Strong Convergence Results for Hierarchical Circularly Iterative Method about Hierarchical Circularly Optimization
Author(s)
Hongbo Liu*
ABSTRACT
An hierarchical circularly iterative method is introduced for solving a system of variational circularly inequalities with set of fixed points of strongly quasi-nonexpansive mapping problems in this paper. Under some suitable conditions, strong convergence results for the hierarchical circularly iterative sequence are proved in the setting of Hilbert spaces. Our scheme can be regarded as a more general variant of the algorithm proposed by Maingé.
Cite this paper
H. Liu, "Strong Convergence Results for Hierarchical Circularly Iterative Method about Hierarchical Circularly Optimization," Advances in Pure Mathematics, Vol. 3 No. 7, 2013, pp. 615-620. doi: 10.4236/apm.2013.37079.
H. Liu, "Strong Convergence Results for Hierarchical Circularly Iterative Method about Hierarchical Circularly Optimization," Advances in Pure Mathematics, Vol. 3 No. 7, 2013, pp. 615-620. doi: 10.4236/apm.2013.37079.
References
[1] D. Kinderlehrer and G. Stampacchia, “An Introduction to Variational Inequalities and Their Applications,” Classics in Applied Mathematics, Vol. 31, SIAM, Philadelphia, 2000. http://dx.doi.org/10.1137/1.9780898719451 [2] E. Buzogány, I. Mezei and V. Varga, “Two-Variable Variational-Hemivariational Inequalities,” Studia Universitatis Babes-Bolyai Mathematica, Vol. 47, No. 3, 2002, pp. 31-41. [3] E. Blum and W. Oettli, “From Optimization and Variational Inequalities to Equilibrium Problems,” Mathematics Student-India, Vol. 63, No. 1, 1994, pp. 123-145. [4] R. U. Verma, “Projection Methods, Algorithms and A New System of Nonlinear Variational Inequalities,” Computers & Mathematics with Applications, Vol. 41, No. 7-8, 2001, pp. 1025-1031.
http://dx.doi.org/10.1016/S0898-1221(00)00336-9 [5] D. Kinderlehrer and G. Stampacchia, “An Introduction to Variational Inequalities and Their Applications,” Academic Press, New York, 1980. [6] M. A. Noor and K. I. Noor, “Sensitivity Analysis of Quasi Variational Inclusions,” Journal of Mathematical Analysis and Applications, Vol. 236, No. 2, 1999, pp. 290-299. http://dx.doi.org/10.1006/jmaa.1999.6424 [7] S. S. Chang, “Set-Valued Variational Inclusions in Banach Spaces,” Journal of Mathematical Analysis and Applications, Vol. 248, No. 2, 2000, pp. 438-454.
http://dx.doi.org/10.1006/jmaa.2000.6919 [8] S. S. Chang, “Existence and Approximation of Solutions of Set-Valued Variational Inclusions in Banach Spaces,” Nonlinear Analysis, Vol. 47, No. 1, 2001, pp. 583-494.
http://dx.doi.org/10.1016/S0362-546X(01)00203-6 [9] V. F. Demyanov, G. E. Stavroulakis, L. N. Polyakova and P. D. Panagiotopoulos, “Quasidifferentiability and Nonsmooth Modeling in Mechanics, Engineering and Economics,” Kluwer Academic, Dordrecht, 1996. [10] P. E. Maingé and A. Mouda, “Strong Convergence of an Iterative Method for Hierarchical Fixed Point Problems,” Pacific Journal of Optimization, Vol. 3, No. 3, 2007, pp. 529-538. [11] R. Kraikaew and S. Saejung, “On Maingé’s Approach for Hierarchical Optimization Problems,” Journal of Optimization Theory and Applications, Vol. 154, No. 1, 2012, pp. 71-87. http://dx.doi.org/10.1007/s10957-011-9982-4 [12] S. S. Zhang, Joseph H. W. Lee and C. K. Chan, “Algorithms of Common Solutions for Quasi Variational Inclusion and Fixed Point Problems,” Applied Mathematics and Mechanics, Vol. 29, No. 5, 2008, pp. 1-11.
http://dx.doi.org/10.1007/s10483-008-0101-7 [13] T. L. Hicks and J. D. Kubicek, “On the Mann Iteration Process in a Hilbert Space,” Journal of Mathematical Analysis and Applications, Vol. 59, No. 3, 1977, pp. 498-504. http://dx.doi.org/10.1016/0022-247X(77)90076-2 [14] W. Takahashi, “Introduction to Nonlinear and Convex Analysis,” Yokohama Publishers, Yokohama, 2009.
