Abstract
In this paper, we study parafirmly nonexpansive mappings with Bregman distance and weak convergence of underrelaxed sequential parafirmly nonexpansive mappings.
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Von neumann, J., Functional Operators, II: The Geometry of Orthogonal Spaces, Princeton University Press, Princeton, New Jersey, 1950.
Halperin, I., The Product of Projection Operators, Acta Scientiarum Mathematicarum, Vol. 23, pp. 96–99, 1962.
Deutsch, F., Best Approximation in Inner Product Spaces, CMS Books in Mathematics, Springer, New York, NY, Vol. 7, 2001.
Deutsch, F., The Method of Alternating Orthogonal Projections, Approximation Theory, Spline Functions, and Applications (Maratea, 1991); Edited by S. P. Singh, Kluwer Academic Publishers, Dordrecht, Holland, pp. 105–121, 1992.
Bregman, L. M., The Method of Successive Projections for Finding a Common Point of Convex Sets, Soviet Mathematics Doklady, Vol. 6, pp. 688–692, 1965.
Bauschke, H. H., and Borwein, J. M., On Projection Algorithms for Solving Convex Feasibility Problems, SIAM Review, Vol. 38, pp. 367–426, 1996.
Censor, Y., and Zenios, S. A., Parallel Optimization: Theory, Algorithms, and Applications, Oxford University Press, New York, NY, 1997.
Gubin, L. G., Polyak, B. T., and Raik, E. V., The Method of Projections for Finding the Common Point of Convex Sets, USSR Computational Mathematics and Mathematical Physics, Vol. 7, pp. 1–24, 1967.
Bregman, L. M., The Relaxation Method for Finding the Common Point of Convex Sets and Its Application to the Solution of Convex Programming, USSR Computational Mathematics and Mathematical Physics, Vol. 7, pp. 200–217, 1967.
Censor, Y., and Elfving, T., A Mutiprojection Algorithm Using Bregman Projections in a Product Space, Numerical Algorithms, Vol. 8, pp. 221–239, 1994.
Censor, Y., and Lent, A., An Iterative Row-Action Method for Interval Convex Programming, Journal of Optimization Theory and Applications, Vol. 34, pp. 321–353, 1981.
Censor, Y., and Reich, S., Iterations of Paracontractions and Parafirmly Nonexpansive Operators with Applications to Feasibility and Optimization, Optimization, Vol. 37, pp. 323–339, 1996.
Bauschke, H. H., and Borwein, J. M., Legendre Functions and the Method of Random Bregman Projections, Journal of Convex Analysis, Vol. 4, pp. 27–67, 1997.
Alber, Y., and Butnariu, D., Convergence of Bregman Projection Methods for Solving Consistent Convex Feasibility Problems in Reflexive Banach Spaces, Journal of Optimization Theory and Applications, Vol. 92, pp. 33–61, 1997.
Tseng, P., On the Convergence of the Products of Parafirmly Nonexpansive Mappings, SIAM Journal on Optimization, Vol. 2, pp. 425–434, 1992.
Reich, S., A Weak Convergence Theorem for the Alternating Method with Bregman Distances, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Dekker, New York, NY, pp. 313–318, 1996.
Bauschke, H. H., Borwein, J. M., and Lewis, A. S., The Method of Cyclic Projections for Closed Sets in Hilbert Space, Recent Developments in Optimization Theory and Nonlinear Analysis (Jerusalem, 1995); Contempory Mathematics, American Mathematical Society, Providence, Rhode Island, Vol. 204, pp. 1–38, 1997.
Vladimirov, A. A., Nesterov, Y. E., and Chekanov, Y. N., Uniformly Convex Functionals, Vestnik Moskovskogo Universiteta, Seriya XV, Vychislitelprime naya Matematika i Kibernetika, No. 3, pp. 12–23, 1978 (in Russian).
Bauschke, H. H., Borwein, J. M., and Combettes, P. L., Essential Smoothness, Essential Strict Convexity, and Legendre Functions in Banach Spaces, Communications in Contemporary Mathematics, Vol. 3, pp. 615–647, 2001.
Bauschke, H. H., Borwein, J. M., and Combettes, P. L., Bregman Monotone Optimization Algorithms, SIAM Journal on Control and Optimization, Vol. 42, pp. 596–636, 2003.
Butnariu, D., and Iusem, A. N., Totally Convex Functions for Fixed Points Computation and Infinite-Dimensional Optimization, Applied Optimization, Kluwer Academic Publishers, Dordrecht, Holland, Vol. 40, 2000.
Bauschke, H. H., and Lewis, A. S., Dykstra's Algorithm with Bregman Projections: A Convergence Proof, Optimization, Vol. 48, pp. 409–427, 2000.
Kreyszig, E., Introductory Functional Analysis with Applications, John Wiley and Sons, New York, NY, 1978.
Bauschke, H. H., and Combettes, P. L., Construction of Best Bregman Approximations in Reflexive Banach Spaces, Proceedings of the American Mathematical Society, Vol. 131, pp. 3757–3766, 2003.
Censor, Y., Row-Action Methods for Huge and Sparse Systems and Their Applications, SIAM Review, Vol. 23, pp. 444–466, 1981.
Censor, Y., and Herman, G. T., Block-Iterative Algorithms with Underrelaxed Bregman Projection, SIAM Journal on Optimization, Vol. 13, pp. 283–297, 2002.
Bauschke, H. H., Projection Algorithms and Monotone Operators, PhD Thesis, Simon Fraser University, Burnaby, British Columbia, Canada, 1996.
Van tiel, J., Convex Analysis: An Introductory Text, John Wiley and Sons, New York, NY, 1984.
Mhaskar, H. N., and Pai, D. V., Fundamental of Approximation Theory, CRC Press, Boca Raton, Florida, 2000.
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Lee, M.B., Park, S.H. Convergence of Sequential Parafirmly Nonexpansive Mappings in Reflexive Banach Spaces. Journal of Optimization Theory and Applications 123, 549–571 (2004). https://doi.org/10.1007/s10957-004-5723-2
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DOI: https://doi.org/10.1007/s10957-004-5723-2