- Research Article
- Open Access
On Mappings with Contractive Iterate at a Point in Generalized Metric Spaces
© Gajić and and Z. Lozanov-Crvenković. 2010
- Received: 7 September 2010
- Accepted: 29 December 2010
- Published: 4 January 2011
Using the setting of generalized metric space, the so-called G-metric space, fixed point theorems for mappings with a contractive and a generalized contractive iterate at a point are proved. These results generalize some comparable results in the literature. A common fixed point result is also proved.
- Positive Integer
- Continuous Mapping
- Differential Geometry
- Symmetric Space
- Arbitrary Point
Sehgal in  proved fixed point theorem for mappings with a contractive iterate at a point and therefore generalized a well-known Banach theorem.
Guseman  extended Sehgal's result by removing the condition of continuity of and weakening (1.1) to hold on some subset of such that , where, for some , contains the closure of the iterates of . Further extensions appear in [3, 4]. Our aim in this study is to show that these results are valid in more general class of spaces.
In 1963, S. Gähler introduced the notion of 2-metric spaces but different authors proved that there is no relation between these two function and there is no easy relationship between results obtained in the two settings. Because of that, Dhage  introduced a new concept of the measure of nearness between three or more object. But topological structure of so called -metric spaces was incorrect. Finally, Mustafa and Sims  introduced correct definition of generalized metric space as follows.
Definition 1.2 (see ).
Example 1.4 (see ).
Definition 1.5 (see ).
Definition 1.6 (see ).
Proposition 1.7 (see ).
Definition 1.8 (see ).
Proposition 1.9 (see ).
and that in general these inequalities cannot be improved.
Proposition 1.10 (see ).
Since need not be less then 1 we can use metric fixed point results only for . On the other side, using the concept of -metric space, we are going to prove the result, if the first case for any , and in the second one for . This means that our results are real generalization in the case of nonsymmetric -metric spaces.
and result follows from Theorem 2 in .
Let us note that this result is very close to Theorem 2.1 in .
The next theorems are generalizations of Ćirić fixed point results in .
for all . Then the result follows from Theorem 2.1 in  and it is true for all .
Thus by induction we obtain (2.32).
Let us prove that is a Cauchy sequence. Let , , , and we define inductively a sequence of integers and a sequence of points in as follows: , and , . Evidently, is a subsequence of the orbit . Using this sequence we will prove that is a Cauchy sequence.
Now, we are going to prove Hadžić  fixed point theorem in 2-metric space, in a manner of -metric spaces.
Let be a complete -metric space, and one to one continuous mappings, continuous mapping commutative with and . Suppose that there exists a point such that is complete and that the following conditions are satisfied:
which is contradiction!
The authors are thankful to professor B. E. Rhoades, for his advice which helped in improving the results. This work was supported by grants approved by the Ministry of Science and Technological Development, Republic of Serbia, for the first author by Grant no. 144016, and for the second author by Grant no. 144025.
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