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THEOREM (Takahashi [57]). Let (~,~) be a metrically convex, metrically compact space satisfying the condition of Paragraph 5, ~ Be a nonempty closed bounded convex set with normal structure, ~: A--+-A be a nonexpansive map. Then there exists a fixed point of the map Proof. T . We consider the family of all nonempty closed bounded convex sets contained in and invariant with respect to one can find a smallest set variant with respect to T T . By virtue of the convex compactness, in this family ~ , and it must be a singleton, since otherwise ~ and B c c ~ , Bc=~ B, which contradicts the choice of would be inB .

Then~(T~)=T~{T~E)=T~, , and by virtue of the uniqueness of such points, The theorem is proved. It is easy to note that an analogous assertion will be true when the map T~ commutes with all the maps of some set /~ of maps of X into itself and has a unique fixed point. One can note also that these maps can also have other fixed points. one can take the identity map of 2. into itself. We consider another characteristic example of a theorem on common fixed points. THEOREM. Let ~, for any ~ and ~ from T~ he maps of the complete metric space (~,~) into itself such that X and fixed ~, Then there exists a point for X For example, as one of them ~ ~(~<~.

For T ) set ~ of fixed points. the set ~(mJ is a compact, connected set. a singleton, or uncountable, while if it is a singleton, then(~$)~=O This set is empty, converges (to a fixed point). The sequence (~(I~Z,~))~=0 is a nonnegative, nonincreasing, Proof. Let ~{~ ~(Tnx,~)=~ . sequence. and hence convergent ~ -- ~. , ~(T~,[)--~(~,F), that if~(~c):# ~, ~$~ theorem of Paragraph i. 3. (~) whence ~ e [ and hence e~-----~(~,~)----0, if h(~):# ~. , the theorem being proved is a consequence of the The theorem is proved.

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