A monotone principle of fixed points

Author:
M. R. Tasković

Journal:
Proc. Amer. Math. Soc. **94** (1985), 427-432

MSC:
Primary 54H25; Secondary 54C60

MathSciNet review:
787887

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Abstract: In this paper we formulate a new principle of fixed points, and we call it "*monotone principle of fixed points*".

A fixed point theorem for set-valued mappings in a complete metric space and some theorems on fixed points in arbitrary topological spaces are presented in this paper. Also, we describe a class of conditions sufficient for the existence of a fixed point which generalize several known results. We introduce the concept of a contraction principle and CS-convergence. With such an extension, a very general fixed point theorem is obtained to include a recent result of the author, which contains, as special cases, some results of J. Dugundji and A. Granas, F. Browder, D. W. Boyd and J. S. Wong, J. Caristi, T. L. Hicks and B. E. Rhoades, B. Fisher, W. Kirk and M. Krasnoselskij.

**[1]**D. W. Boyd and J. S. W. Wong,*On nonlinear contractions*, Proc. Amer. Math. Soc.**20**(1969), 458–464. MR**0239559**, 10.1090/S0002-9939-1969-0239559-9**[2]**Felix E. Browder,*On the convergence of successive approximations for nonlinear functional equations*, Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math.**30**(1968), 27–35. MR**0230180****[3]**James Caristi,*Fixed point theorems for mappings satisfying inwardness conditions*, Trans. Amer. Math. Soc.**215**(1976), 241–251. MR**0394329**, 10.1090/S0002-9947-1976-0394329-4**[4]**J. Dugundji and A. Granas,*Weakly contractive maps and elementary domain invariance theorem*, Bull. Soc. Math. Grèce (N.S.)**19**(1978), no. 1, 141–151. MR**528510****[5]**Brian Fisher,*Results on fixed points*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**25**(1977), no. 12, 1253–1256 (English, with Russian summary). MR**0515244****[6]**T. L. Hicks and B. E. Rhoades,*A Banach type fixed-point theorem*, Math. Japon.**24**(1979/80), no. 3, 327–330. MR**550217****[7]**W. A. Kirk,*Caristi’s fixed point theorem and metric convexity*, Colloq. Math.**36**(1976), no. 1, 81–86. MR**0436111****[8]**M. Krasnoselskij et al.,*Approximate solution of operator equations*, Wolters-Noordhoff, Groningen, p. 172.**[9]**Milan R. Tasković,*Some results in the fixed point theory. II*, Publ. Inst. Math. (Beograd) (N.S.)**27(41)**(1980), 249–258. MR**621957****[10]**-,*Some theorems on fixed point and its applications*, Ann. Soc. Math. Polon. Ser. I Comment. Math. Prace Mat.**24**(1984), 151-162.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1985-0787887-X

Keywords:
Fixed point theorems,
contraction and nonexpansive mappings,
complete metric space,
topological space

Article copyright:
© Copyright 1985
American Mathematical Society