Korean J. Math.  Vol 27, No 2 (2019)  pp.297-327
DOI: https://doi.org/10.11568/kjm.2019.27.2.297

A new mapping for finding a common solution of split generalized equilibrium problem, variational inequality problem and fixed point problem

Mohammad Farid, Kaleem Raza Kazmi

Abstract


In this paper, we introduce and study a general iterative algorithm to approximate a common solution of split generalized equilibrium problem, variational inequality problem and fixed point problem for a finite family of nonexpansive mappings in real Hilbert spaces. Further, we prove a strong convergence theorem for the sequences generated by the proposed iterative scheme. Finally, we derive some consequences from our main result. The results presented in this paper extended and unify many of the previously known results in this area.

Keywords


Split generalized equilibrium problem; Variational inequality problem; Fixed-point problem; Nonexpansive mapping

Subject classification

49J30, 47H10, 47H17, 90C99

Sponsor(s)



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References


H.H. Bauschke and P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, London, (2011). (Google Scholar)

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1994), 123–145. (Google Scholar)

C. Byrne, Y. Censor, A. Gibali and S. Reich, Weak and strong convergence of algorithms for the split common null point problem, J. Nonlinear Convex Anal., 13 (4) (2012), 759–775. (Google Scholar)

L.C. Ceng and J.C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math., 214 (2008), 186–201. (Google Scholar)

Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Physics in Medicine and Biology, 51 (2006), 2353–2365. (Google Scholar)

Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301–323. (Google Scholar)

P.L. Combettes and S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117–136. (Google Scholar)

B. Djafari Rouhani, K. R. Kazmi and Mohd Farid, Common solutions to some systems of variational inequalities and fixed point problems, Fixed Point Theory, 18 (1) (2017), 167–190. (Google Scholar)

P. Hartman and G. Stampacchia, On some non-linear elliptic differential functional equation, Acta Mathematica, 115 (1966), 271–310. (Google Scholar)

A. Kangtunyakarn and S. Suantai, A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings, Nonlinear Analysis, 71 (10) (2009), 4448–4460. (Google Scholar)

K.R. Kazmi, A. Khaliq and A. Raouf, Iterative approximation of solution of generalized mixed set-valued variational inequality problem, Math. Inequal. Appl., 10 (2007), 677–691. (Google Scholar)

K.R. Kazmi and S.H. Rizvi, Iterative algorithms for generalized mixed equilibrium problems, J. Egyptian Math. Soc., 21 (3) (2013), 340–345. (Google Scholar)

K.R. Kazmi and S.H. Rizvi, Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem, J. Egyptian Math. Soc., 21 (2013), 44–51. (Google Scholar)

K.R. Kazmi and S.H. Rizvi, An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping, Optimization Let- ters, 8 (3) (2014), 1113–1124. (Google Scholar)

G. Marino and H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318 (2006), 43–52. (Google Scholar)

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

A. Moudafi, The split common fixed point problem for demicontractive mappings, Inverse Problems, 26 (5) (2010), 6 Pages. (Google Scholar)

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (4) (1967), 595–597. (Google Scholar)

R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Transactions of the American Mathematical Society, 149 (1970), 75-88. (Google Scholar)

T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227–239. (Google Scholar)

S. Takahashi and W. Takahashi, Viscosity approximation method for equilibrium problems and fixed point problems in Hilbert space, J. Math. Anal. Appl., 331 (2007), 506–515. (Google Scholar)

H.K. Xu, Viscosity approximation method for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279–291. (Google Scholar)


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