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.
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 ).
2. Fixed Point Results
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.
3. A Common Fixed Point Result
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.
- Sehgal VM: A fixed point theorem for mappings with a contractive iterate. Proceedings of the American Mathematical Society 1969, 23: 631–634. 10.1090/S0002-9939-1969-0250292-XMathSciNetView ArticleMATHGoogle Scholar
- Guseman LF Jr.: Fixed point theorems for mappings with a contractive iterate at a point. Proceedings of the American Mathematical Society 1970, 26: 615–618. 10.1090/S0002-9939-1970-0266010-3MathSciNetView ArticleMATHGoogle Scholar
- Rhoades BE: A comparison of various definitions of contractive mappings. Transactions of the American Mathematical Society 1977, 226: 257–290.MathSciNetView ArticleMATHGoogle Scholar
- Ćirić L: On Sehgal's maps with a contractive iterate at a point. Publications de l'Institut Mathématique 1983,33(47):59–62.MATHGoogle Scholar
- Dhage BC: Generalised metric spaces and mappings with fixed point. Bulletin of the Calcutta Mathematical Society 1992,84(4):329–336.MathSciNetMATHGoogle Scholar
- Mustafa Z, Sims B: A new approach to generalized metric spaces. Journal of Nonlinear and Convex Analysis 2006,7(2):289–297.MathSciNetMATHGoogle Scholar
- Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorem for mapping on complete -metric spaces. Fixed Point Theory and Applications 2008, Article ID 189870 2008:-12 Pages.Google Scholar
- Abbas M, Rhoades BE: Common fixed point results for noncommuting mappings without continuity in generalized metric spaces. Applied Mathematics and Computation 2009,215(1):262–269. 10.1016/j.amc.2009.04.085MathSciNetView ArticleMATHGoogle Scholar
- Mustafa Z, Shatanawi W, Bataineh M: Fixed point theorem on uncomplete -metric spaces. Journal of Matehematics and Statistics 2008,4(4):196–210.View ArticleGoogle Scholar
- Shatanawi W: Fixed point theory for contractive mappings satisfying -maps in -metric spaces. Fixed Point Theory and Applications 2010, Article ID 181650 2010:-9 Pages.Google Scholar
- Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete -metric spaces. Fixed Point Theory and Applications 2009, Article ID 917175 2009:-10 Pages.Google Scholar
- Dehghan Nezhad A, Mazaheri H: New results in -best approximation in -metric spaces. Ukrainian Mathematical Journal 2010,62(4):648–654. 10.1007/s11253-010-0377-8MathSciNetView ArticleMATHGoogle Scholar
- Saadati R, Vaezpour SM, Vetro P, Rhoades BE: Fixed point theorems in generalized partially ordered -metric spaces. Mathematical and Computer Modelling 2010,52(5–6):797–801. 10.1016/j.mcm.2010.05.009MathSciNetView ArticleMATHGoogle Scholar
- Mustafa Z, Shatanawi W, Bataineh M: Existence of fixed point results in -metric spaces. International Journal of Mathematics and Mathematical Sciences 2009, Article ID 283028 2009:-10 Pages.Google Scholar
- Manro S, Bhatia SS, Kumar S: Expansion mapping theorems in -metric spaces. International Journal of Contemporary Mathematical Sciences 2010,5(51):2529–2535.MathSciNetMATHGoogle Scholar
- Chugh R, Kadian T, Rani A, Rhoades BE: Property P in -metric spaces. Fixed Point Theory and Applications 2010, Article ID 401684 2010:-12 Pages.Google Scholar
- Mustafa Z, Awawdeh F, Shatanawi W: Fixed point theorem for expansive mappings in -metric spaces. International Journal of Contemporary Mathematical Sciences 2010,5(50):2463–2472.MathSciNetMATHGoogle Scholar
- Abbas M, Nazir T, Radenović S: Some periodic point results in generalized metric spaces. Applied Mathematics and Computation 2010,217(8):4094–4099. 10.1016/j.amc.2010.10.026MathSciNetView ArticleMATHGoogle Scholar
- Park S: A unified approach to fixed points of contractive maps. Journal of the Korean Mathematical Society 1980,16(2):95–105.MATHMathSciNetGoogle Scholar
- Hadžić O: On common fixed point theorems in 2-metric spaces. Zbornik Radova Prirodno-Matematičkog Fakulteta. Serija za Matemati 1982, 12: 7–18.MATHMathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.