© Hindawi Publishing Corp. COMMON PERIODIC POINTS FOR A CLASS OF CONTINUOUS COMMUTING MAPPINGS ON AN INTERVAL

The existence of common periodic points for a family of continuous 
commuting self-mappings on an interval is proved and two 
illustrative examples are given in support of our theorem and 
definition.


Introduction and preliminaries.
All mappings considered here are assumed to be continuous from the interval I = [u, v] to itself. Let F(f ) and P (f ) be the set of fixed and periodic points of f , respectively, and let P (f ) be the closure of P (f ). Denote L(x, f ) by the set of limit points of the sequence {f n (x)} ∞ n=0 . By Schwartz's theorem [4], it is easy to show that L(x, f )∩P (f ) ≠ ∅ for each x in I. Obviously, F(f ) is a closed set and ∅ ≠ F(f ) ⊂ P (f ). Define the classes of mappings (1.1) The following definition was introduced by Cano [2].
where T is any subset of A ∪ B composed of commuting mappings and h is any mapping which commutes with the elements of T .
Boyce [1] and Huneke [3] showed that if f and g are two commuting selfmappings of I, then f and g need not have a common fixed point in I. Cano [2] proved the following theorem. In this note, we consider a larger class of mappings which has the common periodic point property and properly contains the class H considered by Cano. Two illustrative examples are given in support of our theorem and definition.
We first introduce the following definition.

Definition 1.3. A class of mappings T is said to be a C-class if T = T ∪{h} and T is a commuting family of mappings, where T is any subset of A∪D and h is any mapping.
Obviously, B ⊂ D. The following example proves that B is a proper subset of D.
Remark 1.5. Clearly, H-class is C-class, but the converse is not true.

Main results.
Our main result is as follows.
Theorem 2.1. There is a common periodic point for every C-class in I.
Proof. Let T be a C-class and T 1 a finite subset of T . We can write T 1 as where f 1 ∈ A, i = 1, 2,...,n, and h is a possible arbitrary mapping that commutes with the elements of T , g j ∈ D, j = 1, 2,...,m. Suppose that there are dif- Without loss of generality, we can assume a k > a i . Since f i and f k commute and  takes [a, b] into [a, b], and so, it must have a fixed point z ∈ [a, b]. Now, {g n 1 (z)} ∞ n=0 has a limit point z 1 ∈ P (g 1 ) because P (g 1 ) is a closed set. Clearly, there exists a subsequence {g by (2.3), we have (2.4) From (2.4), we have f i (z 1 ) ∈ F(f i ). Using the same method, we can show that z 1 ∈ F(h). So, Similarly, {g n j (z j−1 )} ∞ n=0 , j = 2, 3,...,m, has a limit point Thus, by the compactness of I. When T contains no such h, T ∩A = ∅, or T ∩D = ∅, we have the same result from the above proof. This completes the proof.
We at last give an example in which Theorem 2.1 holds but Theorem 1.2 is not applicable.
Let h be a continuous mapping and commute with f and g. It is easy to see that that is, f ∈ D, f ∈ B, and g ∈ A. Clearly, f and g are continuous and