New algorithm method for solving the variational inequality problem in Hilbert space

  • Authors

    • Zena Hussein Maibed Baghdad University
    2019-11-10
    https://doi.org/10.14419/gjma.v7i2.28900
  • Resolven, Mapping, Non-Spreading, Common, FixediPoint, Strong Convergence.

  • Abstract

    Theipurpose of,thisipaper,is toiintroduce,aiconcept of generalizedinon_spreading,and define a new algorithm,for infinite,families of generalizedinon_spreading,and finite families of resolvent,mappings. Also, We study,the existence,solution of variational inequality,to a commonifixedipoint in Hilbertispaces. The main,results in this paper extendiand generalized,of many knowniresults initheiliterature.

     

     

  • References

    1. [1] K.P.R. Sastry, G.V.R. Babu, Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point, Czechoslovak Math. J. 55 (2005) 817–826. https://doi.org/10.1007/s10587-005-0068-z.

      [2] F.E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, in: Proc. Symp. Pure Math., vol. 18, Amer. Math. Soc., Providence, RI, 1976.

      [3] W.A. Kirk, A fixed point theorem for mappings which do not increase distance, Amer. Math. Monthly 72 (1965) 1004–1006. https://doi.org/10.2307/2313345.

      [4] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974) 147–150. https://doi.org/10.1090/S0002-9939-1974-0336469-5.

      [5] W.R. Mann Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506–510. https://doi.org/10.1090/S0002-9939-1953-0054846-3.

      [6] J.M. Borwein and J.D. Vanderwerff, Convex Functions, Cambridge University Press, 2010.

      [7] R.S. Burachik and A.N. Iusem, Set-Valued Mappings and Enlargements of Monotone Operators, Springer-Verlag, 2008. 24.

      [8] S. Simons, From Hahn-Banach to Monotonicity, Springer-Verlag, 2008.

      [9] C. Z ˘alinescu , Convex Analysis in General Vector Spaces, world Scientific Publishing, 2002. https://doi.org/10.1142/9789812777096.

      [10] H.K.Xu," a nother control condition in an iterative method for nonexpansive mappings, bull, austral. Math. soc, 65 (2002), 109-113. https://doi.org/10.1017/S0004972700020116.

      [11] H.K.Xu," Iterative Algorithm for nonlinear operators",J. london Math.soc .66(2002) 240-256 https://doi.org/10.1112/S0024610702003332.

      [12] A.Moudafi, "Viscosity approximation method for fixed point problems", Journal of Mathematical Analysis and Applications, 241(2000) 46-55.

      [13] H.K.Xu, "Viscosity approximation methods for non-expansive mapping ", J. Math .Anal.Appl. 298 (2004) 279-291. https://doi.org/10.1016/j.jmaa.2004.04.059.

      [14] S. Kamimura, W. Takahashi, "Approximating solutions of maximal monotone operators in Hilbert spaces", J. Approx. Theory 106 (2000) 226–240. https://doi.org/10.1006/jath.2000.3493.

      [15] Z. H. Mabeed, "Strongly Convergence Theorems of Ishikawa Iteration Process with Errors in Banach Space" Journal of Qadisiyah Computer Science and Mathematics, 3(2011)1-8.

      [16] Z. H. Mabeed, Some Convergence Theorems for The Fixed Point in Banach Spaces, Journal of university of Anbar for pure science, 2) 2013(7.2.

      [17] Z. H. Maibed, Strong Convergence of Iteration Processes for Infinite Family of General Extended Mappings, IOP Conf. Series: Journal of Physics: Conf. Series 1003 (2018) 012042. https://doi.org/10.1088/1742-6596/1003/1/012042.

      [18] Zeidler ,"Nonlinear Functional Analysis and Application" New York, (1986). https://doi.org/10.1007/978-1-4612-4838-5.

      [19] H.H. Bauschke and P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer-Verlag, 2011. https://doi.org/10.1007/978-1-4419-9467-7.

      [20] B.E. Rhoades, Comments on two fixed point iteration methods, J. Math. Anal. Appl. 56 (1976) 741–750. https://doi.org/10.1016/0022-247X(76)90038-X.

      [21] W.G. Dotson Jr., On the Mann iterative process, Trans. Amer. Math. Soc. 149 (1970)65–73. https://doi.org/10.1090/S0002-9947-1970-0257828-6.

      [22] R.L. Franks, R.P. Marzec, A theorem on mean value iterations, Proc. Amer. Math. Soc. 30 (1971) 324–326. https://doi.org/10.1090/S0002-9939-1971-0280656-9.

      [23] C.W. Groetsch, A note on segmenting Mann iterates, J. Math. Anal. Appl. 40 (1972) 369–372. https://doi.org/10.1016/0022-247X(72)90056-X.

      [24] S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976) 65–71. https://doi.org/10.1090/S0002-9939-1976-0412909-X.

      [25] A.K. Kalinde, B.E. Rhoades, Fixed point Ishikawa iterations, J. Math. Anal. Appl. 170 (1992) 600–606. https://doi.org/10.1016/0022-247X(92)90040-K.

      [26] K.K. Tan, H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993) 301–308. https://doi.org/10.1006/jmaa.1993.1309.

      [27] M.K. Ghosh, L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings, J. Math. Anal. Appl. 207 (1997) 96–103. https://doi.org/10.1006/jmaa.1997.5268.

      [28] C.E. Chidume, S.A. Mutangadura, An example of the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc. 129 (2001) 2359–2363. https://doi.org/10.1090/S0002-9939-01-06009-9.

      [29] Z. H. Maibed,generalized tupled common fixed point theorems for weakly compatibile mappings in fuzze metric space,(IJCIET)10(2019)255-273.

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  • How to Cite

    Hussein Maibed, Z. (2019). New algorithm method for solving the variational inequality problem in Hilbert space. Global Journal of Mathematical Analysis, 7(2), 15-18. https://doi.org/10.14419/gjma.v7i2.28900

    Received date: 2019-04-19

    Accepted date: 2019-07-06

    Published date: 2019-11-10