## Pacific Journal of Mathematics

- Pacific J. Math.
- Volume 127, Number 2 (1987), 373-387.

### $u$-mappings on trees.

**Full-text: Open access**

#### Article information

**Source**

Pacific J. Math., Volume 127, Number 2 (1987), 373-387.

**Dates**

First available in Project Euclid: 8 December 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.pjm/1102699568

**Mathematical Reviews number (MathSciNet)**

MR881765

**Zentralblatt MATH identifier**

0584.54030

**Subjects**

Primary: 54F20

Secondary: 54B25 54F50: Spaces of dimension $\leq 1$; curves, dendrites [See also 26A03] 54H25: Fixed-point and coincidence theorems [See also 47H10, 55M20]

#### Citation

Marsh, M. M. $u$-mappings on trees. Pacific J. Math. 127 (1987), no. 2, 373--387. https://projecteuclid.org/euclid.pjm/1102699568

#### References

- [rop. 7] , it follows that / is universal. Suppose that X hasexactly m interior edges which arefolded by/. We assume that whenever /': Z -* Y is a w-mapping of a tree Z ontoY such that Z hasfewer than m interior edges which arefolded by/',then /' is universal. By way of contradiction, we assume that / is notuniversal. Let g: X -> Y bea mapping such that f(x) g(x) for each x e X.
- [3,r2] ,r2] ,r2] ,r2] -* U,F] In a manner similar to that outlined in Example 2, it is easy to check that / has the desired properties. We also notice that a restriction of the mapping / would yield an example of a non-universal mapping which satisfies properties (1),(2),(3), and (5),but not (4). Let Xf = X - (39 b2]. Then the mapping f\x,: X' -> Y has the desired properties. Examples of non-universal mappings which do not satisfy property (1) or do not satisfy property (5) can also be given.
- [1] D. Bellamy, A tree-like continuum without the fixed point property, Houston J. Math., 6 (1979), 1-14.
- [2] R. H. Bing, The elusive fixed point property, Amer. Math. Monthly, 76 (1969), 119-132.
- [3] C. A. Eberhart and J. B. Fugate, Weakly confluent maps on trees, General Topology and Modern Analysis, Academic Press, NewYork (1981), 209-215.
- [4] O. H. Hamilton, A fixed point theorem for pseudo arcs and certain other metric continua, Proc. Amer. Math. Soc,2 (1951), 173-174.
- [5] W. Holsztynski, Universal mappings andfixed point theorems, Bull. Acad. Polon. Sci., XV (1967), 433-438.
- [6] M. M. Marsh, Fixed Point Theorems for Certain Tree-Like Continua, Dissertation, Unversity of Houston (1981).
- [7] M. M. Marsh, A fixed point theorem for inverse limits of fans, Proc. Amer. Math. Soc, 91 (1984), 139-142.
- [8] S. B. Nadler, Jr., Universal mappings and weakly confluent mappings, CX (1980), # 3 , 221-235.Mathematical Reviews (MathSciNet): MR82h:54057

#### Pacific Journal of Mathematics, A Non-profit Corporation

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