Summary
In this paper we consider several classes of mappings related to the class of contraction mappings by introducing a convexity condition with respect to the iterates of the mappings. Several fixed point theorems are proved for such mappings. Further, in a similar way we consider a related class of mappings satisfying a convexity condition with respect to diameters of bounded sets. In the last part we consider classes of mappings on PM- spaces (probabilistic metric spaces of K. Menger) and some fixed point theorems are given for such classes.
Article PDF
Similar content being viewed by others
References
S. Banach,Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fund. Math.,3 (1922), pp. 133–181.
M. Edelstein,On fixed and periodic points under contractive mappings, J. London Math. Soc.,37 (1962), pp. 74–79.
R. J. Egbert,Products and quotients of probabilistic metric spaces, Pacif. J. Math.,24 (1968), pp. 437–455.
V. I.Istraţescu,Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters, II (in preparation).
V. I.Istraţescu,Fixed Point Theory. An Introduction, D. Reidel Publ. Comp. (1981).
V. I.Istraţescu,Strict convexity and complex strict convexity (in preparation).
A. A. Ivanov,Fixed points of mappings of metric spaces, J. Sov. Math.,12 (1) (1979), pp. 1–65.
W. A. Luxemburg,On the convergence of successive approximations in the theory of ordinary differential equations. III, Niew. Archief. Wisc.,6 (1958), pp. 93–98.
K. Menger,Statistical metrics, Proc. Nat. Acad. Sci. U.S.A.,28 (1942), pp. 535–537.
V. V. Nemytski,The method of fixed points in analysis, Uspechi Math. Nauk,1 (1936), pp. 141–174.
V. M.Sehgal,Some fixed point theorems in functional analysis and probability, Ph. D. Dissertation, Wayne State Univ. (1966).
V. M. Sehgal,A fixed point theorem for mappings with a contractive iterate, Proc. A.M.S.,23 (1969), pp. 631–634.
H. Sherwood,Complete probabilistic metric spaces, Z. Wahr. verw. Geb.,20 (1971), pp. 117–128.
C. S. Wong,Fixed points of certain self maps on an interval, Proc. A.M.S.,42 (1974), pp. 234–235.
Rights and permissions
About this article
Cite this article
Istraţescu, V.I. Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters. — I. Annali di Matematica pura ed applicata 130, 89–104 (1982). https://doi.org/10.1007/BF01761490
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01761490