. Nonlinear Convex Anal. 15, 809-818 (2014). MR3222909
[32] A. Moudafi, E. Al-Shemas, Simultaneous iterative methods for split equality problems and application. Trans. Math. Program. Appl., 1, 1-11 (2013).
[33] P.M.Pardalos, T.M. Rassias, A.A.Khan, Nonlinear Analysis and Variational Problems. Springer, Berlin (2010).
[34] R.T.Rockafellar, R.J-B.,Wets, Variational Analysis, 2nd printing. Springer, New York (2004).
[35] Y. Shehu, O. S. Iyiola, C. D. Enyi, An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces. Numerical Algorithms. 72, 835-864 (2016). MR3529823
[36] Y. Shehu, F.U.Ogbuisi, O.S. Iyiola, Convergence Analysis of an iterative algorithm for fixed point problems and split feasibility problems in certain Banach spaces. Optimization. 65, 299-323 (2016). MR3438112
[37] W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space. J. Nonlinear Convex Anal. 11, 79-88 (2010). MR2729999
[38] W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118, 417-428 (2003). MR2006529
[39] H. K. Xu, Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240-256 (2002). MR1911872
[40] H. K. Xu, Iterative methods for split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Problem. 26, 105018 (2010). MR271977
[41] I. Yamada, The hybrid steepest descent method for the variational inequality problem of the intersection of fixed point sets of nonexpansive mappings. Studies Comput. Math. 8, 473-504 (2001). MR1853237
[42] J. Zhao, Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms. Optimization. 64, 2619-2630 (2015). MR3411824

Home Riv.Mat.Univ.Parma