Convergence theorems for nonexpansive mappings and feasibility problems. (English) Zbl 0971.47040

The authors introduce an iteration scheme given by finite nonexpansive mappings in Banach spaces and prove some convergence theorems which are connected with the problem of image recovery. Using these results, they study common fixed points of finite nonexpansive operators.


47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J25 Iterative procedures involving nonlinear operators
65J15 Numerical solutions to equations with nonlinear operators
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[1] Crombez, G., Image recovery by convex combinations of projections, J. math. anal. appl., 155, 413-419, (1991) · Zbl 0752.65045
[2] Kitahara, S.; Takahashi, W., Image recovery by convex combinations for sunny nonexpansive retractions, Topol. methods nonlinear anal., 2, 333-342, (1993) · Zbl 0815.47068
[3] Takahashi, W.; Tamura, T., Limit theorems of operators by convex combinations for nonexpansive retractions in Banach spaces, J. approximation theory, 91, 386-397, (1997) · Zbl 0904.47045
[4] Takahashi, W., Fixed point theorems and nonlinear ergodic theorems for nonlinear semigroups and their applications, Nonlinear analysis, 30, 1283-1293, (1997) · Zbl 0897.47049
[5] W. Takahashi, Weak and strong convergence theorems for families of nonexpansive mappings and their applications, In Proceedings of the Workshop on Fixed Point Theory, Kazimiers Dolny, Poland (to appear). · Zbl 1012.47029
[6] W. Takahashi and T. Tamura, Convergence theorems for a pair of nonexpansive mappings, J. Convex Analysis (to appear). · Zbl 0916.47042
[7] Das, G.; Debata, J.P., Fixed points of quasi-nonexpansive mappings, Indian J. pure appl. math., 17, 1263-1269, (1986) · Zbl 0605.47054
[8] Kirk, W.A., A fixed point theorem for mappings which do not increase distances, Amer. math. monthly, 72, 1004-1006, (1965) · Zbl 0141.32402
[9] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. amer. math. soc., 73, 591-597, (1967) · Zbl 0179.19902
[10] Edelstein, M.; O’Brien, R.C., Nonexpansive mappings, asymptotic regularity and successive approximations, J. London math. soc., 17, 547-554, (1978) · Zbl 0421.47031
[11] Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. math. anal. appl., 67, 274-276, (1979) · Zbl 0423.47026
[12] Lau, A.T.; Takahashi, W., Weak convergence and non-linear ergodic theorems for reversible semigroups of nonexpansive mappings, Pacific J. math., 126, 277-294, (1987) · Zbl 0587.47058
[13] W. Takahashi and G.E. Kim, Approximating fixed points of nonexpansive mappings in Banach spaces, Math. Japonica (to appear). · Zbl 0913.47056
[14] Takahashi, W.; Park, J.Y., On the asymptotic behavior of almost orbits of commutative semigroups in Banach spaces, (), 271-293
[15] Schu, S., Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. austral. math. soc., 43, 153-159, (1991) · Zbl 0709.47051
[16] Browder, F.E., ()
[17] Takahashi, W., Fixed point theorems for families of nonexpansive mappings on unbounded sets, J. math. soc. Japan, 36, 543-553, (1984) · Zbl 0599.47091
[18] Takahashi, W., Nonlinear functional analysis, (1988), Kindai-Kagakusha Tokyo, (in Japanese)
[19] Bruck, R.E., Properties of fixed-point sets of nonexpansive mappings in Banach spaces, Trans. amer. math. soc., 179, 251-262, (1973) · Zbl 0265.47043
[20] Bruck, R.E., A common fixed point theorem for a commuting family of nonexpansive mappings, Pacific J. math., 53, 59-71, (1974) · Zbl 0312.47045
[21] Hirano, N.; Kido, K.; Takahashi, W., The existence of nonexpansive retractions in Banach spaces, J. math. soc. Japan, 38, 1-7, (1986) · Zbl 0594.47052
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