# zbMATH — the first resource for mathematics

A new class of nonexpansive type mappings and fixed points. (English) Zbl 1003.54024
Summary: A new class of self-mappings on metric spaces which satisfy a nonexpansive type condition is introduced and investigated. The main result is that such mappings have a unique fixed point. Also, a remetrization theorem which is converse to the Banach contraction principle is given.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
Full Text:
##### References:
 [1] J. Bogin: A generalization of a fixed point theorem of Goebel, Kirk and Shimi. Canad. Math. Bull 19 (1976), 7-12. · Zbl 0329.47021 [2] S.K. Chatterjea: Applications of an extension of a theorem of Fisher on common fixed point. Pure Math. Manuscript 6 (1987), 35-38. · Zbl 0664.54029 [3] Lj. B. Ćirić: On a common fixed point theorem of a Greguš type. Publ. Inst. Math. (Beograd) (N.S.) 49 (63) (1991), 174-178. · Zbl 1082.47507 [4] Lj. B. Ćirić: On some discontinuous fixed point mappings in convex metric spaces. Czechoslovak Math. J. 43 (118) (1993), 319-326. · Zbl 0814.47065 [5] Lj. B. Ćirić: On Diviccaro, Fisher and Sessa open questions. Arch. Math. (Brno) 29 (1993), 145-152. · Zbl 0810.47051 [6] Lj. B. Ćirić: Nonexpansive type mappings and fixed point theorems in convex metric spaces. Rend. Accad. Naz. Sci. XL Mem. Mat. (5), Vol. XIX (1995), 263-271. · Zbl 0941.47041 [7] Lj. B. Ćirić: On some nonexpansive type mappings and fixed points. Indian J. Pure Appl. Math. 24 (3) (1993), 145-149. · Zbl 0796.47042 [8] Lj. B. Ćirić: On same mappings in metric space and fixed points. Acad. Roy. Belg. Bull. Cl. Sci. (5) T.VI (1995), 81-89. · Zbl 1116.54300 [9] D. Delbosco, O. Ferrero and F. Rossati: Teoremi di punto fisso per applicazioni negli spazi di Banach. Boll. Un. Mat. Ital. A. (6) 2 (1993), 297-303. · Zbl 0532.47046 [10] M. L. Diviccaro, B. Fisher and S. Sessa: A common fixed point theorem of Greguš type. Publ. Math. (Debrecen) 34 (1987), 83-89. · Zbl 0634.47051 [11] Sh. T. Dzhabbarov: A generalized contraction mapping principle and infinite systems of differential equations. Differentsialnye Uravneniya 26 (1990), no. 8, 1299-1309, 1467 · Zbl 0715.34115 [12] B. Fisher: Common fixed points on a Banach space. Chung Yuan J. 11 (1982), 19-26. [13] B. Fisher and S. Sessa: On a fixed point theorem of Greguš. Internat. J. Math. Sci. 9 (1986), 23-28. · Zbl 0597.47036 [14] M. Greguš: A fixed point theorem in Banach spaces. Boll. Un. Mat. Ital. A (5) 17 (1980), 193-198. · Zbl 0538.47035 [15] K. Iseki: On common fixed point theorems of mappings. Proc. Japan Acad. Ser. A Math. Sci. 50 (1974), 408-409. · Zbl 0312.54045 [16] G. Jungck: On a fixed point theorem of Fisher and Sessa. Internat. J. Math. Sci. 13 (1990), 497-500. · Zbl 0705.54034 [17] B. Y. Li: Fixed point theorems of nonexpansive mappings in convex metric spaces. Appl. Math. Mech. (English 10 (1989), 183-188. · Zbl 0752.47022 [18] P. R. Meyers: A converse to Banach’s contraction theorem. J. Res. Nat. Bur. Standards, Sect. B 71 (1967), 73-76. · Zbl 0161.19803 [19] R. N. Mukherjee and V. Verma: A note on a fixed point theorem of Greguš. Math. Japon. 33 (1988), 745-749. · Zbl 0655.47047 [20] P. Omari, G. Villari and F. Zanolin: A survey of recent applications of fixed point theory to periodic solutions of the Lienard equations, Fixed point theory and its applications. (Berkley, CA 1986), 171-178, Contemp. Math. 72, Amer. Math. Soc., Providence, RI, 1988. · Zbl 0656.34031 [21] B. E. Rhoades: A generalization of a fixed point theorem of Bogin. Math. Sem. Notes, Kobe Univ. 6 (1978), 1-7. · Zbl 0387.47040 [22] B. E. Rhoades: Some applications of contractive type mappings. Math. Sem. Notes, Kobe Univ. 5 (1977), 137-139. · Zbl 0362.54044 [23] Md. Shahabuddin: Study of applications of Banach’s fixed point theorem. Chittagong Univ. Stud., Part II Sci. 11 (1987), 85-91. [24] A. Weiczorek: Applications of fixed point theorems in game theory and mathematical economics (Polish). Wiadom. Mat. 28 (1988), 25-34.
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.