Ding, X. P.; He, Y. R. Fixed point theorems for metrically weakly inward set-valued mappings. (English) Zbl 0949.47045 J. Appl. Anal. 5, No. 2, 283-293 (1999). The paper is devoted to the existence of a fixed point of set-valued mappings satisfying some metrically inward conditions. The concept of metrically weakly inward set-valued mapping, introduced by the authors is compared to other inward type conditions, in Section 2. The main results of the paper, Theorem 3.1 and Theorem 3.2 in Section 3, are proved by an argument based on Caristi’s fixed point theorem. Theorem 3.1 shows that any u.s.c. set-valued mapping \(T\) of \(K\) into \(CB(X)\) satisfying both a inward and a contraction condition, does possess a fixed point (\(K\) is a nonempty complete subset of a metric space \((X, d)\), while \(CB(X)\) denotes the family of all nonempty bounded closed subsets of \(X\)). The statement of Theorem 3.2 is as follows: Let \(K\) be a complete subset of a metric space \((X, d)\) and \(T\) a metrically weakly inward set-valued \(k\)-contraction mapping of \(K\) into \(CB(X)\). Then \(T\) has a fixed point in \(K\). These results and their corollaries improve and generalize various other results in the literature, as shown by means of Example 3.1. Reviewer: Vasile Berinde (Baia Mare) Cited in 5 Documents MSC: 47H10 Fixed-point theorems 47H04 Set-valued operators 54C60 Set-valued maps in general topology 54E40 Special maps on metric spaces Keywords:metric space; metrically weakly inward set-valued mapping; fixed point theorem; Caristi’s fixed point theorem PDFBibTeX XMLCite \textit{X. P. Ding} and \textit{Y. R. He}, J. Appl. Anal. 5, No. 2, 283--293 (1999; Zbl 0949.47045) Full Text: DOI References: [1] Banach S., Fund. Math. 3 pp 133– (1922) [2] Caristi J., Trans. Amer. Math. Soc. 215 pp 241– (1976) · doi:10.1090/S0002-9947-1976-0394329-4 [3] Downing D., Math. Japon. 22 pp 99– (1977) [4] Halpern B.R., Trans. Amer. Math. Soc. 62 pp 353– (1968) · doi:10.1090/S0002-9947-1968-0221345-0 [5] Jachymski J., J. Math. Anal. Appl. 227 pp 55– (1998) · Zbl 0916.47044 · doi:10.1006/jmaa.1998.6074 [6] Martinez-Yanez C., Nonlinear Anal. 16 pp 847– (1991) · Zbl 0735.47032 · doi:10.1016/0362-546X(91)90148-T [7] Nadler S.B., Pacific J. Math. 30 pp 475– (1969) [8] Reich S., J. Math. Anal. Appl. 62 pp 104– (1978) · Zbl 0375.47031 · doi:10.1016/0022-247X(78)90222-6 [9] Song W., Appl. Math. J. Chinese Univ. Ser. B 10 pp 463– (1995) · Zbl 0862.47039 · doi:10.1007/BF02662502 [10] Yi H.W., J. Math. Anal. Appl. 183 pp 613– (1994) · Zbl 0815.47072 · doi:10.1006/jmaa.1994.1167 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.