Split equality problem with equilibrium problem, variational inequality problem, and fixed point problem of nonexpansive semigroups
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2014
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Authors
Abdul Latif
- Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Mohammad Eslamian
- Department of Mathematics, University of Science and Technology of Mazandaran, P. O. Box 48518-78413, Behshahr, Iran.
Abstract
In this paper, we present a new algorithm for the split equality problem for finding a common element of solution of
equilibrium problem, solution of variational inequality problem for monotone and Lipschitz continuous operators, and common
fixed point of nonexpansive semigroups. We establish strong convergence of the algorithm in an infinite dimensional Hilbert
spaces. Our results improve and generalize some recent results in the literature.
Share and Cite
ISRP Style
Abdul Latif, Mohammad Eslamian, Split equality problem with equilibrium problem, variational inequality problem, and fixed point problem of nonexpansive semigroups, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 6, 3217--3230
AMA Style
Latif Abdul, Eslamian Mohammad, Split equality problem with equilibrium problem, variational inequality problem, and fixed point problem of nonexpansive semigroups. J. Nonlinear Sci. Appl. (2017); 10(6):3217--3230
Chicago/Turabian Style
Latif, Abdul, Eslamian, Mohammad. "Split equality problem with equilibrium problem, variational inequality problem, and fixed point problem of nonexpansive semigroups." Journal of Nonlinear Sciences and Applications, 10, no. 6 (2017): 3217--3230
Keywords
- Split equality problem
- equilibrium problem
- variational inequality
- nonexpansive semigroups
- fixed point.
MSC
References
-
[1]
A. Aleyner, Y. Censor, Best approximation to common fixed points of a semigroup of nonexpansive operators, J. Nonlinear Convex Anal., 6 (2005), 137–151.
-
[2]
P. N. Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optimization, 62 (2013), 271–283.
-
[3]
H. Attouch, J. Bolte, P. Redont, A. Soubeyran, Alternating proximal algorithms for weakly coupled convex minimization problems, Applications to dynamical games and PDE’s, J. Convex Anal., 15 (2008), 485–506.
-
[4]
H. Attouch, A. Cabot, P. Frankel, J. Peypouquet, Alternating proximal algorithms for linearly constrained variational inequalities: application to domain decomposition for PDE’s, Nonlinear Anal., 74 (2011), 7455–7473.
-
[5]
D. P. Bertsekas, E. M. Gafni, Projection methods for variational inequalities with application to the traffic assignment problem, Nondifferential and variational techniques in optimization, Lexington, Ky., (1980), Math. Programming Stud., 17 (1982), 139–159.
-
[6]
E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123–145.
-
[7]
F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A., 54 (1965), 1041–1044.
-
[8]
C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441–453.
-
[9]
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103–120.
-
[10]
Y. Censor, T. Bortfeld, B. Martin, A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353–2365.
-
[11]
Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221–239.
-
[12]
Y. Censor, A. Gibali, S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301–323.
-
[13]
Y. Censor, A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587–600.
-
[14]
R.-D. Chen, Y.-Y. Song, Convergence to common fixed point of nonexpansive semigroups, J. Comput. Appl. Math., 200 (2007), 566–575.
-
[15]
C. E. Chidume, S. A. Mutangadura, An example of the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc., 129 (2001), 2359–2363.
-
[16]
P. L. Combettes, S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117–136.
-
[17]
Q.-L. Dong, S.-N. He, J. Zhao, Solving the split equality problem without prior knowledge of operator norms, Optimization, 64 (2015), 1887–1906.
-
[18]
M. Eslamian, Hybrid method for equilibrium problems and fixed point problems of finite families of nonexpansive semigroups, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 107 (2013), 299–307.
-
[19]
M. Eslamian, General algorithms for split common fixed point problem of demicontractive mappings, Optimization, 65 (2016), 443–465.
-
[20]
M. Eslamian, A. Abkar, Viscosity iterative scheme for generalized mixed equilibrium problems and nonexpansive semigroups, TOP, 22 (2014), 554–570.
-
[21]
M. Eslamian, A. Latif, General split feasibility problems in Hilbert spaces, Abstr. Appl. Anal., 2013 (2013), 6 pages.
-
[22]
M. Eslamian, J. Vahidi, Split common fixed point problem of nonexpansive semigroup, Mediterr. J. Math., 13 (2016), 1177–1195.
-
[23]
F. Facchinei, J.-S. Pang, Finite-dimensional variational inequalities and complementarity problems, Springer Series in Operations Research, Springer-Verlag, New York (2003)
-
[24]
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 (2009), 1881–1893.
-
[25]
D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London (1980)
-
[26]
G. M. Korpelevič, An extragradient method for finding saddle points and for other problems, (Russian) Èkonom. i Mat. Metody, 12 (1976), 747–756.
-
[27]
A. Latif, M. Eslamian, Strong convergence and split common fixed point problem for set-valued operators, J. Nonlinear Convex Anal., 17 (2016), 967–986.
-
[28]
A. Latif, J. Vahidi, M. Eslamian, Strong convergence for generalized multiple-set split feasibility problem, Filomat, 30 (2016), 459–467.
-
[29]
A. T.-M. Lau, N. Shioji, W. Takahashi, Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces, J. Funct. Anal., 161 (1999), 62–75.
-
[30]
G. López, V. Martín-Márquez, F.-H. Wang, H.-K. Xu, Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Problems, 27 (2012), 18 pages.
-
[31]
P. E. Maingé, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499–1515.
-
[32]
P. E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899–912.
-
[33]
A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Anal., 79 (2013), 117–121.
-
[34]
A. Moudafi, Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal., 15 (2014), 809–818.
-
[35]
A. Moudafi, E. Al-Shemas, Simultaneous iterative methods for split equality problem, Trans. Math. Program. Appl., 1 (2013), 1–11.
-
[36]
A. Moudafi, M. Théra, Proximal and dynamical approaches to equilibrium problems, Ill-posed variational problems and regularization techniques, Trier,/ (1998), Lecture Notes in Econom. and Math. Systems, Springer, Berlin, 477 (1999), 187–201.
-
[37]
P. M. Pardalos (ed.), T. M. Rassias (ed.), A. A. Khan (ed.), Nonlinear analysis and variational problems, In honor of George Isac, Springer Optimization and Its Applications, Springer, New York (2010)
-
[38]
R. T. Rockafellar, R. J.-B. Wets, Variational analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin (1998)
-
[39]
G. Rodé, An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space, J. Math. Anal. Appl., 85 (1982), 172–178.
-
[40]
H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256.
-
[41]
H. Zegeye, N. Shahzad, Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings, Comput. Math. Appl., 62 (2011), 4007–4014.
-
[42]
J. Zhao, Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms, Optimization, 64 (2014), 2619–2630.