Limit theorems of operators by convex combinations of nonexpansive retractions in Banach spaces. (English) Zbl 0904.47045

The paper deals with the so-called convex feasibility problem in Banach space setting. Roughly speaking this problem is formulated as follows: Let \(C\) be a nonempty convex closed subset of a Banach space \(E\) and let \(C_1,C_2,\dots, C_r\) be nonexpansive retracts of \(C\) such that \(\bigcap^r_{i= 1}C_i\neq\emptyset\). Assume that \(T\) is a mapping on \(C\) given by the formula \(T= \sum^r_{i= 1}\alpha_i T_i\), where \(\alpha_i\in(0, 1)\), \(\sum^r_{i= 1}\alpha_i= 1\) and \(T_i= (1-\lambda_i)I+ \lambda_i P_i\), where \(\lambda_i\in(0, 1)\) and \(P_i\) is a nonexpansive retraction of \(C\) onto \(C_i\) \((i= 1,2,\dots,r)\). One can find assumptions concerning the space \(E\) which guarantee that the set \(F(T)\) of fixed points of the mapping \(T\) can be represented as \(F(T)= \bigcap^r_{i= 1}C_i\) and for every \(x\in C\) the sequence \(\{T^nx\}\) converges weakly to an element of \(F(T)\). The authors prove that if \(E\) is a uniformly convex Banach space with a Fréchet differentiable norm (or a reflexive and strictly convex Banach space satisfying the Opial condition) then the convex feasibility problem has a positive solution. Apart from that the problem of finding a common fixed point for a finite commuting family of nonexpansive mappings in a strictly convex and reflexive Banach space is also considered.


47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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[1] 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
[2] 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
[3] Bruck, R.E., A simple proof of the Mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. math., 32, 107-116, (1979) · Zbl 0423.47024
[4] Crombez, G., Image recovery by convex combinations of projections, J. math. anal. appl., 155, 413-419, (1991) · Zbl 0752.65045
[5] Edelstein, M.; O’Brien, R.C., Nonexpansive mappings, asymptotic regularity and successive approximations, J. London math. soc., 17, 547-554, (1978) · Zbl 0421.47031
[6] 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
[7] Ishikawa, S., Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. amer. math. soc., 59, 65-71, (1976) · Zbl 0352.47024
[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] Kitahara, S.; Takahashi, W., Image recovery by convex combinations of sunny nonexpansive retractions, Topol. methods nonlinear anal., 2, 333-342, (1993) · Zbl 0815.47068
[10] 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
[11] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. amer. math. soc., 73, 591-597, (1967) · Zbl 0179.19902
[12] Reich, S., A limit theorem of projections, Linear multilinear algebra, 13, 281-290, (1983) · Zbl 0523.47040
[13] Takahashi, W., Fixed point theorems for families of nonexpansive mappings on unbounded sets, J. math. soc. Japan, 36, 543-553, (1984) · Zbl 0599.47091
[14] Takahashi, W., Nonlinear functional analysis, (1988), Kindai – kagakusha Tokyo
[15] Takahashi, W.; Park, J.Y., On the asymptotic behavior of almost orbits of commutative semigroups in Banach spaces, (), 271-293
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