## Multivalued nonexpansive mappings in Banach spaces.(English)Zbl 0988.47034

Using the technique of asymptotic centers a few fixed point theorems are proved for nonexpansive multivalued mappings acting from a nonempty closed bounded convex subset $$E$$ of a Banach space $$X$$ into the family of nonempty compact convex subsets of $$X$$. It is assumed that considered mappings satisfy an extra condition expressed in terms of the so-called inwardness.

### MSC:

 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H04 Set-valued operators 47H10 Fixed-point theorems
Full Text:

### References:

 [1] Caristi, Ca.J., Fixed point theorems for mappings satisfying inwardness conditions, Trans. amer. math. soc., 215, 241-251, (1976) · Zbl 0305.47029 [2] K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992. · Zbl 0760.34002 [3] Downing, D.; Kirk, W.A., Fixed point theorems for set-valued mappings in metric and Banach spaces, Math. japonica, 22, 99-112, (1977) · Zbl 0372.47030 [4] Edelstein, M., The construction of an asymptotic center with a fixed point property, Bull. amer. math. soc., 78, 206-208, (1972) · Zbl 0231.47029 [5] Goebel, K., On a fixed point theorem for multivalued nonexpansive mappings, Ann. univ. M. Curie-sklowdska, 29, 70-72, (1975) [6] Huff, R.E., Banach spaces which are nearly uniformly convex, Rocky mountain J. math., 10, 743-749, (1980) · Zbl 0505.46011 [7] J.L. Kelly, General Topology, van Nostrand, Princeton, NJ, 1955. [8] W.A. Kirk, Nonexpansive mappings in product spaces, set-valued mappings and k-uniform rotundity, in: F.E. Browder (Ed.), Non-linear Functional Analysis and Applications, Proceedings of the Symposium Pure Mathematics, Vol. 45, Part 2, American Mathematical Society, Providence, RI, 1986, pp. 51-64. · Zbl 0594.47048 [9] Kirk, W.A.; Massa, S., Remarks on asymptotic and Chebyshev centers, Houston J. math., 16, 357-364, (1990) · Zbl 0729.47053 [10] Kuczumow, T.; Prus, S., Compact asymptotic centers and fixed points of multivalued nonexpansive mappings, Houston J. math., 16, 465-468, (1990) · Zbl 0724.47033 [11] Lami Dozo, E., Multivalued nonexpansive mappings and Opial’s condition, Proc. amer. math. soc., 38, 286-292, (1973) · Zbl 0268.47060 [12] Lim, T.C., A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space, Bull. amer. math. soc., 80, 1123-1126, (1974) · Zbl 0297.47045 [13] Lim, T.C., Remarks on some fixed point theorems, Proc. amer. math. soc., 60, 179-182, (1976) · Zbl 0346.47046 [14] Nadler, S.B., Multivalued contraction mappings, Pacific J. math., 30, 475-488, (1969) · Zbl 0187.45002 [15] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. amer. math. soc., 73, 595-597, (1967) · Zbl 0179.19902 [16] Reich, S., Approximate selections, best approximations, fixed points, and invariant sets, J. math. anal. appl., 62, 104-113, (1978) · Zbl 0375.47031 [17] Sullivan, F., A generalization of uniformly rotund Banach spaces, Can. J. math., 31, 628-636, (1979) · Zbl 0422.46011 [18] Xu, H.K., Inequalities in Banach spaces with applications, Nonlinear anal., 16, 1127-1138, (1991) · Zbl 0757.46033 [19] E. Zeidler, Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems, Springer, New York, 1986. · Zbl 0583.47050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.