On a generalization of a Greguš fixed point theorem.(English)Zbl 1079.47509

Summary: Let $$C$$ be a closed convex subset of a complete convex metric space $$X$$. In this paper, a class of selfmappings on $$C$$, which satisfy the nonexpansive type condition $d(Tx,Ty)\leq a\max \{d(x,y),c[d(x,Ty)+d(y,Tx)]\}+b\max \{d(x,Tx),d(y,Ty)\}$ (where $$0<a<1,\;a+b=1,\;c\leq \frac {4-a}{8-a}$$), is introduced and investigated. The main result is that such mappings have a unique fixed point.

MSC:

 47H10 Fixed-point theorems 54H25 Fixed-point and coincidence theorems (topological aspects)
Full Text:

References:

 [1] LJ. B. Ćirić: On some discontinuous fixed point mappings in convex metric spaces. Czechoslovak Math. J. 43(118) (1993), 319-326. · Zbl 0814.47065 [2] LJ. B. Ćirić: A generalization of Banach’s contraction principle. Proc. Amer. Math. Soc. 45 (1974), 267-273. · Zbl 0291.54056 [3] LJ. B. Ćirić: On a common fixed point theorem of a Greguš type. Publ. Inst. Math (Beograd) (49)63 (1991), 174-178. · Zbl 0753.54023 [4] M. L. Diviccaro, B. Fisher, S. Sessa: A common fixed point theorem of Greguš type. Publ. Math. Debrecen 34 (1987), 83-89. · Zbl 0634.47051 [5] B. Fisher: Common fixed points on a Banach space. Chung Yuan J. 11 (1982), 19-26. [6] B. Fisher, S. Sessa: On a fixed point theorem of Greguš. Internat. J. Math. Math. Sci. 9 (1986), no. 1, 23-28. · Zbl 0597.47036 [7] M. Greguš: A fixed point theorem in Banach space. Boll. Un. Mat. Ital. A 5 (1980), 193-198. · Zbl 0538.47035 [8] B. Y. Li: Fixed point theorems of nonexpansive mappings in convex metric spaces. Appl. Math. Mech. (English 10 (1989), 183-188. · Zbl 0752.47022 [9] R. N. Mukherjea, V. Verma: A note on a fixed point theorem of Greguš. Math. Japon. 33 (1988), 745-749. · Zbl 0655.47047 [10] W. Takahashi: A convexity in metric space and nonexpansive mappings I. Kodai Math. Sem. Rep. 22 (1970), 142-149. · Zbl 0268.54048
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.