## Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces.(English)Zbl 1123.47047

The paper gives convergence theorems for approximating fixed points of multivalued nonexpansive nonself mappings by means of viscosity type methods. The main result (Theorem 1) goes as follows. Let $$E$$ be a uniformly convex Banach space with uniformly Gâteaux differentiable norm, $$C$$ be a nonempty closed convex subset of $$E$$ and $$T:C\rightarrow \mathcal{K}(E)$$ be a nonself nonexpansive multivalued mapping (here, $$\mathcal{K}(E)$$ denotes the set of all nonempty compact subsets of $$E$$). Suppose that $$C$$ is a nonexpansive retract of $$E$$ and $$T$$ has only strict fixed points, that is, $$T(y)=\{y\}$$ for all fixed points $$y$$ of $$T$$. For each $$u\in C$$ and $$t\in (0,1)$$, consider the multivalued contraction $$G_t:C\rightarrow \mathcal{K}(E)$$ defined by $G_t=tTx+(1-t)u,\;x\in C,$ and assume that $$G_t$$ has a fixed point $$x_t\in C$$. Then $$T$$ has a fixed point if and only if $$x_t$$ remains bounded as $$t\rightarrow 1$$ and, in this case, $$x_t$$ converges strongly as $$t\rightarrow 1$$ to a fixed point of $$T$$. Several other related results are obtained in the same way or as corollaries.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H04 Set-valued operators 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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### References:

 [1] Barbu, V.; Precupanu, Th., Convexity and optimization in Banach spaces, (1978), Editura Academiei R. S. R. Bucharest · Zbl 0379.49010 [2] Browder, F.E., Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces, Arch. ration. mech. anal., 24, 82-90, (1967) · Zbl 0148.13601 [3] Day, M.M., Normed linear spaces, (1973), Springer-Verlag Berlin, New York · Zbl 0268.46013 [4] Deimling, K., Multivalued differential equations, (1992), Walter de Gruyter Berlin · Zbl 0760.34002 [5] Goebel, K.; Reich, S., Uniform convexity, hyperbolic geometry and nonexpansive mappings, (1984), Marcel Dekker New York, Basel · Zbl 0537.46001 [6] Ha, K.S.; Jung, J.S., Strong convergence theorems for accretive operators in Banach spaces, J. math. anal. appl., 147, 330-339, (1990) · Zbl 0712.47045 [7] Halpern, B., Fixed points of nonexpansive maps, Bull. amer. math. soc., 73, 957-961, (1967) · Zbl 0177.19101 [8] Jung, J.S.; Kim, S.S., Strong convergence theorems for nonexpansive nonself-mappings in Banach spaces, Nonlinear anal., 33, 321-329, (1998) · Zbl 0988.47033 [9] Jung, J.S.; Kim, T.H., Strong convergence of approximating fixed points for nonexpansive nonself-mappings in Banach spaces, Kodai math. J., 21, 259-272, (1998) · Zbl 0928.47040 [10] Kim, G.E.; Takahashi, W., Strong convergence theorems for nonexpansive nonself-mappings in Banach spaces, Nihonkai math. J., 7, 63-72, (1996) · Zbl 0997.47512 [11] Kim, T.H.; Jung, J.S., Approximating fixed points of nonlinear mappings in Banach spaces, Annal. univ. mariae Curie-skłodowska, sect. A math. l LI.2, 14, 149-165, (1997) · Zbl 1012.47034 [12] López Acedo, G.; Xu, H.K., Remarks on multivalued nonexpansive mappings, Soochow J. math., 21, 107-115, (1995) · Zbl 0826.47037 [13] Nadler, S., Multivalued contraction mappings, Pacific J. math., 30, 475-488, (1969) · Zbl 0187.45002 [14] Pietramala, P., Convergence of approximating fixed points sets for multivalued nonexpansive mappings, Comment. math. univ. carolinae., 32, 697-701, (1991) · Zbl 0756.47039 [15] Reich, S., Approximate selections, best approximations, fixed points, and invariant sets, J. math. anal. appl., 62, 104-113, (1978) · Zbl 0375.47031 [16] Reich, S., Product formulas, nonlinear semigroups and accretive operators, J. funct. anal., 36, 147-168, (1980) · Zbl 0437.47048 [17] Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. math. anal. appl., 75, 287-292, (1980) · Zbl 0437.47047 [18] Reich, S., Nonlinear semigroups, holomorphic mappings, and integral equations, (), 307-324, Part 2 [19] Sahu, D.R., Strong convergence theorems for nonexpansive type and non-self multi-valued mappings, Nonlinear anal., 37, 401-407, (1999) · Zbl 0938.47039 [20] Singh, S.P.; Watson, B., On approximating fixed points, Proc. symp. pure math., 45, 2, 393-395, (1988) [21] Takahashi, W.; Ueda, Y., On reich’s strong convergence theorems for resolvents of accretive operators, J. math. anal. appl., 104, 546-553, (1984) · Zbl 0599.47084 [22] Takahashi, W.; Jeong, D.H., Fixed point theorem for nonexpansive semigroups on Banach spaces, Proc. amer. math. soc., 122, 1175-1179, (1994) · Zbl 0818.47055 [23] Takahashi, W.; Kim, G.E., Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces, Nonlinear anal., 32, 447-454, (1998) · Zbl 0947.47049 [24] Xu, H.K., Inequalities in Banach spaces with applications, Nonlinear anal., 16, 1127-1138, (1991) · Zbl 0757.46033 [25] Xu, H.K., Approximating curves of nonexpansive nonself mappings in Banach spaces, C. R. acad. sci. Paris, ser. I., 325, 179-184, (1997) [26] H.K. Xu, Metric Fixed Point Theory for Multivalued Mappings, Dissertationes Math. 389, 2000 [27] Xu, H.K., Multivalued nonexpansive mappings in Banach spaces, Nonlinear anal., 43, 693-706, (2001) · Zbl 0988.47034 [28] Xu, H.K.; Yin, X.M., Strong convergence theorems for nonexpansive nonself-mappings, Nonlinear anal., 24, 223-228, (1995) · Zbl 0826.47038
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