[1] D. Kinderlehrer and G. Stampacchia, “An Introduction to Variational Inequalities and Their Applications,” Classics in Applied Mathematics, Vol. 31, SIAM, Philadelphia, 2000. http://dx.doi.org/10.1137/1.9780898719451 [2] E. Buzogány, I. Mezei and V. Varga, “Two-Variable Variational-Hemivariational Inequalities,” Studia Universitatis Babes-Bolyai Mathematica, Vol. 47, No. 3, 2002, pp. 31-41. [3] E. Blum and W. Oettli, “From Optimization and Variational Inequalities to Equilibrium Problems,” Mathematics Student-India, Vol. 63, No. 1, 1994, pp. 123-145. [4] R. U. Verma, “Projection Methods, Algorithms and A New System of Nonlinear Variational Inequalities,” Computers & Mathematics with Applications, Vol. 41, No. 7-8, 2001, pp. 1025-1031.
http://dx.doi.org/10.1016/S0898-1221(00)00336-9 [5] D. Kinderlehrer and G. Stampacchia, “An Introduction to Variational Inequalities and Their Applications,” Academic Press, New York, 1980. [6] M. A. Noor and K. I. Noor, “Sensitivity Analysis of Quasi Variational Inclusions,” Journal of Mathematical Analysis and Applications, Vol. 236, No. 2, 1999, pp. 290-299. http://dx.doi.org/10.1006/jmaa.1999.6424 [7] S. S. Chang, “Set-Valued Variational Inclusions in Banach Spaces,” Journal of Mathematical Analysis and Applications, Vol. 248, No. 2, 2000, pp. 438-454.
http://dx.doi.org/10.1006/jmaa.2000.6919 [8] S. S. Chang, “Existence and Approximation of Solutions of Set-Valued Variational Inclusions in Banach Spaces,” Nonlinear Analysis, Vol. 47, No. 1, 2001, pp. 583-494.
http://dx.doi.org/10.1016/S0362-546X(01)00203-6 [9] V. F. Demyanov, G. E. Stavroulakis, L. N. Polyakova and P. D. Panagiotopoulos, “Quasidifferentiability and Nonsmooth Modeling in Mechanics, Engineering and Economics,” Kluwer Academic, Dordrecht, 1996. [10] P. E. Maingé and A. Mouda, “Strong Convergence of an Iterative Method for Hierarchical Fixed Point Problems,” Pacific Journal of Optimization, Vol. 3, No. 3, 2007, pp. 529-538. [11] R. Kraikaew and S. Saejung, “On Maingé’s Approach for Hierarchical Optimization Problems,” Journal of Optimization Theory and Applications, Vol. 154, No. 1, 2012, pp. 71-87. http://dx.doi.org/10.1007/s10957-011-9982-4 [12] S. S. Zhang, Joseph H. W. Lee and C. K. Chan, “Algorithms of Common Solutions for Quasi Variational Inclusion and Fixed Point Problems,” Applied Mathematics and Mechanics, Vol. 29, No. 5, 2008, pp. 1-11.
http://dx.doi.org/10.1007/s10483-008-0101-7 [13] T. L. Hicks and J. D. Kubicek, “On the Mann Iteration Process in a Hilbert Space,” Journal of Mathematical Analysis and Applications, Vol. 59, No. 3, 1977, pp. 498-504. http://dx.doi.org/10.1016/0022-247X(77)90076-2 [14] W. Takahashi, “Introduction to Nonlinear and Convex Analysis,” Yokohama Publishers, Yokohama, 2009.