Abstract
Using the concept of variational inequality method, we introduce the iterative scheme for finding a common element of the set of fixed point of a \(\kappa \)-strictly pseudo-contractive map** and four sets of solutions of variational inequality problems. Furthermore, by using our main result, we give the numerical examples for supporting our results.
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References
Lions, J.L., Stampacchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20, 493–517 (1967)
Kinderlehrer, D., Stampaccia, G.: An Iteration to Variational Inequalities and Their Applications. Academic Press, New York (1990)
Geobel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)
Iiduka, H., Takahashi, W.: Strong convergence theorems for nonexpansive nonself-map**s and inverse-strongly-monotone map**s. J. Convex Anal. 11(1), 69–79 (2004)
Kangtunyakarn, A.: A new iterative scheme for fixed point problems of infinite family of \(\kappa _{i}\)-pseudo-contractive map**s, equilibrium problem, variational inequality problems. J. Glob. Optim. (2012). https://doi.org/10.1007/s10898-012-9925-0
Kangtunyakarn, A.: Convergence theorem of \(\kappa \)-strictly pseudo-contractive map** and a modification of generalized equilibrium problems. Fixed Point Theory Appl. 2012, 89 (2012)
Opial, Z.: Weak convergence of the sequence of successive approximation of nonexpansive map**s. Bull. Am. Math. Soc. 73, 591–597 (1967)
Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
Xu, H.K.: An iterative approach to quadric optimization. J. Optim. Theory Appl. 116, 659–678 (2003)
Acedo, G.L., Xu, H.K.: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 67, 2258–2271 (2006)
Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive map**s and monotone map**s. J. Optim. Theory Appl. 118, 417–428 (2003)
Iiduka, H.: Fixed point optimization algorithm and its application to network bandwidth allocation. J. Comput. Appl. Math. 236, 1733–1742 (2012)
Herman, G.T.: Fundamentals of Computerized Tomography: Image Reconstruction from Projections, vol. 2. Springer, New York (2009)
Stark, H., Yang, Y.: Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics. Wiley, New York (1998)
Li, X.: Fine-granularity and spatially adaptive regularization for projection based image deblurring. IEEE Trans. Image Process. 20(4), 971–983 (2011)
Beck, A., Teboulle, M.: Fast gradient-based algorithms for constrained total variation de-noising and deblurring problems. IEEE Trans. Image Process. 18(11), 2419–2434 (2009)
Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrium Problems: Nonsmooth Optimization and Variational Inequalities Models, vol. 58. Springer, Berlin (2002)
Pardalos, P.M., Rassias, T.M., Khan, A.A.: Nonlinear Analysis and Variational Problems. Springer, New York (2010)
Floudas, C.A., Pardalos, P.M.: Encyclopedia of Optimization, 2nd edn. Springer, Berlin (2009)
Ceng, L.C., Yao, J.C.: Iterative algorithm for generalized set-valued strong nonlinear mixed variational-like inequalities. J. Optim. Theory Appl. 124, 725–738 (2005)
Yao, J.C., Chadli, O.: Pseudomonotone Complementarity Problems and Variational Inequalities. Handbook of Generalized Convexity and Monotonicity, pp. 501–558. Kluwer Academic, Dordrecht (2005)
Wang, G.Q., Cheng, S.S.: Fixed point theorems arising from seeking steady states of neural networks. Appl. Math. Model. 33, 499–506 (2009)
Roberts, J.S.: Artificial Neural Networks. McGraw-Hill Company, Singapore (1997)
Haykin, S.: Neural Networks: A Comprehensive Foundation. Macmillan Company, Eaglewood Cliffs, NJ (1994)
Acknowledgements
This research is supported by the Research Administration Division of King Mongkut’s Institute of Technology Ladkrabang and the Thailand Research Fund under the research project RTA578007.
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Suwannaut, S., Suantai, S. & Kangtunyakarn, A. The method for solving variational inequality problems with numerical results. Afr. Mat. 30, 311–334 (2019). https://doi.org/10.1007/s13370-018-0649-2
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DOI: https://doi.org/10.1007/s13370-018-0649-2