zbMATH — the first resource for mathematics

Common fixed points for commuting and compatible maps on compacta. (English) Zbl 0661.54043
Selfmaps f, g of a metric space (X,d) are said to be compatible iff $$\lim_{n}d(fgx_ n,gfx_ n)=0$$ when $$(x_ n)$$ is a sequence such that $$\lim_{n}fx_ n=\lim_{n}gx_ n=t$$ for some $$t\in X.$$
This generalizes the notion of commuting maps and weakly commuting maps introduced by S. Sessa [Publ. Inst. Math., Nouv. Sér. 32(46), 149-153 (1982; Zbl 0523.54030)]. The author proves that two continuous selfmaps on a compact metric space are compatible iff they commute on their set of coincidence points. (This is a corollary from a more general result for introduced by the author “proper maps”). Then he proves several fixed point theorems for compatible and commuting selfmaps of a metric space which generalize many earlier results.
Reviewer: J.Matkowski

MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 54E45 Compact (locally compact) metric spaces
Zbl 0523.54030
Full Text:
References:
 [1] Chêng Chun Chang, On a fixed point theorem of contractive type, Comment. Math. Univ. St. Paul. 32 (1983), no. 1, 15 – 19. · Zbl 0526.54031 [2] Shih Sen Chang, A common fixed point theorem for commuting mappings, Proc. Amer. Math. Soc. 83 (1981), no. 3, 645 – 652. · Zbl 0473.54037 [3] C. E. Chidume, Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings, Proc. Amer. Math. Soc. 99 (1987), no. 2, 283 – 288. · Zbl 0646.47037 [4] G. Das and J. P. Debata, A note on fixed points of commuting mappings of contractive type, Indian J. Math. 27 (1985), no. 1-3, 49 – 51 (1986). · Zbl 0632.54027 [5] K. M. Das and V. Naik, Common fixed point theorems for commuting maps on metric spaces, Proc. Amer. Math. Soc. 77 (1979), 269-373. · Zbl 0418.54025 [6] Brian Fisher, A common fixed point theorem for four mappings on a compact metric space, Bull. Inst. Math. Acad. Sinica 12 (1984), no. 3, 249 – 252. · Zbl 0544.54037 [7] Brian Fisher, Common fixed points of four mappings, Bull. Inst. Math. Acad. Sinica 11 (1983), no. 1, 103 – 113. · Zbl 0515.54029 [8] M. S. Khan and Brian Fisher, Some fixed point theorems for commuting mappings, Math. Nachr. 106 (1982), 323 – 326. · Zbl 0501.54031 [9] Gerald Jungck, Periodic and fixed points, and commuting mappings, Proc. Amer. Math. Soc. 76 (1979), no. 2, 333 – 338. · Zbl 0416.54025 [10] Gerald Jungck, Commuting mappings and fixed points, Amer. Math. Monthly 83 (1976), no. 4, 261 – 263. · Zbl 0321.54025 [11] Gerald Jungck, A common fixed point theorem for commuting maps on \?-spaces, Math. Japon. 25 (1980), no. 1, 81 – 85. · Zbl 0432.54039 [12] -, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. 9 (1986), 771-779. · Zbl 0613.54029 [13] -, Compatible mappings and common fixed points (2), Internat. J. Math. Math. Sci. 9 (1986), 285-288. · Zbl 0647.54035 [14] Solomon Leader, Uniformly contractive fixed points in compact metric spaces, Proc. Amer. Math. Soc. 86 (1982), no. 1, 153 – 158. · Zbl 0507.54040 [15] R. P. Pant, Common fixed points of two pairs of commuting mappings, Indian J. Pure Appl. Math. 17 (1986), no. 2, 187 – 192. · Zbl 0581.54031 [16] Sehie Park and Jong Sook Bae, Extensions of a fixed point theorem of Meir and Keeler, Ark. Mat. 19 (1981), no. 2, 223 – 228. · Zbl 0483.47040 [17] Barada K. Ray, Remarks on a fixed-point theorem of Gerald Jungck, J. Univ. Kuwait Sci. 12 (1985), no. 2, 169 – 172 (English, with Arabic summary). · Zbl 0576.54044 [18] Salvatore Sessa, On a weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math. (Beograd) (N.S.) 32(46) (1982), 149 – 153. · Zbl 0523.54030 [19] Salvatore Sessa and Brian Fisher, On fixed points of weakly commuting mappings in compact metric spaces, Jñānābha 15 (1985), 79 – 91. · Zbl 0583.54031 [20] S. L. Singh and S. P. Singh, A fixed point theorem, Indian J. Pure Appl. Math. 11 (1980), no. 12, 1584 – 1586. · Zbl 0461.54034 [21] B. M. L. Tivari and S. L. Singh, A note on recent generalizations of Jungck contraction principle, J. Uttar Pradesh Gov. Colleges Acad. Soc. 3 (1986), no. 1, 13 – 18 (English, with Hindi summary). · Zbl 0638.54040
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.